Table Of ContentMath Phys Anal Geom (2009) 12:1–18
DOI 10.1007/s11040-008-9049-4
A Geometric Interpretation of the Second-Order
Structure Function Arising in Turbulence
Vladimir N. Grebenev · Martin Oberlack
Received: 25 February 2008 / Accepted: 22 October 2008 / Published online: 20 November 2008
© Springer Science + Business Media B.V. 2008
Abstract We primarily deal with homogeneous isotropic turbulence and use
a closure model for the von Kármán-Howarth equation to study several geo-
metric properties of turbulent fluid dynamics. We focus our attention on the
application of Riemannian geometry methods in turbulence. Some advantage
of this approach consists in exploring the specific form of a closure model for
the von Kármán-Howarth equation that enables to equip a model manifold (a
cylindrical domain in the correlation space) by a family of inner metrics (length
scales of turbulent motion) which depends on time. We show that for large
Reynolds numbers (in the limit of large Reynolds numbers) the radius of this
manifold can be evaluated in terms of the second-order structure function and
the correlation distance. This model manifold presents a shrinking cylindrical
domain as time evolves. This result is derived by using a selfsimilar solution of
the closure model for the von Kármán-Howarth equation under consideration.
We demonstrate that in the new variables the selfsimilar solution obtained
coincides with the element of Beltrami surface (or pseudo-sphere): a canonical
surface of the constant sectional curvature equals −1.
V. N. Grebenev (B)
Institute of Computational Technologies, Russian Academy of Science,
Lavrentjev ave. 6, Novosibirsk 630090, Russia
e-mail:
2 V.N. Grebenev, M. Oberlack
Keywords Beltrami surface · Closure model for the von Kármán-Howarth
equation · Homogeneous isotropic turbulence · Riemannian metric ·
Two-point correlation tensor · Length scales of turbulent motion
Mathematics Subject Classifications (2000) 76F05 · 76F55 · 53B21 ·
53B50 · 58J70
1 Introduction
Turbulent fluid dynamics is characterized by ranking turbulent motions in
size from scales ∼ l of the flow under consideration to much smaller scales
which become progressively smaller as the Reynolds number increases. One
of a fundamental problem of turbulent fluid dynamics consist of studying the
shape dynamics of a fluid volume. The first concept in Richardson point of
view is that the turbulence can be considered to compose eddies (a turbulent
motion localized within a region of size l) of different sizes. Richardson’s
notion is that the eddies are evolved in time, transferring their energy to
smaller scale motions. These smaller eddies undergo a similar cascade process,
and transfer their energy to yet smaller eddies in the inertial range and so
on—continuous until the Reynolds number is sufficiently small that molec-
ular viscosity is effective in dissipating the kinetic energy. The characteristic
features of turbulence—its distribution of eddy sizes, shapes, speeds, vorticity,
circulation, and viscous dissipation—may all be captured within the statistical
approach to fully developed turbulence, and several questions can be posed.
What are the sizes of the eddies which are generated in Richardson scenario?
As time increases, how the shape of eddies is deformed? While there are
many efforts in this direction, the aim of this paper is to present an approach
that is based on the use of methods of Riemannian geometry for studying
the shape dynamics of eddies, in particular, on the interaction between the
deformation of geometric quantities (shape form, curvature and other) of a
manifold (a singled out fluid volume) equipped with a family of Riemannian
metrics (length scales of turbulent motion) and the deformation of these
Riemannian metrics in time t. Our approach is conceptually similar to the
Ricci flow ideas [1]. The Ricci flow is an evolution differential equation on
the space of Riemannian metrics, the behavior of smooth Riemannian metrics
which evolves under the flow may serve as a model to tell us something about
the geometry of an underlying manifold. The advantage of this approach is
that we can control the deformation of geometric quantities of the manifold
under consideration in time, and often a Ricci flow deforms an initial metric to
a canonical metric and a key point is to control the so-called injectivity radius
of the metrics.
A well-known example of the above is a Ricci flow that is starting from
N
a round sphere S with an initial metric gmn(x⃗, 0) = g(0) such that Rmn =
λgmn(x⃗, 0), λ ∈ R where Rmn is the Ricci tensor. This metric is known as
Interpretation of the Second-Order Structure Function 3
Einstein metric. The sphere shrinks homothetically flow to a point in a finite
time under the Ricci flow
∂
gmn = −2Rmn,
∂t
where the evolving metrics are given by the formula
2
gmn(x⃗, t) = ρ (t)gmn(0) ≡ (1 − 2λt)gmn(0), λ = N − 1,
and the sphere shrinks homothetically to a point as t → T = 1/2(N − 1).
Another example of this type would be if g0 is a hyperbolic metric or an
Einstein metric of negative scalar curvature. Then the manifold will expand
homothetically for all times. Indeed if Rmn = −λgmn(x⃗, 0) then ρ(t) satisfies
dρ λ
= ,
dt ρ
with the solution
2
ρ (t) = 1 + 2λt.
2
Hence the evolving metrics gmn(x⃗, t) = ρ (t)gmn(x⃗, 0) exists and expands ho-
mothetically for all times.
These illustrative examples give us a feeling how the Ricci flow can deform a
manifold equipped with an initial Riemannian metric g(0). In the general case,
the Ricci flow behaves more wildly.
In this paper, we deal with homogeneous isotropic turbulence and emphasis
is placed on the use of the specific form of a closure model [2, 3] for the
von Kármán-Howarth equation [4] to introduce into consideration a family
of Riemannian metrics. Inspired by the Ricci flow idea, we study the behavior
of Riemannian metrics constructed and as a consequence, the deformation of
some metric quantities of an underlying Riemannian manifold can be deter-
mined. In order to equip a model manifold (a singled out fluid volume within
turbulent flow) by a family of Riemannian metrics (length scales of turbulent
motion), we rewrite this model in the form of an evolution equation and show
that the right-hand side of this evolution equation coincides with the so-called
radial part of a Laplace-Beltrami type operator. This enables to construct
Riemannian metrics (length scales of turbulent motion) compatible with the
specific form of this closure model. We recall that the Laplace-Beltrami oper-
ator contains a metric tensor of a Riemannian manifold where this operator
is defined on. This is a crucial peculiarity of this operator that makes its
possible to investigate geometric characteristics of an underlying Riemannian
manifold. Using the selfsimilar solution obtained of the closure model for the
von Kármán-Howarth equation under consideration, we calculate explicitly
the deformation of this family of metrics in time. As a remarkable fact, we note
that the above-mentioned selfsimilar solution coincides in the new variables
with the element of Beltrami surface (or pseudo-sphere). This is a canonical
surface of the constant (sectional) curvature equals −1 [5].
Examining length scales of turbulence motion, we can see that some scales
analyzed are based on the use of Euclidian metric to measure a distance.
4 V.N. Grebenev, M. Oberlack
However, it is not so clear why we use Euclidian metric in turbulence to define
a length scale of turbulent motion without taking into account the geometry
of turbulent pattern. The well-known example, where we need a correction of
(linear) length scale, is the use of Prandtl’s mixing-length scale lm [6] in the
problem of decaying fluid oscillations near a wall. In this problem, a modifica-
tion of Prandtl’s mixing-length scale is taken in the following (nonlinear) form:
lm = κr(1 − exp(−r/A)) [6]. The length scale lm plays the role of a measure
of the transversal displacement of fluid particles under turbulent fluctuations.
Although the above example comes from the theory of wall turbulent flows,
nevertheless this fact reflects understanding to make a correction of some
(linear) length scales.
We note that even in the case of homogeneous isotropic turbulence there
is a relatively small number of publications devoted to numerical modeling
isotropic homogeneous turbulence [7] and there are very few results devoted to
mathematical analysis of the von Kármán-Howarth equation for the isotropic
two-point correlation function. We only mention here the paper [8] wherein
this unclosed equation was studied in the framework of the group classification
problem of differential equations [9]. We do not discuss the details of the
Kolmogorov theory (which tell us that the statistical properties of small scales
depend only on the mean rate of energy dissipation ϵ and the correlation
distance r) but remark, however, that still are many discussions on whether
small scale fluctuations are isotropic or not and that the Richardson scenario
may not be valid. Consequently, the velocity statistics in the inertial sub-range
may have nonuniversal features. The notion of intermittency is attributed to
the violation of local homogeneity of turbulence. This phenomenon leads to
the anomalous scaling and reflects a symmetry breaking in the case of ν → 0.
From a physical point of view as the viscosity tends to zero turbulence become
highly intermittent, and vorticity is concentrated on sets of a small measure
and scenario of turbulent motion is complicated significantly.
Here we do not review the papers based on the methods of Lagrangian
formalism (i.e. the description of turbulent motion of fluids particles) for the
stochastic description of turbulence since our approach lies in another field
of mathematical investigations of this phenomenon. The difference between
the application of Lagrangian formalism method for turbulence (exhaustive
reviews on this topic can be found in [10, 11]) and the approach presented
here is the same as using Lagrangian and Euler variables in hydrodynamics.
We do not look at how a marked fluid particle or an ensemble of marked
fluid particles (the separation distance between marked particles) is traveled
in turbulent flow but we prefer to observe entirely the deformation of length
scales of turbulent motion localized within a singled out fluid volume of this
flow in time.
The paper is organized as follows. Section 2 is devoted to a closure model for
the von Kármán-Howarth equation. Observe that this model holds (see [2]) for
a wide range of well accepted turbulence theories for homogeneous isotropic
turbulence as there is Kolmogorov first and second similarity hypothesis.
In Section 3, we show how to equip a model manifold (a singled out fluid
Interpretation of the Second-Order Structure Function 5
volume) by a family of Riemannian metrics (length scales of turbulent motion)
exploring the specific form of the above-mentioned closure model for the von
Kármán-Howarth equation limited to sufficiently large Reynolds numbers.
Moreover, we give a geometric interpretation of the second-order structure
function DLL. At the end of Section 3 we present the results [12] of group
analysis of the von Kármán-Howarth equation (in its inviscid form) and
indicate two scaling symmetries admitted by this equation that enable us to
find a whole class of selfsimilar solutions. We show that one implicit self-
similar solution, which corresponds to Loitsyansky decay low [13], coincides
(in the new variables) with the element of Beltrami surface (or pseudo-
sphere). Negativity of the curvature of Beltarmi surface means a stochastic
behavior of geodesic curves located on this surface [14]. As was noted by
Arnold [14], this property leads to the so-called exponential instability of the
geodesic flow. Here we do not develop this topic. Appendix includes a formal
derivation of the closure relationship [2] (the algebraic approximation for the
triple correlation function) for the von Kármán-Howarth equation limited
to sufficiently large Reynolds numbers in the framework of the method of
differential constraints [15]. In concluding remarks, we provide the results
obtained by physical comments to some extent.
2 Closed Model for the Von Kármán-Howarth Equation
We begin with basic notions of homogeneous isotropic turbulence.
2.1 Two-Point Velocity Correlation Tensor
Traditional Eulerian turbulence models employ the Reynolds decomposition
to separate the fluid velocity u⃗ at a point x⃗ into its mean and fluctuating
⃗ ′
components as u⃗ = u¯ + u⃗ where the symbol (¯·) denotes the Eulerian mean
sometimes also called Reynolds averaging.
In particular, the concept of two- and multi-point correlation functions was
born out of the necessity to obtain length-scale information on turbulent flows.
At the same time the resulting correlation equations have considerably less
unknown terms at the expense of additional dimensions in the equations. In
each of the correlation equations of tensor order n an additional tensor of
the order n + 1 appears as unknown term, see for details [16]. The first of the
infinite sequence of correlation functions is the two-point correlation tensor
defined as
′ ′ ′ ′ ′ ′ ′
Bij(x⃗, x⃗ ; tc) = (u
i(x⃗; tc) − ui(x⃗ ; tc))(u j(x⃗; tc) − u j(x⃗ ; tc)), (2.1)
′ ′ ′
where u⃗ (x⃗; tc) and u⃗ (x⃗ ; tc) are fluctuating velocities at the points (x⃗; tc) and
′ ′
(x⃗ ; tc) for each fixed tc ∈ R+. Therefore, Bij(x⃗, x⃗ ; t) defines a tensor field of
′
the independent variables x⃗, x⃗ and t on a domain D of the Euclidian space
6
R+ × R .
6 V.N. Grebenev, M. Oberlack
The assumption of isotropy and homogeneity of a turbulent flow (invariance
with respect to rotation, reflection and translation) implies that this tensor may
be written in the form [13]
′ ′
Bij(r⃗, tc) = u
i(x⃗; tc)u j(x⃗ + r⃗; tc), (2.2)
3 3 3
which acts in the so-called correlation space K ≡ {r⃗ = (r1, r2, r3)}, K ≃ R
′
for each tc, where r⃗ = x⃗ − x⃗ . Moreover, for isotropic turbulence Bij(r⃗, tc)
is a symmetric tensor which depends only on the length |r⃗| of the vector
′ ′ 6
r⃗ = r⃗(x, x , tc), (x, x ) ∈ R , and the correlations Bij can be expressed by using
only the longitudinal correlational function BLL(|r⃗|, tc) and the transversal
correlation function BNN(|r⃗|, tc).
2.2 Closure of the Von Kárman-Howarth Equation
The correlation functions directly connect the concept of length scales with
the result of an actual flow measurement. However, the two-point correlation
functions yield no information on the energy, that is contained in a given
interval of separation r⃗. The third-order correlations function BLL,L pro-
vides information about the energy fluxes between scales. The von Kármán-
Howarth equation relates the time derivative of the component BLL of the
two-point correlation tensor to the divergences of the third-order correlation
function BLL,L and has the following form
( )
∂ BLL 1 ∂
4 ∂ BLL
= r BLL,L + 2ν , (2.3)
4
∂t r ∂r ∂r
where ν is the kinematic viscosity coefficient, r = |r⃗|. This equation directly
follows from the Navier-Stokes equations [13].
Originally, the invariance theory of isotropic turbulence was introduced by
von Kármán and Howarth [4] and refined by Robertson [17], who reviewed
this equation in the light of classical tensor invariant theory. Arad, L’vov and
Procaccia [18] extended these fundamental results by considering projections
of the fluid velocity correlation dynamics onto irreducible representation of
the SO(N) symmetry group.
Equation 2.3 is not closed since it contains two unknowns BLL and BLL,L
which cannot be defined from (2.3) alone without the use of additional
hypotheses. The simplest assumption is the Kármán–Howarth hypothesis on
the similarity of the correlation functions BLL and BLL,L which is
′2 ′2 3/2
BLL(r, t) = u (t) f (η), BLL,L(r, t) = (u (t)) h(η), η = r/L(t), (2.4)
′2 ′2 3/2
where u (t) is the velocity scale for the turbulent kinetic energy, (u (t)) is
the scale for the turbulent transfer and L(t) is a single global length scale of
the turbulence. Substituting these hypothesized expressions into Eq. 2.3, it is
Interpretation of the Second-Order Structure Function 7
straightforward to demonstrate that this equation admits a complete similarity
′2 1/2
solution of type (2.4) only when the Reynolds number Re = u (t) L(t)/ν is
′2
a finite constant. In fact, this directly relates u (t) to L(t). It is known that this
condition is normally not satisfied in experimental measurements of decaying
isotropic turbulence at finite Reynolds numbers.
Batchelor and Townsend [19] carried out a similarity analysis of this
problem in Fourier space and showed that a similarity solution under this
constraint could be found during the final period of decay when the nonlinear
terms become negligible. Millionshchikov in [20] outlined a more general
hypotheses which produces parametric models of isotropic turbulence based
on a closure procedure for von the Kármán–Howarth equation. The essence
of these hypotheses is that BLL,L is given by the following relation of
gradient-type
∂ BLL
BLL,L = 2K , (2.5)
∂r
where K has the dimension of the turbulent kinematic viscosity which is
characterized by a single length and velocity scale. Millionshchikov’s hypothe-
ses [20] assumes that
1/2
′2 ′2
K = κ1u r, u = BLL(0, t), (2.6)
where κ1 denotes an empirical constant. An initial-boundary value problem
for the Millionshtchikov closure model has been studied in [21] wherein the
theory of contractive semigroups was applied to find a solution to the problem
by the use of a Chorin-type formula.
A way of closing the von Kármán–Howarth equation was suggested by
Oberlack in [2] which connects the two-point correlation functions of the third-
order BLL,L and the second order BLL by using the gradient type hypothesis,
that according to [2, 3] takes the form
√
1/2 2
′2
K = κ2rD
LL, DLL = 2[u − κ0 BLL(r, t)], κ0 = 1, κ2 = 3/2 , (2.7)
5C
where C is the Kolmogorov constant. The Millionshchikov hypotheses is a
consequence of the above formula in the case of κ0 = 0.
Comparison with experimental data was done calculating the triple corre-
lation h (the normalized triple-correlation function) out of measured values
of the normalized double correlation function f using the model (2.5), (2.7).
The normalized double correlation function f was recovered simultaneously
with the triple correlation h in Stewart/Taunsend experiments [22]. Good
agreement between measured and computed values of h was achieved within
the range of the reliable data [2].
In [23], isotropic homogeneous turbulence dynamics was described by a
closure system of partial differential equations for the two-point double- and
8 V.N. Grebenev, M. Oberlack
triple correlation functions coming from using the finite-dimensional probabil-
ity density equation. The following system of equations was written:
( )
∂ BLL 1 ∂
4 ∂ BLL
= r BLL,L + 2ν , (2.8)
4
∂t r ∂r ∂r
∂ BLL,L ∂ BLL ∂ 1 ∂
4 a1
− 1/4DLL = ν r BLL,L − 3 BLL,L, (2.9)
4
∂t ∂r ∂r r ∂r τ˜
where the first equation coincides with the von Kármán-Howarth equation,
τ˜ is the quantity which characterizes the correlation time. Applying the so-
called local equilibrium approximation to the second equation, the closure
relationship (2.5), (2.7) can be obtained but as it was noted by Chorin [24],
such approach is based only on a physical hypothesis. In the Appendix to
this paper, we give a formal derivation of this formula based on studying the
Riemannian invariants of characteristics of system (2.8),(2.9). This enables to
find an invariant manifold admitted by (2.8),(2.9) and to construct a reduced
system. Conceptually, this procedure is a similar to the approach suggested
in [25]. The principle difference is that we apply the method of differential
constraints [15] worked out by Cartan and Yanenko to study overdetermined
systems.
Finally, we note that it was in fact Hasselman [26] who was the first to
hypothesize a connection between the correlation functions of the second- and
third-order. His model for isotropic turbulence contains one empirical constant
and a rather complicated expression for the turbulent viscosity coefficient.
3 A Model Manifold Defined by Closure of the Von Kármán–Howarth
Equation
First we review certain definitions and statements from Riemannian geometry.
Then we construct the so-called model manifold by exploring the closure
model (2.5), (2.7) for the von Kármán-Howarth equation and give a geometric
interpretation of the second-order structure function DLL. To study explicitly
the deformation of a family of Riemannian metrics constructed in time, we
use a selfsimilar solution of the closure model for the von Kármán–Howarth
equation.
3.1 Laplace-Beltrami Operator
We recall the definition of some operators on a Riemannian manifold U.
n
Consider a vector field F = Fn∂/∂x on U. The operator div is determined
by the formula
N
∑
1 ∂ √
div F = √ ( gFn),
n
g ∂x
n=1
Interpretation of the Second-Order Structure Function 9
th ⃗
where g = det ‖gmn‖, and the m component of the operator ∇ is defined
according to the formula
N
∑
∂ f
m nm
(∇ f ) = g ,
n
∂x
n=1
nm −1
here g are elements of the matrix ‖gnm‖ . Further we denote
= div ∇
the Laplace-Beltrami operator. The Laplace-Beltrami operator with a positive
smooth weighted function σ(x⃗) is defined in a similar way using the following
formula
N
∑
1 ∂ √
div F = √ (σ(x⃗) gFn).
n
σ (x⃗) g ∂x
n=1
Here σ(x⃗) presents the density of a Borel measure μ on U. If μ is the
Riemannian volume, then σ(x⃗) ≡ 1.
Let Z be a Riemannian manifold which is isometric to
Z ≃ X × Y,
where X is an arbitrary manifold of dim X = N1 and Y is a compact N2-
2
dimensional manifold. Then a metric dz⃗ on Z is determined by
2 2 2 2
dz⃗ = dx⃗ + γ (x⃗)dy , (3.10)
2 2
where γ (x⃗) is a positive smooth function and dx⃗ , dy⃗ are metrics on X, Y
correspondingly.
We assume that the density σ (z⃗) of a Borel measure μ on Z can be written
as σ (z⃗) = τ(x⃗)η(y⃗). Then the Laplace-Beltrami operator given on Z takes the
form
−2
Z = A + γ B, (3.11)
where A is the Laplace-Beltrami operator on X with the weighted function
N2
γ τ and B denotes the Laplace-Beltrami operator defined on Y with the
weighted function η [27].
As an elementary example which illustrates the above construction we
consider the Laplace operator
2 2 2
∂ ∂ ∂
= + +
2 2 2
∂x ∂y ∂z
written in the spherical coordinates r, ϕ, ψ (x = r sin ϕ cos ψ, y = r sin ϕ sin ψ,
z = r cos ϕ)
1 ∂ ∂ 1
2
= r + 2,
2 2
r ∂r ∂r r
10 V.N. Grebenev, M. Oberlack
where
2
1 ∂ ∂ 1 ∂
2 = sin ψ +
2 2
sin ψ ∂ψ ∂ψ sin ψ ∂ϕ
2
denotes the inner Laplacian on the unite sphere S . Then a Riemannian metric
2 2
dz⃗ of Z = R+ × S is defined by the formula
2 2 2 2 2 2 2 2
dz⃗ = dr + r dθ , dθ = dϕ + sin ϕdψ , (3.12)
1
2 2 2
where dθ is the standard inner metric of S and dz⃗ is equivalent to the usual
1
2
Euclidian metric. If we substitute a function g (r), g(0) = 0, g(r) ⩾ 0 instead
2
of r into the second term of (3.12), then the corresponding Laplace-Beltrami
operator given on Z which is equipped by the metric
2 2 2 2
dz⃗ = dr + g (r)dθ
2
takes the form
1 ∂ ∂ 1
2
Z,dz⃗2
2 = 2 g (r) + 2 2.
g (r) ∂r ∂r g (r)
2
Here Z is a cylindrical domain of the radius γ = g(r) and dz⃗ determines
2
another inner metric on Z . Therefore the definition of the Laplace-Beltrami
operator on the cross product of Riemannian manifolds and the above exam-
ples show us that the so-called radial part A of the operator Z completely
defines the form of a Riemannian metric dz⃗ given on Z ≃ X × Y.
3.2 Metric Properties of a Model Manifold Determined by the Model
Limited to Sufficiently Large Reynolds Number
2
Let us consider a cylindrical domain Z = R+ × S in the correlation space
3 3
K (≃ R ). In order to equip this cylindrical domain by an inner metric, we
explore the specific form of the right-hand side of the closure model for the
von Kármán–Howarth equation.
So, assuming the Reynolds number to be large, the first order O(1) of Eq. 2.3
with the closure relationships (2.5), (2.7) reduces to its inviscid form
∂ BLL 2κ2 ∂
5 1/2 ∂ 3
=
4 r DLL BLL, r = |r⃗|, r⃗ ∈ K . (3.13)
∂t r ∂r ∂r
1/2 ˜ ˜ ′2 ˜
Let q = 2r , BLL(q, t) ≡ BLL(r, t) and DLL = 2[u − BLL]. Then Eq. 3.13
can be rewritten in the form
˜
∂ BLL κ2 ∂
9 ˜ 1/2 ∂ ˜
=
9 q DLL BLL. (3.14)
∂t q ∂q ∂q
2
Further, let Z = R+ × S be a manifold with the metric
2 2 2 2
dz⃗ = dq + γ˜ (q, tc)dθ (3.15)
