Anisotropy Properties of Turbulence - WSEAS · Anisotropy Properties of Turbulence SANKHA BANERJEE Massachusetts Institute of Technology Department of Mechanical Engineering U.S.A

Anisotropy Properties of Turbulence

SANKHA BANERJEEMassachusetts Institute of TechnologyDepartment of Mechanical Engineering

U.S.A

OEZGUER ERTUNC & FRANZ DURSTUniversity of Erlangen-NuerembergInstitute of Chemical Engineering

Germany

Abstract: In the literature, anisotropy-invariant maps are being proposed to represent a domain within whichall realizable Reynolds stress invariants must lie. It is shown that the representation proposed by Lumley andNewman has disadvantages owing to the fact that the anisotropy invariants(II, III) are nonlinear functions ofstresses. In the current work, it is proposed to use an equivalent linear representation of the anisotropy invariantsin terms of eigenvalues. A novel barycentric map, based on the convex combination of scalar metrics dependenton eigenvalues, is proposed to provide a non-distorted visual representation of anisotropy in turbulent quantities.This barycentric map provides the possibility of viewing the Reynolds stress and any anisotropic stress tensor.Additionally the barycentric map provides the possibility of quantifying the weighting for any point inside it interms of the limiting states (one component, two component, three component). The mathematical basis for thebarycentric map is derived using the spectral decomposition theorem for second-order tensors. In this way, ananalytical proof is provided thatAll turbulence lies along the boundaries or inside the barycentric map. It isproved that the barycentric map and the anisotropy-invariant maps are one to one uniquely interdependent and asa result satisfy the requirement of realizability.

Key–Words:Nonlinearity, Barycentric map, Quantification of anisotropy.

1 Introduction

The momentum transport at a point in a complexthree-dimensional turbulent flows is not uniform inall directions, since, this process is characterized bya tensor, namely Reynolds stress tensor. This tensoruiuj can be identified as normal (wheni = j) andtangential (wheni 6= j) turbulent stresses. It is verydifficult to gain complete information contained in theReynolds stress tensor using only one scalar, so, it isnecessary, to extract information about, the differentaspects of the tensor using different scalars.

The Reynolds stress tensor carry informationabout the componentality of the turbulence (i.e the rel-ative strengths of different velocity fluctuation compo-nents). From DNS studies of turbulent flows, it can beconcluded that each large scale structure tends to or-ganize spatially the fluctuating motion in its vicinity,and in so doing, it tries to eliminate the gradients ofthe fluctuation fields in certain directions (the direc-tions in which the spatial structure is significant), andit enhances the gradients of fluctuations in other direc-tions. Thus associated with each eddy in a turbulentmomentum transport are local axes of dependence andindependence. In the undeformed state of isotropicturbulence the various axes of dependence and inde-pendence due to individual eddies are oriented ran-

domly. The velocity fluctuation field which can bethought to be ensemble of all the eddies has gradientsin all three directions.

In turbulent flows, the mean flow gradient cre-ates structural anisotropy, since it stretches and alignsthe energy-containing turbulent eddies. This inturn aligns the axes of dependence and indepen-dence of individual eddies and creates directions inwhich, in astatistical sense, the gradients of energy-containing fluctuations are weak. Thus by acting onthe energy-containing structure mean flow gradientcreates axes of independence, which are reflected inauto-correlations of gradients of the fluctuation fields.Thus there are regions in a turbulent flow, where thereis one such direction of independence, and the turbu-lence becomes two-dimensional. However the direc-tion of independence, says nothing about the intensityof the fluctuations in that direction relative to the to-tal intensity. Thus theanisotropy in the dimension-ality of the turbulence is in general distinct from theanisotropy of its componentality.

So characterization of the more general state ofthe turbulence requires information about the dimen-sionality of the turbulence (i.e the relative unifor-mity of structures in different directions) as well asthe componentality of the turbulence (i.e the rela-tive strengths of different velocity fluctuation com-

Proceedings of the 13th WSEAS International Conference on APPLIED MATHEMATICS (MATH'08)

ISSN: 1790-2769 26 ISBN: 978-960-474-034-5

ponents). Turbulence models carrying only compo-nentality information (e.g.standard Reynolds stressmodels carrying only Reynolds stress tensor) cannotpossibly satisfy conditions associated with the dimen-sionality of the turbulence or even reflect the differ-ences in dynamic behaviour associated with structuresof different dimensionality (i.e nearly isotropic tur-bulence vs. turbulence with strongly organized two-dimensional structures as observed in stratified flows).This was the central message put forward from thework of Kassinos & Reynolds (1994)[1].

In the current work, the additional tensors intro-duced by Kassinos & Reynolds (1994)[1] will be plot-ted in the barycentric map and in themetricsplot in-troduced by Banerjee et al. (2007)[2]. Kinematic re-lationships will be derived between the additional ten-sors and the anisotropic Reynolds stress tensor to seehow their information can add to the existing knowl-edge of turbulence modelling. Our analyis will be lim-ited to the domain of irrotational homogeneous flows,so the inhomogenity tensorCij and the stropholysistensorQijk are not taken into account.

Kassinos & Reynolds (1994)[1] showed that threekinds of information are important in determining theevolution of the turbulent stresses in homogeneousturbulence;

1. Componentality information embodied inuiuj .

2. Dimensionality information available throughdij andfij (definitions given below).

3. Information about the breaking of reflectionalsymmetry by mean or frame rotation carried bythe stropholysis tensorQ∗

ijk.

The additional tensorsdij andfij are defined in thefollowing from Kassinos & Reynolds (1994)[1].

Dimensionality tensorThe structure dimensionality tensorDij , its as-

sociated normalized tensordij and dimensionalityanisotropy tensordij are defined as

Dij = ψ′

n,iψ′

n,j dij =Dij

Dkkdij = dij −

δij3

(1)

where the turbulence stream function vectorψ′

i, is de-fined by

u′

= ǫitsψ′

s,t ψ′

i,i = 0 ψ′

i,nn = −ω′

i (2)

whereω′

i denotes the turbulent vorticity vector. TheDij tensor reveals the level oftwo-dimensionalityofthe turbulence. For example, when the turbulence isindependent ofx3 direction thend33 = 0 because

none of the stream-function components varies in thatdirection.

Another definition ofDij can be arrived at, byconsidering the inviscid rapid distortion theory of ho-mogeneous turbulence, for which the evolution equa-tions for the Reynolds stresses,Rij = uiuj, are (seefor example Reynolds & Kassinos (1994)[1])

dRij

dt= −GikRkj −GjkRki + Πr

ij (3)

whereGij = Ui,j is the mean velocity gradient tensorand Πr

ij is the rapid pressure strain-rate term, whichresults from the familiar splitting of the pressure fluc-tuations into ‘slow’ and ‘rapid’ parts (see equation??for detailed understanding) where

Mijkl =

∫κkκl

κ2Eij(κ)d

3κ (4)

whereEij(κ) = ui(κ)u∗j (κ) is the velocity spectrum

tensor,κ is the wavenumber vector, the hats denoteFourier coefficients and the∗ denotes a complex con-jugate.Mijkl is a fourth-rank tensor that must be mod-elled in terms of the tensor variables in the one-pointmodel. The two non-zero contractions ofMijkl are

Rij = Mijkk =

∫Eij(κ)d

3κ

Dij = Miikl =

∫κkκl

κ2Eij(κ)d

3κ (5)

Circulicity tensor The structure circulicity ten-

sorFij , and its associated normalized tensorfij andthe circulicity anisotropy tensorfij are defined by

Fij = ψ′

i,nψ′

j,n fij =Fij

Fkkfij = fij −

δij3

(6)

The Fij tensor describes the large-scale structure ofthe vorticity field. When, most of the large-scale cir-culation is about thex3 axis then the large-scale circu-lation is concentrated about thex3 axis, i.ef33 → 1.

It has to be noted from the above definitions, thatthe trace ofFij andDij are equal. The tensorsuiuj,Dij andFij are all positive semi-definite, and haveonly two independent anisotropy invariants. Whenconsidered together,Dij andFij give a fairly detaileddescription of the turbulence structure. The second-rank dimensionality and circulicity tensors representone-point correlations that carry non-local informa-tion about the structure of a turbulent flow. This isdue to the fact,ψ

′

i satisfies the poisson equation.From equations (1) and (6) for homogeneous tur-

bulence it can be derived that

ukuk = Dkk = Fkk = q2 (7)


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and a fundamentalconstitutive equationin homoge-neous flow can be derived of the form

uiuj +Dij + Fij = q2δij (8)

From equation (8) it can be found that only two ten-sors are linearly independent in the context of homo-geneous turbulence. By carrying onlyuiuj as themodel variable, Reynolds stress models effectivelylump together information found in the dimensional-ity Dij and circulicityFij into uiuj. It is because ofthis fact, the Reynolds stress models are not sensitiveto the differences in the dynamic behaviour associatedwith the different dimensionality in homogeneous tur-bulence.

1.1 Properties of the Reynolds Stress and itsAnisotropy Tensor

The symmetry property of the Reynolds stress ten-sor uiuj ensures diagonalizability of its matrix. Inthe diagonal form, all tensor entities ofuiuj are thematrix’s eigenvalues, which are non-negative by def-inition. The individual components of the Reynoldsstress tensor, have to satisfy the physical realizabilityconstraints (9) noted by Schumann[3]

uµuµ ≥ 0, uµuµ + uνuν ≥ 2|uµuν |,det (uiuj) ≥ 0, µ, ν = {1, 2, 3}. (9)

From equation (9), it can be concluded that theReynolds stress tensor is required to be a positivesemi-definite matrix. From equation (??) and the con-dition of non-negativeness for the main diagonal ele-ments in equation (9) the following relationship holdsfor the corresponding anisotropy tensorbij

bµµ =uµuµ

2k− 1

3, µ = {1, 2, 3}. (10)

It can be seen that the diagonal component ofaµµ

of the anisotropy tensor takes its minimal value ifuµuµ = 0. The maximal value corresponds to thesituation whenuµuµ = 2k. For the off-diagonal com-ponentaµν of the anisotropy tensor, one can write asimilar expression:

bµν =uµuν

2k, µ 6= ν, µ, ν = {1, 2, 3}. (11)

Because of the positive semi-definiteness ofuiuj, theoff-diagonal elementaµν reaches its minimal value inthe case whenuµuν = −k and its maximal value ifuµuν = k. Applying these conditions to equations(10) and (11), intervals between which the anisotropytensor components belong can be calculated:

−1/3 ≤ bµµ ≤ 2/3, −1/2 ≤ bµν ≤ 1/2,

µ 6= ν, µ, ν = {1, 2, 3}. (12)

1.2 Canonical Form of the Anisotropy Ten-sor

The anisotropy tensor of Reynolds stress is also sym-metric i.e, bij=bji. Its eigenvalues are real and theeigenvectors associated with each eigenvalue can bechosen to be mutually orthogonal. Hence the follow-ing orthogonality condition holds for the eigenvectors:

xki x

li = δkl (13)

Furthermore, if a tensorXij consists of the eigenvec-tors of bij such that the orthogonality condition (13)holds, then the tensorbij can be diagonalized by sim-ilarity transformation:

XikbklXlj = ∆ij (14)

where

∆ij = λiδij (15)

is a diagonal tensor with the eigenvalues ofbij on itsdiagonal. The coordinate system made of the eigen-vectors is termed the principal axes in whichbij is di-agonal. The relationship between the Reynolds stresstensor and the anisotropic stress tensor is given by therelation (??). Let λri be an eigenvalue ofuiuj andλi

be the corresponding eigenvalue ofbij. Let the corre-sponding eigenvector bexi. Then

bijxi =uiuj

2k xi − δij

3 xi

λixi = λri

2k xi − δij

3 xi ri, i = {1, 2, 3}(16)

whereλi = λri

2k − 13 , with eigenvectors ofuiuj and

bij being xi. It is important to note that the eigen-vectors ofuiuj and bij are identical. This informa-tion makes the trajectory of the normalized Reynoldsstress and the anisotropic Reynolds stress identical in-side the barycentric map for any kind of flow.

In the current work, the following notation for theeigenvalues of the anisotropy tensor after diagonaliza-tion (i.e in principal coordinate system(pcs)) are ap-plied:

λ1 = maxµ

(bµµ|pcs), λ2 = maxν 6=µ

(bνν |pcs),

µ, ν = {1, 2, 3}. (17)

The scalar functionsλ1 andλ2 are the two indepen-dent anisotropy eigenvalues that can be used for thecharacterization of the anisotropy stress tensor. Fromthe fact that the anisotropy tensor has a zero trace, onecan represent its smallest eigenvalue by means of theabove introduced notation:

λ3 = −λ1 − λ2 = minχ6=µ,ν

(aχχ|pcs). (18)


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From the notation (17), the following inequalityholds:

λ1 ≥ λ2 ≥ λ3. (19)

Introducing thenon-increasing orderon diagonalelements of the anisotropy tensorbij , it can be consid-ered by the reordered matrixbij . It can be seen thatfor any physically realizable Reynolds stress tensor,it is possible to construct exactly one correspondinganisotropy tensor in its canonical form (20) that canbe written in terms of notation (17) as

bij =

λ1 0 00 λ2 00 0 λ3

. (20)

1.3 Anisotropy-Invariant Maps

The anisotropy of turbulence can be characterized bythe following techniques. The first representation,proposed by Lumley and Newman[4], given in fig-ure (??, (left)), can be constructed from the followingnonlinear relationships based on the two invariants ofthe anisotropy tensor (21), denotedII − III:

I = bii II = bijbji III = bijbinbjn (21)

II ≥ 3

2

(4

3|III|

)2/3

, II ≤ 2

9+ 2III. (22)

The second representation suggested byLumley[5], denotedη − ζ is an alternative rep-resentation ofII − III, denotedη − ζ:

ζ3 = III/2 η2 = −II/2 (23)

This map is depicted in the figure (1), (right).The third representation was originally suggested

by Lumley[6], based on the eigenvalues of theanisotropy tensor, denotedλ1, λ2, depicted in the fig-ure 1, (left). This representation and its relationships,were used in the modelling of turbulence by Teren-tiev. The relationships of the different boundaries ofthis map in terms ofλ1, λ2:

λ1 ≥ (3|λ2| − λ2)/2, λ1 ≤ 1/3 − λ2 (24)

It is easy to see that there exists a direct functionalrelationship between the anisotropy invariants and itscorresponding eigenvalues [i.e.II = w(λ1, λ2)and III = q(λ1, λ2)]. For the transformation fromthe eigenvalue representation(λ1, λ2) to the principalcomponents(III, II), one can directly use the defini-tion for the anisotropy invariants expressed by equa-tion (21) to obtain

II = 2(λ21 + λ1λ2 + λ2

2), III = −3λ1λ2(λ1 + λ2). (25)

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

B

A

Isotropic limit (A=0, B=0)

Two−component axisymmetric limit (A=1/6, B=1/6)

One−component limit (A=2/3, B=−1/3)

Axisymmetric contraction limit (A=(3|B|−B)/2)

Two−component plane−strain limit (A=1/3, B=0)

Axisymmetric expansion limit (A=(3|B|−B)/2)

Plane−strain limit (B=0)

Two−component limit (A=1/3−B)

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

iso

1C

2C, axiaxi, ζ > 0

axi, ζ < 0

2C

ζ

η

Figure 1: Anisotropy-invariant map based on two in-dependent eigenvalues of the anisotropy tensor(A,B)(left) and anisotropy invariant map based on(ζ, η)(right). The eigenvalues(A,B) correspond toλ1, λ2.

1.4 Anisotropy invariant measures

The different aspects of the Reynolds stress tensorwhich can be considered to be informative are listedas follows:

1. The overall magnitude of momentum transport.

2. The degree, with which the rate of momentumtransport is dissimilar, in different orthogonal di-rections. This is referred to as the degree ofanisotropy. Information about the relative mag-nitude of the eigenvalues of the Reynolds stresstensor, i.e the largest, intermediate, and smallest.This is referred to as the eigenvalue order

3. The degree, that the rate of momentum transportin two of the three orthogonal directions differsfrom the rate of momentum transport in the thirddirection, i.e, how symmetrically the momentumtransport is distributed in the orthogonal direc-tions. This can be referred to as the skewness ofthe momentum transport.

The anisotropy and the skewness measures of theReynolds stresses are non-dimensional, i.e they areindependent of scale. The diagonal elements of theReynolds stress tensor, describe the rate of momen-tum transport in the different directions and the off-diagonal elements describe the degree of correlationof the momentum transport along the various axes.As, a first step, in extracting scalar information fromthe Reynolds stress tensor, it is necessary to divideit into two parts, each having complementary infor-mation. Diagonalization of the Reynolds stress tensorseperates it, into its component eigenvalues and eigen-vectors.

uiuj =[V1 V2 V3

]λ1 0 00 λ2 00 0 λ3

V1

V2

V3

The eigenvectorsV1, V2 andV3 are column vec-tors which describe the orientation of the momentum


ISSN: 1790-2769 29 ISBN: 978-960-474-034-5

Set �� for � � �� , then:

��

��

Where

��

��

��

� ��

��

� ��

��

��

��

�(20)

��

��

��

� ��

��

� ��

��

��

��

�(21)

Lemma 2 If the hypothesis 1 are satisfied, then thesystem (12)-(13) is an exponential observer of the sys-tem (4)-(5) for �� and �� are positives.

3.2 Proof of the lemma 2To prove convergence, let us consider the followingequation of LYAPUNOV:

� � �� (22)

where �� and �� .By calculating the derivative of � along the trajecto-ries of �� and ��, we obtains:

��

��

��

��

��

��

� ��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

�� (23)

By taking account of the hypothesis 1, and by intro-ducing the norms, then:

��

��

��

��

��

�

��

��

��

��

��

��

��

��

��

��

��

��

�� (24)

� ��

��

��

��

��

��

��

��

��

��

� �� !�� !�� (25)

where

� � � ��

and � � ��

� �� and ��

�� and �� are the eigenvaluesminimal and maximum of �� respectively.

� �� and ��

��

�.

By rewriting the preceding expression of � (22)according to �� and ��, one obtains:

�� !�� !�� (26)

Where

!� � ��

��

!� � ��

��

��

��

��

��

��

��

��"

If we take:� � �

#� � �� !� � �and#� � �� !� � �

Finally, while taking:

$ � �� #�� #�� (27)

It follows that:

�� $ �� $� (28)

This finishes the proof of convergence of the observer.Thus � is a LYAPUNOV FUNCTION.


ISSN: 1790-2769 30 ISBN: 978-960-474-034-5

the turbulent momentum transport in orthogonal di-rections. But, they are not invariant as,

λ1

λ2

λ3

6=

λ2

λ1

λ3

(27)

Cartesian coordinate with ordered eigenvalues.In this coordinate system, the position of a pointAis specified by the eigenvalues of the Reynolds stresstensor ordered by magnitude.

Aordered(λ1, λ2, λ3) =

λmax

λint

λmin

(28)

whereλmax, λint, andλmin denote the maximum, in-termediate and minimum values of the eigenvalues.This is not a true coordinate system, since it mapssix points from eigenvalue space onto one point inthe ordered eigenvalue space. The six points, whichare mapped onto ordered eigenvalue space, are pointsdefined by the six possible permutations of the eigen-value order. However,Aordered transformation is in-variant.

Rotated cartesian coordinate system. The axes ofthe original (not ordered) Cartesian coordinate systemcan be rotated by taking linear combinations of eigen-values. The rotation matrix must have orthogonal ba-sis vectors in order for the new coordinate system tohave orthogonal axes. If the rotation matrix is taken tobe orthonormal (i.e the orthogonal basis vectors are ofunit length) then there is no resultant dilation or con-traction of the eigenvalue space. When an orthonor-mal transform is used, the resulting measures maintainthe units of the original cartesian system.

Arotated(λ1, λ2, λ3) =

θ11 θ12 θ13θ21 θ22 θ23θ31 θ32 θ33

λmax

λint

λmin

(29)

A particularly useful orthonormal rotation matrix isone which places the new z axis along the eigenvaluetriple identity line,λ1 = λ2 = λ3, with the positive zaxis in the direction of positive eigenvalues. When theturbulent momentum transport in the Reynolds stresstensor is isotropic, i.e the same in all directions, theeigenvalues of the Reynolds stress tensor lie along thistriple identity line.

Ar =

−2√6

1√6

1√6

0 −1√2

1√2

1√3

1√3

1√3

λ1

λ2

λ3

=

xyz

(30)

wherex, y, z > 0.. The eigenvalue triple identityline is an obvious line of symmetry in the eigenvalue

space, since any measure applied to any point alongthe line of triple identity will automatically be invari-ant. The subscriptr in Ar denotes the particular ro-tation of the axes where the z axis is along the tripleidentity line. In this system the x-axis is arbitrarilychosen to be alongλ2 = λ3 plane with the positivexaxis in the direction whereλ1 < λ2, (see figure (2)).The positivey axis is in the planeλ1 = (λ2 + λ3)/2with positivey direction whereλ2 < λ3. As shownin Eqs. (30),z is an invariant, whereasx andy arenot invariant measures. The parameterz has the prop-erties of trace of a matrixuiuj andλbar which is themean of the three eigenvalues.

z =(λ1 + λ2 + λ3)√

3= λbar

√3 (31)

The six points in the eigenvalue space correspond-

p=1

orderedspace

p=2

p=3

p=4

p=5

p=6

lambda2=lambda3

lambda1=lambda2lambda1=lamda3

x axis

y axis

Figure 2: The plane of the six permutations of theeigenvalue order.

ing to the six permutations of the eigenvalues of theReynolds stress tensor, when defined in terms of therotated cartesian coordinate system, all have the samez value. These points defined by the six permutationsof the eigenvalues display an interesting symmetry inthe x − y plane (orthogonal to the z axis). This sixpoints all lie at a fixed distancer from the orign inthex − y plane (see figure(3)). This radial symmetryof the eigenvalue permutations in the rotated cartesiancoordinate system can be utilized in circular cylindri-cal coordinate system described below. The six pointsdefined by the six permutations of the eigenvalues areconsidered to be three sets of two points; each setis symmetrically placed about one of the three linesdefined by the intersection of thex, y plane and theplanes:λ1 = λ2, λ2 = λ3 andλ3 = λ1 in figure (2).

Circular cylindrical coordinate system. This sys-tem is a modification of theAr system, Eq. [30]. The


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x axis

lambda2=lambda3

r

lambda1=lamda2lambda1=lambda3

Figure 3: Thex−−y plane of the rotated Cartesian co-ordinate system showing projection of points definedby the six permutations of the eigenvalue order. Thesepoints all lie in at a distancer from the orign.

z axis is kept the same in theAr system. The othertwo coordinates,r andφ of theAcyl are thex and yof Ar written in polar coordinate form.λbar denotesthe mean of the eigenvalues Eq. [30],r ≥ 0, and0 ≤ φ < 2π.

Pcyl(λ1, λ2, λ3) =

rφz

(32)

The value of the polar angle measure,φ must be de-fined to account for, which quadrant of the x-y plane,the eigenvalue point lies in figure (2).

Case I, Quadrant I: x > 0 andy > 0 then

φ(λ1, λ2, λ3) = arctan

[ √3(λ2 − λ3)

2λ1 − λ2 − λ3

](33)

Case II, Quadrant II and III : x < 0

φ(λ1, λ2, λ3) = π + arctan

[ √3(λ2 − λ3)

2λ1 − λ2 − λ3

](34)

Case IV, Quadrant IV: x > 0 andy < 0

φ(λ1, λ2, λ3) = 2π + arctan

[ √3(λ2 − λ3)

2λ1 − λ2 − λ3

](35)

As in theAr system,z is invariant. Equation [??]demonstrates that the new parameterr is also invari-ant.r can be shown to be proportional to the standard

deviation of the eigenvalues.

sd(λ1, λ2, λ3) =

√(λ1−λbar)2+(λ2−λbar)2+(λ3−λbar)2√

2(36)

= r√2

(37)

wheresd(λ1, λ2, λ3) is the standard deviation of theeigenvalues. The polar angle measureφ, is unitlessbut not invariant. The zero point ofφ is chosen to bex axis, whereλ2 = λ3, in the direction thatλ1 <λ2. φ ranges from0 to 2π. It is important, to notefrom the above measure that a scalar is not invariantby definition.

Another invariant measure, which represents thevariation of the momentum transport in terms of theeigenvalues of the Reynolds stress tensor is

cv(λ1, λ2, λ3) =√

3r√2z

(38)

=

√(λ1−λbar)2+(λ2−λbar)2+(λ3−λbar)2√

2λbar(39)

wherecv(λ1, λ2, λ3) is unitless. Thecv(λ1, λ2, λ3)measures are not orthogonal toz. Thus the values ofcv(λ1, λ2, λ3) andz are neither independent nor com-plementary.

Spherical coordinate system. This coordinatesystem is composed of one radial measure,ρ, and twounitless angle measures,θ and φ. The polar anglemeasureφ of the spherical coordinate system is iden-tical to that of circular cylindrical coordinate system:

As(λ1, λ2, λ3) =

θφρ

(40)

ρ(λ1, λ2, λ3) =√λ2

1 + λ22 + λ2

3 (41)

θ(λ1, λ2, λ3) = arccos

[(λ1+λ2+λ3)√3√

λ2

1+λ2

2+λ2

3

](42)

= arccos[

zρ

](43)

= arctan[

rz

](44)

φ(λ1, λ2, λ3) = γ + arctan

[ √3(λ2 − λ3)

2λ1 − λ2 − λ3

](45)

whereρ > 0, and0 ≤ φ < 2π. The quadrant mod-ification γ was detailed in the Eqs.[33-35]. Since,ρis invariant and is the measure of the total magnitudeof momentum transport.θ is unitless and invariant.It can be related to the coefficient of variation of the


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eigenvalues and thus is a measure of the relative dis-persion. φ is unitless but not an invariant. It is anindicator of the permutation of the order of the eigen-values which is discussed below. The zero point ofθis chosen to coincide with the eigenvalue triple iden-tity line (λ1 = λ2 = λ3). The zero point ofφ is againchosen to coincide with the planeλ2 = λ3 in the di-rection thatλ1 < λ2.

p=1

p=2

p=3

p=4

p=5

p=6

λ1=λ

3

λ2=λ

3

λ1=λ

2

Figure 4: Thex − y plane. The lines indicate theintersection of thex − y plane with theλ1 = λ2,λ2 = λ3, andλ3 = λ1 planes, respectively. The re-gion wherep = 1 corresponds to the projection of theordered eigenvalue space onto thex − y plane. Thebij distribution from data of direct numerical simula-tion of fully developed plane turbulent channel flowfor Reτ = 180 by Kim et al.[7]. The region wherep = 1 corresponds to the projection of the orderedeigenvalue space onto thex− y plane.

The skewness measure, s and permutation indicator, p:The measure of the polar angle,φ, can be dividedinto two components. The first component iss, askewness measure and the second component isp,an indicator of the order of the eigenvalues. Theindicator measure,p can be defined as follows for thefigure (4)

p = 1 ∀ 0 ≤ φ < π/3p = 2 ∀ π/3 ≤ φ < 2π/3p = 3 ∀ 2π/3 ≤ φ < πp = 4 ∀ π ≤ φ < 4π/3p = 5 ∀ 4π/3 ≤ φ < 5π/3p = 6 ∀ 5π/3 ≤ φ < 2π

(46)

where p is an indicator of the permutation of theeigenvalue order. An equivalent definition ofp is as

follows:

p = 1 ∀ λ1 < λ2 ≤ λ3

p = 2 ∀ λ2 ≤ λ1 < λ3

p = 3 ∀ λ2 < λ3 ≤ λ1

p = 4 ∀ λ3 ≤ λ2 < λ1

p = 5 ∀ λ3 < λ1 ≤ λ2

p = 6 ∀ λ1 ≤ λ3 < λ2

(47)

The skewness measure can be defined as

p = 1 s = φp = 2 s = (2π/3) − φp = 3 s = φ− (2π/3)p = 4 s = (4π/3) − φp = 5 s = φ− (4π/3)p = 6 s = 2π − φ

(48)

The skewness parameter, s, can be equivalently be de-fined by ordered eigenvalues,

s(λ1, λ2, λ3) = arctan

[ √3(λint − λmax)

(2λmin − λint − λmax)

](49)

The permutationp, skewnesss or the angleφ canbe used to check, in which barycentric map the flowtrajectory lies. This is a very useful indicator of theeigenvalue order in turbulent stresses.

1.5 General Classification of States of Tur-bulence

The classification of turbulence used in the presentstudy is based on the componentiality of turbulence,namely Reynolds stress tensor. The Reynolds stress’smajor, medium and minor axes are along the tensor’seigenvectors, with scalings along the axes being theeigenvalues. Letλi ≥ −1

3 for i = 1, 2, 3 be the ma-jor, medium and minor eigenvalues of thebij satisfy-ing equation (12) and letxi be an eigenvector corroe-sponding toλi.

From the spectral theorem, any second-order ten-sor can be written as

bij = λ1x1xT1 + λ2x2x

T2 + λ3x3x

T3 (50)

The rank, i.e. number of non-zero eigenvalues, ofthe tensorbij reflects the limiting states of turbulencepresent in the tensorbij. The limiting states of thesecond-order tensor can be characterized by the fol-lowing states:

• One-component limiting state of turbulence –ˆ(b)1c (one eigenvalue is non-zero inuiuj),


ISSN: 1790-2769 33 ISBN: 978-960-474-034-5

• Two-component limiting state of turbulence –ˆ(b)2c (two eigenvalues are non-zero inuiuj),

• Three-component limiting state of turbulence –ˆ(b)3c (three eigenvalues are non-zero inuiuj).

In the current work, componentality reflects thenumber of non-zero velocity fluctuationsui, whichmatches the rank of the correlation matrix under con-sideration, e.g. for any anisotropic stress tensor.

Special States of Turbulence of AnisotropyTensor

One-component limiting state

This case corresponds to rank one tensorbij, whereλ1 > 0 > λ2 ≈ λ3. This is the one-component1 (1-c)state of turbulence with only one turbulence compo-nent. Turbulence in an area of this type is only alongone direction, the direction of the eigenvector corre-sponding to the non-zero eigenvalue:

bij ∼= λ1x1xT1∼= λ1b1c (51)

The basis matrix for the limiting state is given by

b1c =

2/3 0 00 −1/3 00 0 −1/3

(52)

Two-component limiting state

This case corresponds to rank two tensoraij, whereλ1 ≈ λ2 > 0 > λ3. This is the two-component (2-c)isotropic state of turbulence, where one component ofturbulent kinetic energy vanishes with the remainingtwo being equal. Turbulence in an area of this typeis restricted to planes, the plane spanned by the twoeigenvectors corresponding to non-zero eigenvalues:

bij ∼= λ1(x1xT1 + x2x

T2 ) ∼= λ1b2c (53)

The basis matrix for the limiting state is given by

b2c =

1/6 0 00 1/6 00 0 −1/3

(54)

1The introduced barycentric map can represent both compo-nentiality and dimensionality of turbulence

Three-component limiting state

This case corresponds to rank three tensorbij,whereλ1=λ2=λ3. This is the three-component (3-C)isotropic state of turbulence. Turbulence in an areaof this type is restricted to a sphere, the axes of thesphere are spanned by the eigenvectors corroespond-ing to non-zero eigenvalues:

bij ∼= λ1(x1xT1 + x2x

T2 + x3x

T3 ) ∼= λ1b3c (55)

The basis matrix for this limiting state is given by

b3c =

0 0 00 0 00 0 0

(56)

bij = Ca1cb1c + Ca

2cb2c +Ca3cb3c (57)

Ca1c ≥ 0, Ca

2c ≥ 0, Ca3c ≥ 0 (58)

where {Ca1c, C

a2c, C

a3c} are the coordinates of

the anisotropy tensoraij in the tensor basis{b1c, b2c, b3c

}. As described above, the rela-

tionships between the eigenvalues of the anisotropytensor can be used for the classification of thebij byusing the coordinates of the tensor in the new basisb1c, b2c and b3c to measure how closebij is from thegeneric cases of one component, two component andthree component limiting states. To obtain a measureof anisotropy from the anisotropy tensorbij, metricsof three different kinds of anisotropy are provided.

The scalar metrics are chosen in a manner suchthat

Ca1c + Ca

2c + Ca3c = 1 (59)

whereCa1c is a linear measure of anisotropy andCa

2c isthe planar measure of anisotropy. The normalizationis done in such a way that all metrics{Ca

1c, Ca2c, C

a3c}

lie in the range[0, 1]. It has to be noted thatb3c isa null matrix which can make the equation (57) non-unique but the restriction onCa

3c using(59) makes itunique.

In general, any anisotropy tensorbij can be ex-pressed as a convex combination2 (57) of the limitingstates (one component, two component, three compo-nent) by setting

Ca1c = λ1 − λ2

Ca2c = 2(λ2 − λ3)

Ca3c = 3λ3 + 1

(60)

2It means when you draw a line between two points in a givenbounded domain, no part of the line falls out of the domain


ISSN: 1790-2769 34 ISBN: 978-960-474-034-5

An anisotropy measure describing the deviation fromisotropy is achieved as

Caani = Ca

1c + Ca2c

Caani = 1 − Ca

3c

Caani = 3λ3

(61)

Taking this formulation into account a new map tocharacterize turbulence is proposed where the threelimiting states of turbulence (one component, twocomponent, three component) represent the three ver-tices of the map. In the current barycentric map, ateach vertex one of the metrics{Ca

1c, Ca2c, C

a3c} has

coefficient 1 and then it decreases to zero for theline opposite to the vertex. A value of1 for anyof the metrics corresponds to the respective limit-ing states and a value of0 means that the respec-tive limiting state doesnot describe the current state.Any point inside the map has local barycentric coor-dinates{Ca

1c, Ca2c, C

a3c} which helps to know directly

the weighting of the three limiting states of turbulenceat that point and so quantification of the anisotropy ispossible.

Plotting of the Map

The barycentric map can be used as a tool to visualizethe nature of anisotropy in different types of turbu-lent flows. For plotting the barycentric map, we iden-tify the basis matricesb1c, b2c, b3c of the equation (57)as the three vertices of the triangle. The corners ofthe map can be identified as three points(x1c, y1c),(x2c, y2c) and (x3c, y3c) in the Euclidean space rep-resenting the three limiting states. To plot a point,we have to take a convex combination of the limitingstates (60) using the equation (57):

xnew = Ca1cx1c + Ca

2cx2c + Ca3cx3c (62)

ynew = Ca1cy1c + Ca

2cy2c + Ca3cy3c (63)

All flows can be plotted in the barycentric map figure5 using the relations(62, 63) to visualize the natureof anisotropy. An important point which needs to benoted in the construction of the barycentric map is thatall the limiting states (one component, two compo-nent, three component) are joined by lines. The proofthat these limiting states can be joined by lines is givenin Terentiev (2006)[8]. Another important feature ofthe barycentric map is that it can be constructed bytaking three arbitrary basis points such as(x1c, y1c),(x2c, y2c) and (x3c, y3c) so one can choose the basisof figure 1, (left) to obtain the eigenvalue map. Thebarycentric map is depicted in the form of an equilat-eral triangle as all the metrics (60) scale from[0, 1]

and an equilateral triangle doesnot introduce any vi-sual bias of the limiting states (one component, twocomponent, three component).

2 comp

3 comp

1 compTwo component limit

Axi-sy

mmetric

contr

actio

n Axi-symmetric expansionPla

inst

rain

Figure 5: Barycentric map based on scalar metricswhich are functions of eigenvalues of the second-order stress tensor describing turbulence. Theisotropic point has a metricCa

3c = 1, the two-component point hasCa

2c = 1 and the one -componentpoint hasCa

1c = 1.

1.6 Computation of the Barycentric Coordi-nates

The scalar metricsC1c, C2c and C3c are computedfrom relations (60). Figure 6 shows thebij distribu-tion for a fully developed plane turbulent channel flowat Reτ = 180. Figure 6 clearly illustrates that themathematical requirement of axisymmetryb22 = b33is not satisfied, but when it is viewed in theAIM Fig.7, (left) it gives a visual impression of axisymmetry.

0 20 40 60 80 100 120 140 160 180−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

y+

a ij

a11

a22

a33

Figure 6: Thebij distribution from data of direct nu-merical simulation of fully developed plane turbulentchannel flow forReτ = 180 by Kim et al.[7]. Theb33stresses are 200% more than theb22 stresses near thewall of the channel and meet the axisymmetry crite-rion only at the channel centreline.


ISSN: 1790-2769 35 ISBN: 978-960-474-034-5

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x2+=0→

x2+=h→

III

II

x2+=0→

x2+=h→

x2+=0→

x2+=h→

x2+ = 0→

x2+ = h→

2 comp 1 comp

3 comp

Figure 7: Invariant characteristics of the anisotropytensors for the Reynolds stress[◦] - uiuj on theanisotropy-invariant map(III, II) (left) and scalarmetrics (C1c, C2c) based on the eigenvalues ofanisotropy tensor for the Reynolds stress on thebarycentric map (right). Data were extracted from adirect numerical simulation of fully developed planeturbulent channel flow forReτ = 180 by Kim etal.[7].

The barycentric map Fig. 7, (right) is much more con-sistent in preserving this physical information and itshows that the fully developed plane turbulent channelflow for Reτ = 180 is far from axisymmetry. In Fig.7, (right) it can be observed that the barycentric mapoffers a visual advantage in depicting that the chan-nel flow is far from isotropy at the channel centreline.The weighting of the limiting states for plane channelflow data of Kim et al.[7] were computed and it wasfound that the isotropic metricC3c is 0.9 and not1 toindicate full isotropy of the flow at the channel centre-line. Table 1 gives a clear mathematical justificationthat the channel flows are not isotropic at the centre-line as claimed by Kim et al.[9]. The nonlinearity in(II, III) makes the flow tend towards isotropy visu-ally when viewed in theAIM (Fig. 7, left). Similarconclusions can be drawn about pipe flows when theyare viewed in the barycentric map.

From Fig. 8, (left), it is important to observe thatwhen the point P, when projected to any one side of thebarycentric map, say for instance, the two-componentside, the distance from the two-component vertex tothe lineC1c = 0.5 intercepts is the quantitative valueofC1c; similarly the distance from the one-componentvertex to the lineC2c = 0.25 intercepts is the value ofC2c. The value ofC3c can be obtained from the equa-tion (59) and in the map it is the distance between theline joining the one and two-component states and theline whereC3c = 0.25 along the sides of the barycen-tric map. The manner in which the barycentric coor-dinates scale in the barycentric map is shown in Fig.(8), (right). The centre of the barycentric map corre-sponds to the point P where

C1c =1

3, C2c =

1

3, C3c =

1

3(64)

The centre of the barycentric map when mapped back

Points C1c C2c C3c

P 0.5 0.25 0.25

x+2 = h 0.1 0.0 0.9x+

2 = 0 0.55 0.45 0.0

Table 1: Weightings of the point P.

1 comp2 comp

3 comp

← C1c

=0.5

← P

← C2c

=0.25

C3c

=0.25

2comp 1comp

isotropic

C3c C3c

C1cC2c

C2c

C1c

Figure 8: Calculation of the weighting of a point Pfrom data extracted from a direct numerical simula-tion of fully developed plane turbulent channel flowfor Reτ = 180 on the barycentric map (left) and theaxis of the metricsC1c, C2c andC3c on the barycen-tric map (right). The isotropic point corresponds to3comp.

on to the invariant map(AIM) gives a point veryclose to the axi-symmetric expansion curve which canbe verified by inserting the barycentric coordinates(64) into (67). This fact can be visually noticed inthe invariant map(AIM), Fig. (9), (left) and in thebarycentric map Fig. (9), (right). This is a mathe-matical justification about the fact, why channel flowswhich are far away from the axi-symmetric limitingstates in the barycentric map Fig. 7, (right) and are sonear to the axi-symmetric limiting states in theAIMFig. 7, (left).

1.7 Mapping Between the Barycentric andthe Invariant Map

In this section we show that there is a bijection be-tween the barycentric map and the invariant map.

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

III

II

P

P

2 comp 1 comp

3 comp

Figure 9: The centrepoint P of the barycentric map(right) projected onto the invariant map (left). Thepoint P lies closer to the axisymmetric states in theinvariant map.


ISSN: 1790-2769 36 ISBN: 978-960-474-034-5

The domain of the invariant map(AIM) is boundedby the line II = 3

2(43III)

2

3 from below and byII = 2III + 2

9 from above. In other words, a point(III, II) lies inside the invariant map if and only if

3

2(4

3III)

2

3 ≤ II ≤ 2III +2

9. (65)

Let the barycentric coordinatesC1c, C2c andC3c

of the barycentric map be given, and from the defini-tions (60) we obtain

λ1 = C1c +C2c

2+C3c

3− 1

3,

λ2 =C2c

2+C3c

3− 1

3,

λ3 =C3c

3− 1

3. (66)

SubstitutingC3c = 1 − C1c − C2c we deduce

II =(2

3C1c +

1

6C2c

)2+(−1

3C1c +

1

6C2c

)2

+(−1

3C1c −

1

3C2c

)2

=2

3C2

1c +1

3C1cC2c +

1

6C2

2c,

III =(2

3C1c +

1

6C2c

)3+(−1

3C1c +

1

6C2c

)3

+(−1

3C1c −

1

3C2c

)3

=2

9C3

1c +1

6C2

1cC2c −1

12C1c C

22c −

1

36C3

2c.

This means that we can expressII andIII in termsof the barycentric coordinatesC1c, C2c, andC3c andthis defines a mapping(III, II) = Φ(C1c, C2c, C3c)from the barycentric map into theII − III plane ofthe invariant map.

1.8 Range of the Mapping

In order to show that the range ofΦ lies within thebounds of the invariant map (65), letC1c,C2c andC3c

be the barycentric coordinates of some point in thebarycentric map, using the equation (67), we deduce

(2

3II)3

−(4

3III)2

=4

27C4

1cC22c +

4

27C3

1cC32c +

1

27C2

1cC42c ≥ 0

and

2 III +2

9− II =

4

9C3

1c +1

3C2

1cC2c

− 1

6C1cC

22c −

1

18C3

2c −2

3C2

1c −1

3C1c c2C

− 1

6C2

2c +2

9

= C3cC2c

(C1c +

1

2C2c

)+

2

9C2

3c

(1 + 2C1c + 2C2c

)≥ 0.

Therefore, we have32 (43 III)

2

3 ≤ II ≤ 2 III+ 29 and

thus the point(III, II) lies inside the domain of theinvariant map(AIM).

1.9 Mapping from the Invariant Coordinatesto the Barycentric Coordinates

Let (III, II) be a point in the invariant map, fulfillingrelation (65). We have to show that there are barycen-tric coordinatesC1c, C2c, andC3c fulfilling the rela-tion (III, II) = Φ(C1c, C2c, C3c).

Consider the cubic function which defines the re-lation between(III, II) and(λ1, λ2):

f(x) = 3x3 − 3

2II x− III. (67)

The critical points off are atxc1 = −√

16 II and

xc2 = +√

16 II with f(xc1) = +

√16 II

3 − III and

f(xc2) = −√

16 II

3 − III. From relation (65), it

follows thatf(xc1) ≥ 0 and f(xc2) ≤ 0. Therefore,f must have exactly one rootβ with

−√

1

6II ≤ β ≤ +

√1

6II (68)

and from equation (67)

3β3 − 3

2II β = III. (69)

Choosing

α = +

√1

2II − 3

4β2 − 1

2β,

γ = −(α+ β). (70)

then from the relation (68) it follows that

1

2II − 3

4β2 ≥ 3

8II ≥ 0 (71)

Henceα is real andβ ≤ +√

16 II ≤ α. In a similar

way, it can be proved thatγ ≤ −√

16 II ≤ β. The

estimate−13 ≤ γ can be derived fromII ≤ 2 III+ 2

9 .


ISSN: 1790-2769 37 ISBN: 978-960-474-034-5

1.10 Uniqueness in the Mapping from theBarycentric map to the Invariant map

Let (c1c, c2c, c3c) and (c1c, c2c, c3c) be the barycen-tric coordinates of two points in the barycentric map,fulfilling relations (58) and (59) respectively, that aremapped to the same point(III, II) in the anisotropymap byΦ. That is, we have

(III, II) = Φ(c1c, c2c, c3c) = Φ(c1c, c2c, c3c) (72)

with Φ as defined by 67. In order to prove the in-jectivity of Φ, we have to show that(c1c, c2c, c3c) =(c1c, c2c, c3c).

Let

α =2

3c1c +

1

6c2c,

α =2

3c1c +

1

6c2c,

β = −1

3c1c +

1

6c2c,

β = −1

3c1c +

1

6c2c,

γ = −1

3c1c −

1

3c2c,

γ = −1

3c1c −

1

3c2c. (73)

Note that we haveα+β+ γ = 0, α+ β+ γ = 0, andby equation 58α ≥ β ≥ γ, α ≥ β ≥ γ. Using thisdefinition, we can write equation 72 as

II = α2 + β2 + (−α− β)2

= α2 + β2 + (−α− β)2,

III = α3 + β3 + (−α− β)3

= α3 + β3 + (−α− β)3.

(74)

Note that

1

6II − β2 =

1

6c1c c2c ≥ 0 (75)

and thusβ ∈[−√

16 II,+

√16 II

]. Similarly, β ∈

[−√

16 II,+

√16 II

].

Combining equations 74 we obtain

3

2β II + III = 3β3,

3

2β II + III = 3β3. (76)

In other words,β and β are both roots off(x) =3x3 − 3

2 IIx − III. We already checked above that

isotropic

2comp 1comp

AB

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

III

II

A

B

A

B

Figure 10: Linear interpolation between two pointsinside the barycentric map projected onto the invariantmap. The isotropic point corresponds to3comp.

f has only one root in the range[−√

16 II,+

√16 II

]

and therefore we get

β = β. (77)

From equation 74, we obtain

α = ±√

1

2II − 3

4β2 − 1

2β. (78)

Insertion intoγ = −α− β yields

γ = ∓√

1

2II − 3

4β2 − 1

2β. (79)

Observing thatα ≥ γ and using the same argumentson α andγ, we conclude that

α = +

√1

2II − 3

4β2 − 1

2β

= +

√1

2II − 3

4β2 − 1

2β

= α (80)

and finally

γ = −α− β

= −α− β = γ.

(81)

Now we can deduce from equation 73 that

c1c = α− β = α− β = c1c

c2c = 2(β − γ) = 2(β − γ) = c2c

c3c = 3γ + 1 = 3γ + 1 = c3c (82)

This completes the proof of the injectivity ofΦ.

1.11 Important Features of the BarycentricCoordinates

Figure 10 describes an easy way of obtaining the in-terpolation between any two points on theAIM ex-actly. Taking a point having coordinates(II, III) in


ISSN: 1790-2769 38 ISBN: 978-960-474-034-5

theAIM we can find the eigenvalues corresonding toit by using equation (67). Once the eigenvalues areknown, the barycentric metrics (60) can be computed.So if the linear interrelationships between unknownsecond-order tensors are desired, as in Jovanovic etal.[10] then the two points can be joined by a line inthe linear domain of barycentric map and then con-verted back to invariants using equation (67). Thisprovides a consistent and sound mathematical inter-polation schemes between two points in the invariantmap.

Figure 11 plots the barycentric metricsC1c, C2c

and C3c on the y-axis andy+ on the x-axis fromdirect numerical simulation data of fully developedplane turbulent channel flow forReτ = 180[7].This 2D plot provides quantitative information abouthow bij behaves in terms of the limiting states (one-component, two-component and three-component) lo-cally. Figure 11 also provides the information that anypoint inside the barycentric map Fig. 5 bears the samearea ratio (i.e take any point in the barycentric mapor the eigenvalue map of Lumley and join that pointwith the limiting states of the map, the ratios of theareas of the three sub-triangles to the total area of thetriangle is referred to as area ratio) as in the Lumleytriangle Fig. 1, (left). Figure 11 provides the possi-bility of studying the dominance of theC1c andC2c

in the near-wall region and the dominance ofC3c overC2c andC1c at the centreline forbij. It is interesting toobserve that at the channel centerline theC1c metricgoes to zero.

It is also interesting to note that the return toisotropy experiments for plane strain of Choi[5] whenviewed in the barycentric map Fig. (14), (left) theyshow no tendency to get aligned with the axisymmet-ric expansion state as claimed by Choi et al.[5]. Thisfact is further substantiated by investigating the scalarmetricsC1c andC2c in the 2D plot Fig. (14), (right)whereC2c 6= 0 signifies that return to isotropy fromplane strain experiments cannot be claimed have thesame behaviour like axi-symmetric expansion(C1c =0).

Figure 12 shows how the barycentric map cap-tures the channel flow in three-dimensional space[7].This projection of a three-dimensional flow is possi-ble only in linear domain and not in a non-linear do-main such as theAIM . From Fig. 13 it can be ob-served that when a uniform distribution of randomlygenerated anisotropic stress tensorsbij are plotted inthe barycentric map and mapped back to the invari-ant map, the distribution is quite different. This factillustrates that conclusion of physical facts (i.e near-ness to axisymmetric and two-component limits) forany turbulent flow should not be made based on vi-

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

← C1c

← C2c

← C3c

y+

C1c

,C2c

,C3c

Lumley eigenvalue mapBarycentric map

Figure 11: At any point in the flow exact weightingcan be found in terms of the limiting states. Datawere extracted from a direct numerical simulationof fully developed plane turbulent channel flow forReτ = 180 by Kim et al.[7].

01

23

45

67

0

0.5

1

1.50

0.2

0.4

0.6

0.8

1

λ1

λ2

λ 3

Figure 12: Scalar metrics based on the eigenvaluesof anisotropy tensorbij projected onto the barycen-tric map. Data were extracted from a direct numericalsimulation of fully developed plane turbulent channelflow for Reτ = 180 by Kim et al.[7].

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

III

II

1 comp2 comp

isotropic

Figure 13: Randomly generated anisotropic tensorsbij which are uniformly distributed in the linearbarycentric space (right) mapped back to the nonlin-ear space(III, II) (left). The nonlinearity inII, IIIof the invariant space concentrates the points nearisotropy. The isotropic point corresponds to3comp.


ISSN: 1790-2769 39 ISBN: 978-960-474-034-5

2 comp 1 comp

3 comp

aij

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

C1c

→

C2c

→

C3c

→

x/M

C1c

,C2c

,C3c

Figure 14: Scalar metricsC1c, C2c andC3c of the bijfor plane strain (case – plane strain[5]) plotted insidethe barycentric map (left) and asmetricsplot (right).

sual observation.It has to be noted that invariant maps based on

(II, III), (λ1, λ2) and (C1c, C2c) are not sensitive tothe individual energy components of the anisotropicstress tensors, as it can be shown that two totally dif-ferent individual energy component anisotropic stresstensors having trace zero will be mapped to the samepoint in the above anisotropy invariant maps. The in-formation which is preserved between two anisotropicstress tensors mapped to two different points in theanisotropy invariant maps is their distance from thelimiting states (one-component, two-component andthree-component).

1.12 Relationship between Invariants andBarycentric Co-ordinates

In this section, we prove that the axisymmetry and thetwo-dimensional relationships of theAIM can be de-rived from equations (57). The anisotropy tensorbijfor the limiting states of turbulence are described inthe following ways:

b1c =D1c

traceD1c− 1

3(83)

b2c =D2c

traceD2c− 1

3(84)

b3c =D3c

traceD3c− 1

3(85)

D1c =

1 0 00 0 00 0 0

(86)

D2c =

1 0 00 1 00 0 0

(87)

D3c =

1 0 00 1 00 0 1

. (88)

For theone-component limit the invariants canbe found as

II =

(2

3

)2

+

(−1

3

)2

+

(−1

3

)2

(89)

III =

(2

3

)3

+

(−1

3

)3

+

(−1

3

)3

. (90)

For the two-component limit the invariants can befound as

II =

(1

6

)2

+

(1

6

)2

+

(−1

3

)2

(91)

III =

(1

6

)3

+

(1

6

)3

+

(−1

3

)3

. (92)

For theisotropic limit the invariants can be found as

II = 0 (93)

III = 0. (94)

The matrix for the axisymmetric expansion boundaryfrom equation (57) and takingC2c = 0 is given as:

b1c−3c = C1c

2/3 0 00 −1/3 00 0 −1/3

+ C3c

0 0 00 0 00 0 0

(95)

Using relation (59), let us takeC1c = κ and C3c =1 − κ, therefore the invariants can be found as

II =

(2

3κ

)2

+

(−1

3κ

)2

+

(−1

3κ

)2

(96)

III =

(2

3κ

)3

+

(−1

3κ

)3

+

(−1

3κ

)3

(97)

II =

(6

9

)κ2 (98)

III =

(2

9

)κ3. (99)


ISSN: 1790-2769 40 ISBN: 978-960-474-034-5

From the equations (98) and (99), the relationship forthe axisymmetric expansion for the(AIM) can be re-covered:

II =3

2

(4

3III

) 2

3

(100)

The matrix for the axisymmetric contraction bound-ary, takingC1c = 0 is given as

b2c−3c = C2c

1/6 0 00 1/6 00 0 −1/3

+ C3c

0 0 00 0 00 0 0

. (101)


II =

(1

6κ

)2

+

(1

6κ

)2

+

(−1

3κ

)2

(102)

III =

(1

6κ

)3

+

(1

6κ

)3

+

(−1

3κ

)3

(103)

II =

(6

36

)κ2 (104)

III =

(− 6

216

)κ3. (105)

From equations (104) and (105) the relationship forthe axisymmetric contraction for the(AIM) can beobtained:

II =3

2

(−4

3III

) 2

3

(106)

The matrix for the two-component boundary isgiven as

b2c−1c = C2c

1/6 0 00 1/6 00 0 −1/3

+ C1c

2/3 0 00 −1/3 00 0 −1/3

.(107)


II =

(1

6− κ

2

)2

+

(κ

2− 1

3

)2

+

(−1

3

)2

(108)

III =

(1

6− κ

2

)3

+

(κ

2− 1

3

)3

+

(−1

3

)3

(109)

II =

(2

3+κ2

2− κ

)(110)

III =

(2

9− κ

2+κ

2

2). (111)

From the equations (110) and (111) the relation-ship for the two-component boundary of the(AIM)can be obtained:

II =2

9+ 2III (112)

To obtain the limiting case of the plane strain limitwe use the observation about the two matrices (107);while moving from the two-component limit towardsthe one-component limit the middle eigenvalue of thetwo-component matrix has to become zero and con-tribute to the major eigenvalue. Hence the relevantmatrix for the plane strain limit is

bpl−strain =2

3

1/6 0 00 1/6 00 0 −1/3

+1

3

2/3 0 00 −1/3 00 0 −1/3

.(113)

where theC2c andC1c are taken as23 and 13 . From

these relationships, the invariants can be calculated asfollows:

II =

(1

3

)2

+

(−1

3

)2

= 2/9 (114)

III =

(1

3

)3

+

(−1

3

)3

= 0. (115)

1.12.1 Representation form of tensors in turbu-lence

The Reynolds stress tensoruiuj completely deter-mines the influence of turbulence on the mean flow.Therefore, in order to study and model it (and otherturbulence relevant properties) within the appropri-ate framework, one should consider the coordinatesystem determined by the principal directions ofReynolds stress tensor, instead of the coordinate sys-tem introduced by the flow geometry or mean velocity.So in the current work all the turbulent flow quantities


ISSN: 1790-2769 41 ISBN: 978-960-474-034-5

are considered in the coordinate system determined bythe principal directions of the Reynolds stress tensor.One can decompose any general symmetric second-order tensor into its diagonal and off-diagonal partsand then project the diagonal part onto the diagonalpart of the Reynolds stress parts. The diagonal partis of relevance since the turbulent transport along thethree axis is dominant compared to the off-diagonalpart which correlates the transport along the three axisand is small in flows of our interest. The following no-tation can be introduced for this purpose for a generaltensorgij :

gij = gij + gij . (116)

The aligned partgij has the same principal coordinatesystem as the matrixuiuj . The misaligned partgij

represents the difference in the principal directions be-tween the original tensor and the Reynolds stress. It isobvious that for such decomposition in the principalcoordinate system of the Reynolds stress tensoruiuj,the aligned part of any tensorgij has a diagonal struc-ture whereas the misaligned partgij has only zeros onthe diagonal. This decomposition can be applied tothe mean strain rateSij and the turbulent dissipationanisotropy tensoreij:

Sij = Sij + Sij, eij = eij + eij . (117)

Let us consider the components ofSij3 and eij

aligned with the Reynolds stress. For irrotational axi-symmetric homogeneous flows, the tensors posses adiagonal structure,bij, Sij andeij have the same or-thogonal and orthonormal vectors which can be con-firmed by calculating the eigenvectors by any alge-braic software package, so these tensors in their com-mon principal coordinate systembij , Sij and eij canbe denoted in a ordered manner asbij , Sij andeij andwritten with same basis matrices, following the termi-nology introduced by Banerjee et al. (2007)[11] as

bij = Ca1cb1c + Ca

2cb2c (118)

Sij = Cs1cb1c + Cs

2cb2c (119)

eij = Ce1cb1c + Ce

2cb2c (120)

Where the scalar functionsCa1c, C

a2c, C

s1c, C

s2c, C

e1c

andCe2c are the functions of eigenvalues of the corre-

sponding matricesbij , Sij, eij , respectively. It shouldbe noted that, since all of these tensors have zerotraces, the invariant properties of each of them can becompletely described by only two independent met-ricsC1c andC2c.

3Sij is not a semi-positive definite tensor.

1.12.2 General interrelationship between alignedtensors

In order to construct a functional interrelationshipbetween the tensorsbij , Sij and eij , which is in-dependent of the choice of the coordinate system,one should solve equations (118), (119) and (120).We exclude from consideration the trivial case ofthe isotropic turbulence, where all components of theReynolds stress anisotropy tensor are zero: and itgives us the pointC3c = 1 in the barycentric map(see figure (5)).

From equation (118) and (119), the following re-lationship can be derived betweenbij andSij for axi-symmetric turbulence:Axi-symmetric ContractionCa

1c = 0, Cs1c = 0

bij = Ca2cb2c

Sij = Cs2cb2c

(121)

Axi-symmetric Expansion Ca2c = 0, Cs

2c = 0

bij = Ca1cb1c

Sij = Cs1cb1c

(122)

From equation (118) and (120), the following rela-tionship can be derived betweenbij and eij for axi-symmetric turbulence:Axi-symmetric ContractionCa

1c = 0, Ce1c = 0

bij = Ca2cb2c

eij = Ce2cb2c

(123)

Axi-symmetric Expansion Ca2c = 0, Ce

2c = 0

bij = Ca1cb1c

eij = Ce1cb1c

(124)

This is the exact and the most general linear rela-tionship between the tensorsbij, eij , Sij written interms of the corresponding eigenvalues for axisym-metric turbulence.

1.12.3 Derivation of the algebraic model for theReynolds stress tensor

From equation (121, 122), the following algebraic ex-pression forbij can be derived in terms of the meanstrain rateSij.

bij =Ca

1c

Cs1cSij Axi-symmetric expansion

bij =Ca

2c

Cs2cSij Axi-symmetric contraction(125)


ISSN: 1790-2769 42 ISBN: 978-960-474-034-5

It is easy to see that the above expression correspondsto the turbulent eddy viscosity formulation assumedby the Boussinesq hypothesis. This formulation al-lows one to close the Reynolds stress tensor by thefollowing linear algebraic equation:

uiuj ≃ −νE

(∂Ui

∂xj+∂Uj

∂xi

)+

2

3kδij ,

νE = −kCa1c

Cs1c

. (126)

Where the subscript1c corresponds to the case of axi-symmetric expansion. Similarly the eddy-viscosityrelationship can be derived based2c, which corre-sponds to the case of axi-symmetric contraction. The“approximately equal” sign in this equation indicatesthat for this closure relationship between the Reynoldsstress anisotropy tensor and the mean strain rate isadopted based on the assumption that the eigenvectorsof these tensors are the same.

It can be shown that the absolute value of the tur-bulent eddy viscosity coefficientνE used in equation(126) corresponds exactly to the result obtained by:

∣∣∣∣Ca

1c

Cs1c

∣∣∣∣ =bαα − bββ

|Sαα − Sββ|=

√√√√ b211 + b222 + b233

S211 + S

222 + S

233

=

√bijbij

Snm Snm

=

√II

IIS. (127)

Since the magnitude of the strain rateIIS = SijSij

is always a non-negative quantity, the sign of theeddy viscosity coefficientνE computed by equationνE ≃

√II/IIS is always fixed. On the other side,

the matrixSij itself does not have the property ofpositive semi-definiteness and its eigenvalues can alsotake negative values. Therefore, the formulation de-rived in this work of the eddy viscosity coefficientνE

expressed by equation (126) differs from the one pro-posed by J. Jovanovic[12] in the sense that the newformulation is able to account for the appropriate signand does not require any explicit corrections.

1.12.4 Modelling of the turbulent eddy viscositycoefficient

The constitutive relationship expressed by equation(126) cannot be used directly for the Reynolds stressmodelling, since it requires the knowledge of the tur-bulent eddy viscosity functionνE. In this work, thefollowing approach is discussed: the modelling of theeddy viscosity coefficientνE by the algebraic rela-tionship.Algebraic model for the eddy viscosity coefficient.

In order to close the turbulence model, one can ap-proximate all values of the effective eddy viscositycoefficientνE using the idea described in detail by J.Jovanovic and I. Otic (2000)[12]. Here, its applicationto the barycentric map figure (5) is shown. Since alleigenvalues of any anisotropy tensor are zeros in theisotropic limiting state of turbulence, the eddy viscos-ity coefficientνE represented by equation (126) is alsohas to be zero. Taking the corresponding limiting val-ues of the Reynolds stress anisotropy tensoraij , onecan specify the functionνE for the one-componentand two-component axisymmetric limiting states ofturbulence4 in the following forms:

νE|iso = 0, νE|1c = −kCa1c

Cs1c

, νE|2c = −kCa2c

Cs2c

.(128)

Figures (15), (16) demonstrate that good agreement

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.110.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

−c2ca .*S

11/c

2cs

b 11

expbanerjee

−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

−c2ca .*S

11/c

2cs

b 11

expbanerjee

Figure 15: A cross plot ofbij versus -ca2cS11/cs2c for

axisymmetric strained turbulence in contraction (case- c1.27), (left) and (case - c3.69), (right). Data usedfrom Ertunc (2007)[13].

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

−c2ca .*S

11/c

2cs

b 11

expbanerjee

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.360.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

−c1ca .*S

11/c

1cs

b 11

expbanerjee

Figure 16: A cross plot ofbij versus -ca2cS11/cs2c for

axisymmetric strained turbulence in contraction (case- c14.69), (left) and betweenbij versus -ca1c. ∗S11/c

s1c

for axisymmetric strained turbulence expansion (case- c3.69), (right). Data used from Ertunc (2007)[13].

exists between linear relationships derived in (128)and the experimental data of axi-symmetric flows of

4In the work of J. Jovanovic and I. Otic (2000)[12], the fol-lowing limiting values of the eddy viscosity coefficientνE wereproposed based on application of the nonlinear anisotropy invari-ant map:

νE |iso = 0, νE|1c =2k√6IIS

, νE|2c =k√6IIS

.


ISSN: 1790-2769 43 ISBN: 978-960-474-034-5

Ertunc (2007). The results of the eddy viscosity modelgiven in equation (126) are shown in Appendix 1.1where they are compared tok − ǫ model results andexperiments. The present eddy viscosity model per-forms better thank−ǫmodel in rapid strain situationsbut not so well slow strain situations. This is becauseof the fact that the present eddy viscosity (see equation(128)) contains mean velocity gradient which makes itperform very well in rapid strain situations where themean flow time scale is smaller than the turbulencetime scale.

1.12.5 Linear interpolation scheme

The scalar quantityV at the three corner points ofthe barycentric map, can be labelled as three compo-nentV |3c, one-componentV |1c and two-componentV |2c turbulence. In order to approximate values of thefunctionV for an arbitrary, but physically realizable,anisotropy state of turbulence, one can apply the fol-lowing interpolation scheme. The above non-collinearpoints in the barycentric map are, where the functionV is knowna priori, and therefore, a linear interpola-tion technique can be efficiently applied.

It is obvious that in order to approximate all val-ues of the functionV on the straight line crossingtwo arbitrary points(M ′,N ′) and (M ′′,N ′′), oneshould introduce an interpolation functionR(M,N).It should belong to the family of the linear functions,for which the following relationships hold in the cho-sen points:

R(M ′,N ′) = 1, R(M ′′,N ′′) = 0. (129)

One is free to use any particular choice of the linearfunctionR(M,N), which satisfies these equation, toobtain the same approximation result computed by thefollowing equation:

V ≃ RV |(M ′,N ′) + (1 −R)V |(M ′′,N ′′). (130)

To construct a two-dimensional linear approxima-tion scheme based on equation (130) for the generalanisotropy state of turbulence, one can introduce thefollowing linear functionsG, H andF , which satisfythe corresponding requirements imposed by equation(129): • The interpolation scheme for the axisymmet-ric case of turbulence:

V |axi ≃

GV |iso + (1 −G)V |1c, C2c = 0,V |iso, C3c = 1,GV |iso + (1 −G)V |2c, C1c = 0.

(131)

G|iso = 1, G|1c = 0, G|2c = 0. (132)

• The interpolation scheme for the two-componentcase of turbulence:

V |2c ≃ HV |1c + (1 −H)V |2c. (133)

H|1c = 1, H|2c = 0. (134)

• The interpolation scheme for the general anisotropystate of turbulence:

V ≃ FV |axi + (1 − F )V |2c. (135)

F |axi = 1, F |2c = 0. (136)

To avoid a singularity in the one-component and two-component axisymmetric points of the barycentricmap, one should explicitly specify the following val-ues for the functionF for both of these cases:

F |1c = 1, F |2c = 1. (137)

1.12.6 Lumley interpolation scheme

The functionJ(II, III) is introduced there as a mea-sure of “two-componentality”5 in the following form:

J(II, III) = 1 − 9

(1

2II − III

). (138)

This function fulfills the requirement imposed byequation (129) with respect to the isotropicJ |iso =1 and two-componentJ |2c = 0 limiting states ofturbulence. Applying the axisymmetric criterion tothe equation (138), one can construct the function6

G(III) = J(II, III)|axi, which also satisfies all re-quired conditions (132):

G(III) = 1 − 9

((3

4III2

)1/3

− III

). (139)

The function F (II, III), which fulfills equations(136) and (137), is derived as the ratio of these twointroduced above functionsJ(II, III) andG(III).

5The functionJ(II, III) can be derived from the relation be-tween the invariants for the two component state of turbulence.

6One can see that the equivalent form of the functionG(III)can be written in terms of the second anisotropy invariantII . Thisform can be found by applying the equality|III | =

√II3/6,

which is valid for the axisymmetric limiting state of turbulence:

G(II) =

1 − 9(II/2 −

√II3/6

), III < 0,

1, III = 0,

1 − 9(II/2 +

√II3/6

), III > 0.


ISSN: 1790-2769 44 ISBN: 978-960-474-034-5

It is used for the interpolation between the two-component and axisymmetric boundaries of theanisotropy invariant triangle:

F (II, III) =J(II, III)

G(III). (140)

Because of a high nonlinearity of the principal com-ponentsII andIII with respect to the original ten-sor elements, the interpolation system, constructed bymeans of these functionsG(III), F (II, III), doesnot produce the required linear approximation of thescalar quantityV . In order to eliminate this inconsis-tency, one can use the linear affine space(C1c, C2c),in which the construction of the appropriate linear in-terpolation scheme becomes a trivial task. The linearinterpolation functionsG(C1c), F (C1c, C2c) can beeasily introduced for the linear anisotropy eigenvaluemap.

1.12.7 Interpolation scheme in affine space

Let us consider the functionJ(C1c, C2c) representingthe degree of the turbulence anisotropy:

J(C1c, C2c) = 1 − (C1c + C2c). (141)

This linear function can be considered as the mag-nitude of “two-componentality” of the original tur-bulence property tensor. It also fulfills all require-ments expressed by equation (129) with respect to theisotropic J |iso = 1 and two-componentJ |2c = 0cases of turbulence. For the interpolation along bothaxisymmetric branches of the anisotropy eigenvaluetriangle, the linear functionG(C1c) can be derived byputtingC2c = 0, into the equation (141). The resul-tant interpolation functionG(C1c) = J(C1c, C2c)|axi

is in agreement with all requirements expressed by(132) and can be written as

G(C1c) = 1 − C1c. (142)

Approximation of the turbulence scalarV over thewhole barycentric map can be computed by using thelinear7 interpolation functionF = F (C1c, C2c) deter-mined in exactly the same way as in equation (140).The linear functionsJ(C1c, C2c),G(C1c) are used:

F (C1c, C2c) =J(C1c, C2c)

G(C1c). (143)

In addition, the corresponding interpolation schemefor the two-component line of the barycentric map

7One can see that the functionF = F (C1c, C2c) is indeedlinear with respect to the interpolation direction along theC1c

axes. The functionF = F (II, III), for example, is quadraticwith respect to this term.

can be constructed with the help of the linear functionH = H(C2c), which also satisfies equation (134):

H(C1c, C2c) = C1c + C2c. (144)

One can clearly see that, in contrast to the nonlin-ear functions based on the principal componentsIIandIII, the application of the linear functions basedon the corresponding anisotropy eigenvaluesA andB results in a mathematically consistent interpola-tion scheme. Therefore, for turbulence modelling, theauthor proposes to use the interpolation scheme ex-pressed by equations (131), (133), (135) with linearfunctions (142), (144), (143) derived for the affinespace in terms of barycentric coordinates.

1.13 Invariant mapping of homogeneous tur-bulent flows

In the current work, the DNS data of Lee & Reynolds(1985 [14], denoted by LR) is used, and two casesof axisymmetric contraction (AXK, AXM), axisym-metric expansion (EXO, EXQ) and plane strain (PXA,PXF) are considered. In all of these cases a constantand uniform mean strain was applied to an initiallyisotropic turbulence.S is the mean strain rate

S = SijSij/2 (145)

The mean strain-rate tensor in the general form

Case Sq20/ǫ0 Sq2/ǫ(min - max)AXK 0.96 1.0 - 2.1

AXM 96.5 38.2 - 96.5

EXO 0.7 0.7 - 3.0

EXM 70.7 47.6 - 70.7

PXA 1.0 1.0 - 3.0

PXF 154.0 63.6 - 154.0

Table 2: Parameters for the simulations of Lee andReynolds (1985) [14].

Sij = ±2

3S

1 0 00 −1

2 00 0 −1

2

. (146)

with the plus sign corresponding to the case of ax-isymmetric contraction and the minus sign to the caseof axisymmetric expansion. Following the notation of


ISSN: 1790-2769 45 ISBN: 978-960-474-034-5

Lee & Reynolds (1985) [14], we define the referencetotal strain by

C = exp

∫ t

0Sdt′ = eSt (147)

and use this expression (147) as the dimensionlesstime variable in the discussion of evolution histories.

1 1.5 2 2.5 3 3.5−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

C

b ij

b11rapid

b11slow

b22rapid

b22slow

1 1.5 2 2.5 3 3.5−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

C

d ij

d11rapid

d11slow

d22rapid

d22slow

Figure 17: DNS results (Lee & Reynolds 1985 [14])showing the components of the Reynolds stressanisotropybij (left) and the dimensionality anisotropydij (right). [· − ·] lines correspond toSq20/ǫ0 =0.96 (case AXK - slow strain) and the [+-]lines toSq20/ǫ0 = 96.5 (case AXM - rapid strain). In thisflow the eddies become aligned with the axial direc-tion of positive mean strainx1. The stretching of axialvorticity results in an increase ofb22 andb33 relativeto b11 in contraction.

2C=2D 1C=1D

3 comp

bij

dij

1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c

Figure 18: Scalar metricsC1c, C2c andC3c of dij andbij for contraction (case AXK – slow strain) plottedinside the barycentric map(left) and asmetrics plot(right).

Axisymmetric contraction

The time histories of thebij (Reynolds stressanisotropy) anddij (dimensionality anisotropy) forthe two cases of axisymmetric contraction (AXK andAXM) listed in table (2) are shown in the figure (17).The anisotropy metricsC1c,C2c andC3c of bij anddij

tensor is plotted in the barycentric map (left) and themetricsplot (right), in figure (18) for the case (AXK)and in figure (19) for the case (AXM). The effect ofthe axisymmetric contraction is to stretch and orient

2C=2D 1C=1D

3 comp

bij

dij

1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c

Figure 19: Scalar metricsC1c, C2c andC3c of the di-mensionality and componentiality of the anisotropicReynolds stress tensor for contraction (case AXM –rapid strain) plotted inside the barycentric map (left)and as metrics plot (right).

the eddies so that they tend to become aligned withthe positive strain(S > 0) in axial direction. Thisis accompanied by a decrease in the axial stressb11and dimensionalityd11 as the axial vorticity is be-ing stretched. There is a strong alignment of elon-gated vortical eddies with the axial direction. Theeddy stretching and alignment are most effective inthe rapid case (AXM), figure (19) where the turbu-lence tries to reach an approximate2D − 2C state.From the figure (17 it can be observed that thedij andbij evolution histories are only weakly dependent onthe rate of straining, and follow each other closely.Also from themetricsplot in the fig. (18, 19), (right)it is found thatC1c = 0 for axisymmetric contrac-tion, case (AXK and AXM) and which confirms theassumption taken in deriving eddy viscosity relation-ships. It has to be noted that no physical conclu-sions can be deduced if the trajectories ofbij anddij

are observed for both cases (AXK and AXM) in thebarycentric map figures (18, 19), (left), however fromthe metricsplot in the figures (18, 19), (right) it canbe deduced that the trajectories ofbij anddij followeach other closely, and consideringdij as an extra pa-rameter in modelling rapid pressure-strain closure foraxisymmetric contraction will increase the computa-tional cost, but not improved predictions drastically.

Axisymmetric expansion

The time histories of thebij and dij for the twocases of axisymmetric expansion (EXO and EXQ) areshown in the figure (20). Flow through axisymmetricexpansion exhibits counter-intuitive behaviour, whichmakes it a challenge to turbulence modelling. Theeffect of the negative axial strain(S < 0) in thesecases is to concentrate the eddies in thin, disk-likeregions normal to the axis of symmetry, while theweaker positive strain causes a mild radial stretch-ing. Sustained axisymmetric expansion produces pan-cake turbulence, in which the symmetry axis becomes


ISSN: 1790-2769 46 ISBN: 978-960-474-034-5

1 1.5 2 2.5 3 3.5−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

C

b ij

b11rapid

b11slow

b22rapid

b22slow

1 1.5 2 2.5 3 3.5−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Cd ij

d11rapid

d11slow

d22rapid

d22slow

d33rapid

d33slow

Figure 20: DNS results (Lee & Reynolds 1985 [14])showing the components of the Reynolds stressanisotropybij (left) and the dimensionality anisotropydij (right). [· − ·] lines correspond toSq20/ǫ0 =0.71 (case EXO – slow strain) and the [+-]lines toSq20/ǫ0 = 70.7 (case EXQ – rapid strain). In this flowthe negative axial and positive lateral mean strain con-centrate the eddies in thin pancake-like structures nor-mal to the axial direction. The result is an increaseof the axial Reynolds stressb11 relative to the lateralstress components. The Reynolds stresses are muchmore anisotropic in the slowly strained case than theyare in the rapidly strained case.

2C=2D 1C=1D

3 comp

bij

dij

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c

Figure 21: Scalar metricsC1c, C2c andC3c of dij andbij for expansion (case EXO – slow strain) plottedinside the barycentric map (left) and as metrics plot(right).

2C=2D 1C=1D

3 comp

bij

dij

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c

Figure 22: Scalar metricsC1c, C2c andC3c of dij andbij for expansion (case EXM – rapid strain) plottedinside the barycentric map (left) and as metrics plot(right).

the direction with the most energetic velocity fluc-tuations, but with very little large-scale circulationaround it. However in the weak strain case, for whichthere is a large disparity between the anisotropy ofthe dimensionality tensor, which is very weak, andthe anisotropy of the Reynolds stress tensor, which isquite strong (see figure (21), (right)). From themet-rics plot in the figure (21), (right) it can be observedthat the anisotropy in dimensionality and Reynoldsstress are only weakly dependent on the rate of strain-ing. Also from themetricsplot in the figures (21, 22),(right) it can be found thatC2c = 0 for axisymmetricexpansion flows (EXO and EXQ) and it confirms theassumption taken in deriving eddy viscosity relation-ships. It has to be noted that no physical conclusionscan be deduced if the trajectories ofbij anddij areobserved in the barycentric map for both cases (EXOand EXQ) in figures (21, 22), (right). Similar, to con-traction it can be observed that the trajectories ofbijanddij follow each other closely, and consideringdij

as an extra parameter in modelling of rapid pressure-strain closure will increase the computational cost, butnot improved predictions drastically .

For irrotational homogeneous flows, such as axi-symmetric flows the tensorsbij, dij can be taken asbij , dij, these have the same orthogonal and orthonor-mal vectors which can be confirmed by calculating theeigenvectors of these tensors by any available soft-ware packages like MATLAB or MAPLE, and canbe written following the terminology introduced byBanerjee et al. (2007) [11] as

bij = Ca1cb1c + Ca

2cb2c (148)

dij = Cd1cb1c + Cd

2cb2c (149)

For axi-symmetric contraction Ca1c = 0, Cd

1c = 0

bij = Ca2cb2c

dij = Cd2cb2c

(150)

For axi-symmetric expansion Ca2c = 0, Cd

2c = 0

bij = Ca1cb1c

dij = Cd1cb1c

(151)

bij =Ca

1c

Cd1c

dij Axi-symmetric expansion

bij =Ca

2c

Cd2c

dij Axi-symmetric contraction

(152)Exact relationships can be derived forbij anddij for the one-component and two-component


ISSN: 1790-2769 47 ISBN: 978-960-474-034-5

limits of the barycentric map (5) as followsFor one component limit Ca

1c = 1, Cd1c = 1

bij = dij (153)

For two component limit Ca2c = 1, Cd

2c = 1

bij = dij (154)

So from the kinematic relationships (153, 154) it canbe concluded that at the two component limit of thebarycentric map (5) the dimensionality and the com-ponentiality of the Reynolds stress tensor are equal.So at the two-component limit (see figure (23)) theproblem of modelling the rapid pressure-strain clo-sure with onlyuiuj as the parameter is a well posedproblem. The difficulty about the dimensionality andthe componentiality of the Reynolds stress tensor isencountered at the three component limit (isotropicpoint) where the turbulence can be one-component,two-component or three-component. This is outlinedin the figure (23).

2C = 2D

3C

1C = 1Dcomponentiality=dimensionality

Axi-sy

mmetric

contr

actio

n Axi-symmetric expansionPla

inst

rain

1D

2D3D

Figure 23: Barycentric map based on scalar metricswhich are functions of eigenvalues of the second-order stress tensor describing dimensionality andcomponentiality at various limiting states. Theisotropic point has a metricCa

3c = 1, the two-component point hasCa

2c = 1 = Cd2c and the one-

component point hasCa1c = 1 = Cd

1c.

Plane Strain

The mean strain rate tensor for the plane strain caseshas the general form

Sij = S

0 0 00 −1 00 0 +1

. (155)

1 1.5 2 2.5 3 3.5 4−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

b11rapid

b22rapid

b33rapid

b11slow

b22slow

b33slow

1 1.5 2 2.5 3 3.5−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

C

d ij

d11rapid

d22rapid

d33rapid

d11slow

d22slow

d33slow

Figure 24: DNS results (Lee & Reynolds 1985 [14])showing the components of the Reynolds stressanisotropybij (left) and the dimensionality anisotropydij (right). [· − ·] lines correspond toSq20/ǫ0 =1.0 (case PXA – slow strain) and the [+-]lines toSq20/ǫ0 = 154.0 (case PXF – rapid strain). In this flowthe vorticity stretching in the direction of the positivemean strainx3 results in an increase in the large scalecirculation with a decrease inb33 andd33. Note thatin the rapidly strained case theb22 has higher valuesthan the slowly strained case, whereas the opposite istrue ford22.

2 comp 1 comp

3 comp

bij

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Ca1c

→

Ca2c

→

Ca3c

→

x/M

Ca 1c

, Ca 2c

, Ca 3c

Figure 25: Scalar metricsC1c, C2c andC3c of thebijfor plane strain (case Choi) plotted inside the barycen-tric map (left) and as metrics plot (right).

2C=2D 1C=1D

3 comp

bij

dij

1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c

Figure 26: Scalar metricsC1c, C2c andC3c of dij andbij for plane strain (case PXA – slow strain) plottedinside the barycentric map (left) and as metrics plot(right).


ISSN: 1790-2769 48 ISBN: 978-960-474-034-5

2C=2D 1C=1D

3 comp

aij

dij

1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c

Figure 27: Scalar metricsC1c, C2c andC3c of dij andbij for plane strain (case PXF – rapid strain) plottedinside the barycentric map (left) and as metrics plot(right).

The evolution histories of the anisotropies of theReynolds stress and dimensionality are shown in thefigure (24) for two different cases (PXA and PXF).Sustained plane strain produces turbulence consist-ing of eddies elongated in the direction of positivestrain (d33 = −1

3), but very little motion along theiraxes (b33 = −1

3 ). This state corresponds to the same2D−2C limiting state of vortical eddies aligned withdirection of positive strain that is produced by axisym-metric contraction. However, plane strain differs fromthe axisymmetric contraction flow in that the limitingstate is reached along a different path in the barycen-tric map (see figure 25). It has to be noted that, inthe literature (Choi and Lumley (2000) [5]), it is of-ten cited that the return to isotropy of plane strainexperiments tend to orient towards axi-symmetric ex-pansion however when the trajectory of the anisotropytensor is plotted in the figure (25), (left) maintains theslope with respect to the plane strain axis and showsno tendency to become axisymmetric. This has mod-elling justification, as the return to isotropy in planestrain cases can be approximated well by linear returnto isotropy model. From themetricsplot in the fig-ure (25), (right) it can be seen that the trajectories ofC1c andC2c of bij are nearly equal in the return toisotropy process, and there is a steady increase ofC3c

of bij which gives a clear indication of the increase inisotropic character of the flow. From themetricsplotin the figures (26, 27), (right) it can be deduced thatthe trajectories ofbij anddij follow each other closely,nearly exactly for the rapid case (27). Hence, consid-ering dij as an extra parameter in modelling will in-crease the computational cost, but not improved pre-dictions drastically in this case.

Homogeneous shear flow

In the present work, we use the case (C128U) fromthe direct numerical simulations of Rogers & Moin(1987) [15]. In the case (C128U), a fully developedstage is reached for10 ≤ St ≤ 16 during which the

0 2 4 6 8 10 12 14 16−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

St

d ij

da11

da22

da33

da12

0 2 4 6 8 10 12 14 16−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

St

b ij

b11

b22

b33

b12

Figure 28: DNS results (Rogers & Moin (1987) [15])showing the profiles ofdij , (left) andbij , (right).

2 comp 1 comp

3 comp

bij

dij

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

St

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c

Figure 29: Scalar metricsC1c, C2c andC3c of thedij

andbij for homogeneous shear plotted as metrics plot(right) and inside the barycentric map (left).

2 comp 1 comp

3 comp

bij

dij

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

St

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c


and bij for high shear plotted as metrics plot (right)and inside the barycentric map (left).


ISSN: 1790-2769 49 ISBN: 978-960-474-034-5

turbulence Reynolds numberReλ reaches values of upto1500 andSq2/ǫ is about11. The evolution historiesfor the normalized second-rank tensorsbij are shownin figure (28), (right) and fordij in figure (28), (left).Theb11 component grows at the expense of the othertwo normal stresses. In this case, the dimensionalitytensor, bothd22 andd33 dominate overd11.

The observations suggest that in equilibrium tur-bulence of homogeneous shear, the flow consistsmainly of elongated structures, roughly aligned alongthe mean flow direction(d11 ≈ 0.20). Sinced22 ≈d33, which suggests that the dimensionality is close tobeing axisymmetric about thex1 axis which can bealso verified by looking at the trajectories ofdij andbij in the barycentric map figure (29), (left) and theCd

1c ≈ 0 in themetricsplot (see figure (29), (right).The near axisymmetry ofdij is also reflected from thefigure (29), (left), where the anisotropy of the dimen-sionality tensor falls on the axisymmetry line movingtowards the two-dimensional point. No approach toaxisymmetry is observed forbij . As observed in fig-ure (29), (right) thebij reach anisotropy levels that areconsiderably higher than those reached bydij . Figure(29), (right) shows a comparison of the scalar metricsof anisotropy tensorbij and dimensionality tensordij ,which shows thatCd

11 ≈ 0 andCa11 >> Cd

11. In theother two metrics (i.eC2c andC3c) of bij anddij , theanisotropy in dimensionality is more stronger.

This is a particular advantage of barycentric met-rics C1c, C2c andC3c outlined in the current workcompared to invariantsII, III in giving more de-tailed quantitative information about the componen-tiality and directionality of anisotropy. These resultsare in general agreement with what was observed inthe case of irrotational strain where, anisotropy mea-sure of dimensionality was found to be lower than theanisotropy measure ofbij . Similar analysis as above,also holds for a different set of data of high shear ofLee & Moin (1990) (see figure (30)) for profiles inthe barycentric map (30), (left) andmetricsplot (30),(right).

Channel Flow

We begin our study of inhomogeneous flows with thedirect numerical simulations of fully developed chan-nel flow by Kim, Moin & Moser (1987) [7]. Thesesimulations have Reynolds numbers based on the wallshear velocityuτ of Reτ = 180 , Reτ = 395 andReτ = 590. In all the cases, we takex1 andx3 tobe the homogeneous streamwise and spanwise direc-tions, andx2 to be the wall-normal direction. The pro-files of bij anddij for Reτ = 385 was not availablefor investigation, but there were significant deviations

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

III

II

Reτ=180

Reτ=395

Reτ=590

Figure 31: Anisotropy invariant map(III, II) asproposed by J. L. Lumley and G. Newman (1977).The coordinatesII andIII are the second and thirdinvariants of the anisotropy matrix of the Reynoldsstress tensor. Invariant characteristics of anisotropytensors for the Reynolds stressuiuj on the anisotropyinvariant map(III, II) for [·] – Reτ = 180, [·] –Reτ = 395, [·] – Reτ = 590.

2 comp 1 comp

isotropic

Reτ=180

Reτ=395

Reτ=590

Figure 32: Barycentric map(C1c, C2c) as proposedby the authors. The coordinatesC1c and C2c arefunction of the eigenvalues of the anisotropy ma-trix of the Reynolds stress. Invariant characteristicsof anisotropy tensors in terms of(C1c, C2c) for theReynolds stressuiuj on the anisotropy invariant map(III, II) for [◦] – Reτ = 180, [◦] – Reτ = 395, [◦]–Reτ = 590.


ISSN: 1790-2769 50 ISBN: 978-960-474-034-5

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

III

II

Reτ=180

Reτ=395

Reτ=590

Figure 33: Anisotropy invariant map(III, II) as pro-posed by J. L. Lumley and G. Newman (1977). ThecoordinatesII and III are the second and third in-variants of the anisotropy matrix of the dissipationtensor. Invariant characteristics of anisotropy tensorsfor the Reynolds stressuiuj on the anisotropy in-variant map(III, II) for [·] – Reτ = 180, [·] –Reτ = 395, [·] – Reτ = 590.

2 comp 1 comp

isotropic

Reτ=180

Reτ=395

Reτ=590

Figure 34: Barycentric map(C1c, C2c) as proposedby the authors. The coordinatesC1c andC2c are func-tion of the eigenvalues of the anisotropy matrix ofthe dissipation. Invariant characteristics of anisotropytensors in terms of(C1c, C2c) for the dissipationǫijon the anisotropy invariant map(IIIe, IIe) for [◦] –Reτ = 180, [◦] – Reτ = 395, [◦] – Reτ = 590.

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

III

II

Reτ=180

Reτ=395

Reτ=590

Figure 35: Anisotropy invariant map(III, II) as pro-posed by J. L. Lumley and G. Newman (1977). ThecoordinatesII and III are the second and third in-variants of the anisotropy matrix of the dissipationtensor. Invariant characteristics of anisotropy tensorsfor the vorticitywij on the anisotropy invariant map(III, II) for [·] – Reτ = 180, [·] – Reτ = 395, [·] –Reτ = 590.

2 comp 1 comp

isotropic

Reτ=180

Reτ=395

Reτ=590

Figure 36: Barycentric map(C1c, C2c) as proposedby the authors. The coordinatesC1c andC2c are func-tion of the eigenvalues of the anisotropy matrix of thevorticity tensor. Invariant characteristics of anisotropytensors in terms of(C1c, C2c) for the vorticity tensorwij on the anisotropy invariant map(IIIw, IIw) for[◦] – Reτ = 180, [◦] – Reτ = 395, [◦] – Reτ = 590.

2 comp 1 comp

isotropic

Reτ=180

Reτ=395

Reτ=590

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y+

C1c

,C2c

,C3c

C1801c

C3951c

C5901c

C1802c

C3952c

C5902c

C1803c

C3953c

C5903c

Figure 37: Scalar metricsC1c, C2c andC3c of thedimensionality and componentiality of the deviatoricpart of the vorticity tensorwij for channel flow at [◦] –Reτ = 180, [◦] – Reτ = 395, [◦] – Reτ = 590 plot-ted as metrics plot (right) and inside the barycentricmap (left).


ISSN: 1790-2769 51 ISBN: 978-960-474-034-5

2 comp 1 comp

isotropic

Reτ=180

Reτ=395

Reτ=590

0 20 40 60 80 100 120 140 160 180−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y+

C1c

,C2c

,C3c

C1801c

C3951c

C5901c

C1802c

C3952c

C5902c

C1803c

C3953c

C5903c

Figure 38: Scalar metricsC1c, C2c andC3c of thedeviatoric part of the dissipation tensor for channelflow at [◦] – Reτ = 180, [◦] – Reτ = 395, [◦] –Reτ = 590 plotted as metrics plot (right) and insidethe barycentric map (left).

2 comp 1 comp

isotropic

Reτ=180

Reτ=395

Reτ=590

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

y+

C1c

,C2c

,C3c

C1c

180

C1c

395

C1c

590

C2c

180

C2c

395

C2c

590

C3c

180

C3c

395

C3c

590

Figure 39: Scalar metricsC1c, C2c andC3c of thebij for [◦] – Reτ = 180, [◦] – Reτ = 395, [◦] –Reτ = 590 plotted as metrics plot (right) and insidethe barycentric map (left).

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

III

II

Reτ=180

Reτ=550

Reτ=950

Reτ=2000

2 comp 1 comp

3 comp

Reτ=180

Reτ=550

Reτ=950

Reτ=2000

Figure 40: Invariant characteristics of anisotropy ten-sors for the Reynolds stressuiuj on the anisotropyinvariant map(III, II) for [·] – Reτ = 180, [·] –Reτ = 550, [·] – Reτ = 950, [·] – Reτ = 2000,(left). Invariant characteristics of anisotropy tensors interms of(C1c, C2c) for the Reynolds stressuiuj on thebarycentric map for [·] –Reτ = 180, [·] –Reτ = 550,[·] – Reτ = 950, [·] – Reτ = 2000, (right).

in theCa1c andCd

1c near the wall tilly+ = 8 which ispredominantly due to the inhomogenities present nearthe wall. From thebij profiles in the invariant mapfigure (31), and in the barycentric map figure (32) it isclear that the barycentric map doesnot give a false im-pression of axi-symmetry which is given by the invari-ant map all through the flow development. Also fromthe vorticity correlationwij trajectories in the invari-ant map figure (33), and in the barycentric map figure(34) and the anisotropic dissipationeij profiles in theinvariant map figure (35), and in the barycentric mapfigure (36), it is clear that the barycentric map does-not give a false impression of axi-symmetry, which isprovided in the invariant map all through the flow de-velopment. Thus the barycentric map provides a muchclearer and consistent basis for viewing the second or-der tensors relavant in turbulence.

Figure 39, (right) clearly illustrates that the math-ematical requirement of axisymmetryC2c = C1c = 0is not satisfied, but when it is viewed in theAIM fig-ure 40, (left) it gives a visual impression of axisym-metry. The barycentric map figure 39, (right) is muchmore consistent in preserving this physical informa-tion and it shows that the fully developed plane turbu-lent channel flow forReτ = 180 is far from axisym-metry. In figure 39, (right) it can be observed that thebarycentric map offers a visual advantage in depictingthat the channel flow is far from isotropy at the chan-nel centreline. The weighting of the limiting states forplane channel flow data of Kim et al. [7] were com-puted and it was found that the isotropic metricC3c is0.9 and not1 to indicate full isotropy of the flow at thechannel centreline. Table 1 gives a clear mathemati-cal justification that the channel flows are not isotropicat the centreline as claimed by Kim et al. [9]. Thenonlinearity in(II, III) makes the flow tend towardsisotropy visually when viewed in theAIM (figure 40,left). Similar conclusions can be drawn about pipeflows when they are viewed in the barycentric map.

The corresponding profiles ofbij from theReτ =180 , Reτ = 395 andReτ = 590 case are shown inthe figure (39) in the barycentric map (left) andmet-rics plot (right). The similar profile behaviour at highreynolds number indicate that with increasingReτ theinertial behavior becomes self-similar away from thewalls as observed in the figure (39), (right) where thescalar metrics nearly match each other. The dominantcomponent as observed in the figure (39), (right) is thescalar metricCa

11 andCd11, corresponding to the frac-

tion of turbulent kinetic energy found in streamwisefluctuations, reaches a maximum aty+ = 8. Fromthe figure (37), (left) again the self similar profiles ofthe anisotropic vorticitywij can be observed, how-ever it can be found that the profile forReτ = 590differs considerably and at abouty+ = 40 the scalar


ISSN: 1790-2769 52 ISBN: 978-960-474-034-5

metric Cw11 andCw

22 tend to be zero andCw33 → 1

(see figure (37), (right)) showing that in a channelflow starting fromy+ = 40 the flow is nearly three-dimensional. From the figure (38), (left) the self sim-ilar profiles of the anisotropic dissipationeij can beobserved and it can be noted that away from the wallsaroundy+ = 8 (see figure (38), (right)) the gradi-ents in the anisotropic dissipation is becoming weak,andCe

11, Ce22 decay andCe

33 becomes dominant withCe

33 → 1. The barycentric map provides a muchclearer visualization of the trajectories ofbij , wij andeij for a channel flow and themetricsplot provides aquantification of anisotropy at every point of the flowand how the componentiality metrics vary in the flow.

2 comp 1 comp

3 comp

bij

dij

−50 −40 −30 −20 −10 0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c


andbij for an unforced wake plotted as metrics plot(right) and inside the barycentric map (left).

2 comp 1 comp

3 comp

bij

dij

−100 −80 −60 −40 −20 0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

C1c

,C2c

,C3c

C1ca

C1cd

C2ca

C2cd

C3ca

C3cd

Figure 42: Scalar metricsC1c, C2c andC3c of thedij and bij for strongly-forced wake plotted as met-rics plot (right) and inside the barycentric map (left).

Self-Similar Plane wake

We consider two simulations of time-developing tur-bulent wakes presented by Moser et al. (1998) [16]. Inone of the simulations, two-dimensional disturbanceswere added initially to mimic two-dimensional forc-ing. This case will be referred to here as the stronglyforced wake figure (42). The unforced wake figure(41), where no two-dimensional disturbances wereadded beyond, what was present in the initial con-ditions, was allowed to evolve long enough to attainself-similarity. In the forced case no extended periodof self-similar growth was obtained, but a period ofapproximate self-similarity was used to generate time-averaged profiles.

The half-width is taken to be the distance betweenthe y-locations at which the mean velocity is half ofthe maximum velocity deficit magnitudeU0. The non-dimensionalization of the time variable is based onthe mass-flux deficit m. and the initial magnitude ofthe velocity deficitUd, and is given byτ = tU2

d/m.The profiles ofbij anddij for the unforced case areshown in for a time during the self-similar period inthe barycentric map figure (41), (42), (left) and in themetricsplot in figure (41), (42), (right). From the pro-files of bij anddij in the barycentric map figure (41),(42), (left) it is not possible to deduce any conclu-sive evidence about the flow behaviour, however whenviewed in themetricsplot figure (41), (42), (right) itcan be immediately found that the metricsC1c, C2c

andC3c of bij anddij follow each other very closely.Since the wake is statistically symmetric, the statis-tical sample in these profiles was effectively doubledby averaging the two sides of the wake together. Thedistribution of the various tensor components is simi-lar to what was obtained in the case of homogeneousshear flow.

The profile for the dimensionality tensordij re-mains relatively constant across the wake in thestrongly forced case (see to the profiles ofC1c, C2c

andC3c figure (42), (right) which are nearly constantfor bothbij anddij). The streamwise componentd11

for both forced and unforced case is slightly smallerthan the cross-stream and spanwise components (d22

andd33) indicating the existence of structures that aresomewhat elongated in the streamwise direction. Thisis also evident from the figure (41), (42), (right) wherethe Cd

11 << Cd22, Cd

33. Similar behaviour is alsonoticed about the scalar metrics ofbij which matchnearly well with the metrics ofdij in the stronglyforced case (42), (right), and qualitatively well in theunforced case (41), (right).

There is a striking difference between the profilesof dij for the forced and the unforced wake cases,which is a manifestation of the differences in thestructure of the turbulence in these two flows. Moseret al. (1998) [16] point out that the unforced wake isbest described as “a slab of turbulence with undulatingboundaries”. In contrast, in the forced wake one canobserve coherent concentrations of large-scale span-wise circulation. These differences are nicely re-flected in the profiles ofd33 for the two cases. Notethat the two-dimensional forcing has suppressedCd

1c,andCa

1c and augmented the circulation, as one wouldexpect in a flow with the principal energy-containingstructure consisting of vortical structures aligned withthe spanwisex3 axis.


ISSN: 1790-2769 53 ISBN: 978-960-474-034-5

2 comp 1 comp

isotropic

bij

dij

−30 −20 −10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

C1c

,C2c

,C3c

Ca1c

Cd1c

Ca2c

Cd2c

Ca3c

Cd3c


and bij for strongly-forced mixing layer (case 5.45)plotted as metrics plot (right) and inside the barycen-tric map (left).

2 comp 1 comp

3 comp

bij

dij

−30 −20 −10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

C1c

,C2c

,C3c

C1c

C1c

C2c

C2c

C3c

C3c


and bij for strongly-forced mixing layer (case 6.70)plotted as metrics plot (right) and inside the barycen-tric map (left).

2 comp 1 comp

isotropic

bij

dij

−30 −20 −10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

C1c

,C2c

,C3c

C1ca

C1cd

C2ca

C2cd

C3ca

C3cd


andbij for unforced mixing layer (case 5.45) plottedas metrics plot (right) and inside the barycentric map(left).

Self-similar turbulent mixing layer

The study of the structure tensordij and anisotropicReynolds stress tensorbij in self-similar turbulentmixing layers (see figure (44)) was also undertaken.For this purpose we used from the direct numeri-cal simulations by Rogers & Moser (1994) [17], twocases (strongly forced and unforced) of a temporallyevolving plane mixing layer.

The trajectories of thebij and dij for the un-forced case, case (5.75) (see figure (45)), (left) andcase (6.70) (see figure (46)), (left). From the trajecto-ries ofbij anddij plotted in the barycentric map, noth-ing conclusive can be drawn about the flow behaviour,however when analysed in themetricsplot it can befound that the dimensionality tensordij closely fol-lows the anisotropy tensorbij (case 5.75) (see figure(45)), (right) and (case 6.70) (see figure (46)), (right).

The trajectories of thebij anddij for the stronglyforced case, (case 5.75) (see figure (43)), (left) and(case 6.70) (see figure (44)), (left). From the trajecto-ries ofbij anddij plotted in the barycentric map, noth-ing conclusive can be drawn about the flow behaviour,however when analysed in themetricsplot it can befound that the dimensionality tensordij closely fol-lows the anisotropy tensorbij (case 5.75) (see figure(43)), (right) and (case 6.70) (see figure (44)), (right).The conclusions reached are similar to those statedfor the case of the self-similar plane wake. In theunforced plane mixing layer the profiles of the ten-sorsbij , dij are relatively uniform and the inhomo-geneity effects are quite small except near the edgesof the layer. In the inhomogeneous region, the turbu-lence structure in the strongly inhomogeneous regionis dominated by large-scale circulation (i.e no vortic-ity) about the spanwise direction. Forcing of the two-dimensional modes has the same effect on the mixinglayer as in the plane wake, that is it produces a flowthat is very close to being2D − 2C and that is dom-inated by vortical structures. As in the plane wake,strong forcing effectively reduces the number of in-dependent components required to describe the turbu-lence.

2 comp 1 comp

isotropic

bij

dij

−40 −30 −20 −10 0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

C1c

,C2c

,C3c

C1ca

C1cd

C2ca

C2cd

C3ca

C3cd


andbij for unforced forced mixing layer (case 6.70)plotted as metrics plot (right) and inside the barycen-tric map (left).


ISSN: 1790-2769 54 ISBN: 978-960-474-034-5

1.14 Transport equations for the turbulencestructure tensors

The transport equations for the dimensionalityDij

and circulicityFij can be obtained starting from thefluctuating momentum equation. Because the basicdefinitions for the structure tensorsDij andFij in-volve correlations between gradients of the streamfunction, the necessary manipulations that lead tothe desired transport equations are most convenientlydone in a frame of reference deforming with the meanmotion, in which all the dependent turbulence statis-tics are homogeneous. The evolution equations for theDij andFij in irrotational homogeneous turbulenceare given as

DDij

Dt= −DikGkj −DjkGki + 2SmnLimnj

− 2SmnMmnij − Πsij

− (ψ′

i,mψ′

n,ju′

m,n + ψ′

j,mψ′

n,iu′

m,n)

− 2νu′

m,iu′

m,j (156)

DFij

Dt= FikGkj − FjkGki − 2SkkFij + 2SnmDnmδij

+ 2SmnLimnj − 2SmnMmnij + (ψ′

i,mψ′

n,ju′

m,n

+ ψ′

j,mψ′

n,iu′

m,n) − 2νω′

iω′

j

In equation (157) the rotational part of the rapidpressure-strain-rate term is set to zero andΠs

ij is theslow pressure-strain-rate term. The fourth-rank tensorL is defined by

Lijpq =

∫κiκjκpκq

κ4Enn(κ)d3κ (157)

The triple correlations in (156) and (157) are trace-free and represent nonlinear intercomponent energytransfer between theDij andFij .

The evolution equations of the normalized tensorsdij andfij , and for the corresponding anisotropiesdij

andfij follow easily from the definitions in the be-ginning of this chapter and inserting them into (156)and (157). Using the evolution equation for thedij

and that for the normalized Reynolds stress tensorrij, one can write down the equation for the tensorial dif-ference as∆ij = rij−dij . For the case of irrotationalmean deformation, this equation is given as

D∆ij

Dt= −(Sik∆kj + Sjk∆ki) + 2Snmrnm∆ij

+ 2[Snm(Mimnj +Mjmni +Mmnij − Lmnij)

+ Πsij −

1

2(ψ

′

i,mψ′

n,ju′

m,n + ψ′

j,mψ′

n,iu′

m,n)

− 2ν(u′

i,ku′

j,k − u′

k,iu′

k,j)]/q2 (158)

When the mean strain rate is rapid (i.e inviscidRDT), (158) reduces to

D∆ij

Dt= −(Sik∆kj + Sjk∆ki) + 2Snm[rnm∆ij

+ (Mimnj +Mjmni +Mmnij − Lmnij)/q2](159)

The rapid terms in (156) involve the fourth-ranktensors M and L, but for a simplified analysis thesemay be modelled with a linear representation indij

and rij. Kassinos and Reynolds have shown thatwhen these two linear models are used together, theresulting representation of the rapid terms is exact forweakly anisotropic turbulence. In fact for the irrota-tional flows considered here, this linearized represen-tation of the rapid effects is quite accurate for slowstrains. With these simplifying assumptions, we ob-tain

D∆ij

Dt= −5

7(Sik∆kj + Sjk∆ki) +

10

21Snmrnm∆ij

2Snmrnm∆ij + 2ǫkkS∗∆ij

+ 2ν(u′

i,ku′

j,k − u′

k,iu′

k,j)/q2

1.15 Linear structure-based models forweakly anisotropic turbulence

Two simple models that make use of the new ten-sors are given here because they provide additionalinsight, but one must keep in mind that more sophis-ticated structure-based models have been constructedby Reynolds & Kassinos (1995) [1]. A model forthe fourth-rank tensorMijpq occurring in the rapidpressure-strain-rate term in terms ofrij alone is fun-damentally wrong in non-equilibrium turbulence. Forweakly anisotropic turbulence, one can easily con-struct a model forMijpq that is linear in the anisotropytensorsrij and dij . We can write the most generalfourth-rank linear tensor function of two second-ranktensorsrij anddij,

Mijpq

q2= C1δijδpq + C2(δipδjq + δiqδjp)

+ C3δijrpq + C4δpqrij

+ C5(δiprjq + δiqrjp + δjpriq + δjqrip)

+ C6δijdpq + C7δpqdij

+ C8(δipdjq + δiqdjp + δjpdiq + δjqdip)


ISSN: 1790-2769 55 ISBN: 978-960-474-034-5

For homogeneous turbulence, continuity and thedefinitions from Rotta all of the coefficients in the lin-ear model, and one finds

Mijpq

q2=

2

15δijδpq −

1

30(δipδjq + δiqδjp)

+4

21δijrpq +

11

21δpqrij

− 1

7(δiprjq + δiqrjp + δjpriq + δjqrip)

+11

21δijdpq +

4

21δpqdij

− 1

7(δipdjq + δiqdjp + δjpdiq + δjqdip)

Similar analysis leads to a model for the fourth-ranktensorLijpq, that is linear in the anisotropydij , andwith all its numerical coefficients determined by anal-ysis:

Lijpq/q2 =

1

15(δijδpq + δipδjq + δiqδjp)

+1

7(δijdpq + δpqdij

+ δipdjq + δiqdjp + δjpdiq + δjqdip)

A detailed discussion on the performance of the lin-ear model (160) and ( 160) is given in Reynolds &Kassinos (1995) [1]. For irrotational deformations themodel is quite accurate for total strains less than 3, buteventually becomes unrealizable as the basic assump-tion of weak anisotropy is violated. The experimentaldata of homogeneous axisymmetric turbulence, sub-jected to irrotational strain, (see Ertunc (2007) [13])has quite high anisotropy so the structure based modelfails to give satisfactory results. For this reason theresults from this model are not reported in the nextchapter.

1.16 Utility of barycentric metrics C1c, C2c

and C3c over II, III

Invariant representation of Reynolds stresses hasproven to be a useful tool in studying departuresfrom isotropy, axisymmetry and two-component tur-bulence. TheII and III invariants introduced byLumley given in (21) are written for rectangular,cylindrical and spherical coordinate system in a gen-eral way.

Rectangular

II = b2xx + b2yy + b2zz + 2(b2xy + b2yz + b2xz)

III = b3xx + b3yy + b3zz + 6bxybxzbyz

+ 3[b2xy(bxx + byy) + b2yz(byy + bzz) + b2xz(bxx + bzz)]

Cylindrical

II = b2rr + b2θθ + b2zz + 2(b2rθ + b2θz + b2rz)

III = b3rr + b3θθ + b3zz + 6brθbrzbθz

+ 3[b2rθ(brr + bθθ) + b2θz(bθθ + bzz) + b2rz(brr + bzz)]

Spherical

II = b2rr + b2θθ + b2φφ + 2(b2rθ + b2θφ + b2rφ)

III = b3rr + b3θθ + b3φφ + 6brθbrφbθφ

+ 3[b2rθ(brr + bθθ) + b2θφ(bθθ + bφφ) + b2rφ(brr + bφφ)]

The difficulty of the above expressions in terms of(II, III) is that they have to be adjusted dependingon the number of nonzero tensor components usingEinstein’s summation rules. In case of the barycen-tric metricsC1c, C2c andC3c the computation is muchmore simpler as one just has to adjust thebij matrixbefore diagonalization.

2 comp 1 comp

3 comp

aij

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

St

C1c

,C2c

,C3c

Ca1c

Ca2c

Ca3c

Figure 47: Scalar metricsC1c, C2c andC3c of bij fora swirling flow in a pipe plotted inside the barycentricmap(left) and asmetricsplot (right).

2 ConclusionFrom the above study, the following conclusions havebeen drawn

1. Thepermutation p, skewnesss or theangleφcan be used to check, in which barycentric mapthe flow trajectory lies and is a very useful indi-cator of the eigenvalue order in turbulent stresses.The analysis in the various coordinate systemgives an indication that visualization of turbulentstresses in terms of eigenvalues gives substantialadvantages in every coordinate system.

2. It has been found that the dimensionality tensordij and the anisotropy tensorbij closely evolveeach other, inaxi-symmetricand plane straincasesespecially in cases when rapid strain is ap-plied. In wakes and mixing layers the evolutionof dij andbij is also found quite similar.

3. The componentialityof turbulence equalsdi-mensionalityof turbulence at the two-component


ISSN: 1790-2769 56 ISBN: 978-960-474-034-5

limit of the barycentric map. In places awayfrom the two-component limit major deviationbetween componentiality and dimensionality ofturbulence was not found.

4. The metrics plot with scalar metricsC1c, C2c

and C3c of bij and dij on the y axis and y+

in the x axis provides better quantitative infor-mation about the flow development than the in-variant maps(III, II) and the barycentric mapCa

1c, Ca2c. The study of the trajectories of the di-

mensionality tensordij and the anisotropy ten-sor bij using DNS datasets of Lee & Reynolds(1985) [14] and Rogers and Moin (1987) [15] inthemetricsplot helps in observing how the com-ponentiality of turbulent stresses evolve.

5. Since the dimensionality tensor evolution in allthe test cases studied, is found to be monotone,the use of the dimensionality tensordij as an ex-tra parameter in the rapid pressure-strain modelis not considered in the model development in thenext chapter in the context of irrotational homo-geneous flows. The practice in traditional secondmoment closure modelling to postulate a closureexpression for theMijkl tensor based onbij forirrotational flows, taking an average value of thedimensionality tensor doesnot induce hugely er-roneous results.

Acknowledgements: The research was supported bythe University of Erlangen-Nuremberg and in the caseof the first author, it was also supported by the fellow-ship from M.I.T.

References:

[1] W. C. Reynolds & M. M. Rogers S. C. Kassinos.One-point turbulence structure tensors.Journalof Fluid Mechanics, 428:213–248, 2001.

[2] F. Durst & Ch. Zenger S. Banerjee, R. Krahl.Presentation of anisotropy properties of turbu-lence (invariants versus eigenvalue approaches).Journal of Turbulence, 2007.

[3] U. Schumann. Realizability of Reynolds stressturbulence models.Physics of Fluids, 20:721–725, 1977.

[4] J. L. Lumley and G. Newman. The return toisotropy of homogeneous turbulence.Journal ofFluid Mechanics, 82:161–178, 1977.

[5] Kwing-So Choi and John Lumley. The return toisotropy of homogenous turbulence.Journal ofFluid Mechanics, 436:59–84, 2001.

[6] J. L. Lumley. Computational modeling of tur-bulent flows. Advances in Applied Mechanics,18:123–176, 1978.

[7] J. Kim, P. Moin, and R. Moser. Turbulencestatistics of a fully developed channel flow at lowReynolds number.Journal of Fluid Mechanics,177:133–166, 1987.

[8] L.Terentiev. The Turbulence closure modelbased on Linear Anisotropy Invariant Analysis.PhD thesis, Technische Fakultat der UniversitatErlangen-Nurnberg, 2005.

[9] J. Kim R. A. Antonia and L. W. B. Browne.Some characteristics of small-scale turbulence ina turbulent duct flow.Journal of Fluid Mechan-ics, 233:369–388, 1991.

[10] J. Jovanovic, I. Otic, and P. Bradshaw. On theanisotropy of axisymmetric-strained turbulencein the dissipation range.Journal of Fluid Engi-neering, 125:401–413, May 2003.

[11] F. Durst S. Banerjee, O. Ertunc. Measurementand modeling of homogenous axisymmetric tur-bulence.To be submitted, 2007.

[12] J. Jovanovic and I. Otic. On the constitutive re-lation for the Reynolds stresses and the Prandtl–Kolmogorov hypothesis of effective viscosityin axisymmetric-strained turbulence.ASME,122:48–50, 2000.

[13] O. Ertunc.Experimental and Numerical Investi-gations of Axisymmetric Turbulence. PhD thesis,Friderich Alexander University, Germany, 2006.

[14] M. J. Lee and W. C. Reynolds. Numerical ex-periments on the structure of homogeneous tur-bulence. Technical Report Report TF-24, Ther-moscience Division, Stanford University, 1985.

[15] M. M. Rogers and P. Moin. The structure of thevorticity field in homogeneous turbulent flows.Journal of Fluid Mechanics, 176:33–66, 1987.

[16] Rogers M. M. & Ewing D. W. Moser, R. D. Self-similarity of time evolving plane wakes.Journalof Fluid Mechanics, 367:255–289, 1998.

[17] R. D Rogers, M. M. & Moser. Direct simulationof a self-similar turbulent mixing layer.Physicsof Fluids, 6:903–923, 1994.


ISSN: 1790-2769 57 ISBN: 978-960-474-034-5

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Anisotropy Properties of Turbulence - WSEAS · Anisotropy Properties of Turbulence SANKHA BANERJEE Massachusetts Institute of Technology Department of Mechanical Engineering U.S.A