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Measurement and Modelling of Homogeneous Axisymmetric Turbulence SANKHA BANERJEE Massachusetts Institute of Technology Department of Mechanical Engineering U.S.A OEZGUER ERTUNC & FRANZ DURST University of Erlangen-Nueremberg Institute of Chemical Engineering Germany Abstract: A new method for modelling the unknown correlations in Reynolds stress transport equations is de- veloped taking the kinematic relationship of turbulence tensors in homogeneous axisymmetric turbulence. Both the rapid-straining and the return-to-isotropy process of homogeneous axisymmetric turbulence are studied with the aim of improving Reynolds stress closures. The partition of the stress dissipation is also studied to assess the possible existence of local isotropy for turbulence at small scales. Homogeneous, axisymmetric turbulence is a simple flow, where the axes of anisotropy of the Reynolds stresses and dissipation tensor are found to be aligned. Using the theory of barycentric coordinates, the relationship between the Reynolds stress and dissipation tensors are derived, satisfying restrictions for the limiting states of turbulence and its assumed behavior for large Reynolds number and arbitrary anisotropy. The role of the anisotropy in constraining models for the turbulent dissipation rate and the pressure-strain correlations are discussed. Comparisons of the resulting closure based on barycentric coordinates with the experimental data for axisymmetric flow measurements of Ertunc (2007) [1] are good within the limitations of the data. Key–Words: Turbulence - transport model, Reynolds stress, realizability, second-order closure. 1 Introduction and Aim of work There has been significant development in the field of the second-order modelling of turbulence in the last three decades (Launder, Reece & Rodi (1975) [2]; Reynolds (1976) [3];Lumley (1978) [4]; Bradshaw, Cebeci & Whitelaw (1981) [5]; Speziale (1991) [6]; Reynolds & Kassinos (1995) [7]). In the second-order modelling, a system of statistically averaged equa- tions is closed at the second moment so that all the third-moment terms appearing in the transport equa- tions for second-moment terms, and one auxiliary equation which determines the length scale, must be modelled. Second-order models certainly have an advantage over lower-order models for flows where the transport of second-moment terms, such as the Reynolds stress and heat flux, play important roles. This is because the physics of the transport of second moment terms can be built into the second-order models. Since, in most turbulent flows the turbulent velocity fluctuations are anisotropic, and, differ in magnitude depending on their orientation. The stress transport models have an advantage over the two-equation transport models for accurate prediction of turbulence anisotropy. Axisymmetric turbulence is the simplest type of turbulence where redistribution of energy, related to pressure-strain and anisotropic dissipation, exists. The lowest level of turbulence models at which such effects enter explicitly is that in which transport equa- tions are formulated for the individual Reynolds stress components. To study these effects experimentally, highly anisotropic axisymmetric turbulence was gen- erated in an axi-symmetric duct using medium to high contraction (depending on inlet/outlet area ratio) in the wind-tunnel at LSTM Erlangen. This experimen- tal work is reported in detail in the dissertation of Er- tunc (2007) [1]. The theory of axisymmetric turbulence was first analysed by Batchelor [8] followed by Chandrasekhar (1950) [9] and recently by Lindborg (1995) [10]. Lindborg used a representation that makes use of the cylindrical” symmetry properties and considerably simplified the earlier theory which used a represen- tation more suited to the isotropic “spherical” case. The historical development of the theory of turbulence passing a contraction appears to have begun with the ideas of Prandtl (1930) [11] and was extended by Tay- lor (1935a) [12], who addressed already, at that time, well known effect that a contraction reduces the longi- tudinal fluctuations. Experimental studies of axisym- metric strained turbulence began with the measure- 1st WSEAS International Conference on MARITIME and NAVAL SCIENCE and ENGINEERING (MN'08) Malta, September 11-13, 2008 ISSN:1790-2769 135 ISBN: 978-960-474-004-8

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Page 1: Measurement and Modelling of Homogeneous Axisymmetric ...wseas.us/e-library/conferences/2008/malta/fb-mn/mn01.pdf · where aˆij is the anisotropic Reynolds stress tensor and ˆeij

Measurement and Modelling of Homogeneous AxisymmetricTurbulence

SANKHA BANERJEEMassachusetts Institute of TechnologyDepartment of Mechanical Engineering

U.S.A

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

Germany

Abstract: A new method for modelling the unknown correlations in Reynolds stress transport equations is de-veloped taking the kinematic relationship of turbulence tensors in homogeneous axisymmetric turbulence. Boththe rapid-straining and the return-to-isotropy process of homogeneous axisymmetric turbulence are studied withthe aim of improving Reynolds stress closures. The partition of the stress dissipation is also studied to assess thepossible existence of local isotropy for turbulence at small scales. Homogeneous, axisymmetric turbulence is asimple flow, where the axes of anisotropy of the Reynolds stresses and dissipation tensor are found to be aligned.Using the theory of barycentric coordinates, the relationship between the Reynolds stress and dissipation tensorsare derived, satisfying restrictions for the limiting states of turbulence and its assumed behavior for large Reynoldsnumber and arbitrary anisotropy. The role of the anisotropy in constraining models for the turbulent dissipationrate and the pressure-strain correlations are discussed. Comparisons of the resulting closure based on barycentriccoordinates with the experimental data for axisymmetric flow measurements of Ertunc (2007) [1] are good withinthe limitations of the data.

Key–Words:Turbulence - transport model, Reynolds stress, realizability, second-order closure.

1 Introduction and Aim of work

There has been significant development in the field ofthe second-order modelling of turbulence in the lastthree decades (Launder, Reece & Rodi (1975) [2];Reynolds (1976) [3];Lumley (1978) [4]; Bradshaw,Cebeci & Whitelaw (1981) [5]; Speziale (1991) [6];Reynolds & Kassinos (1995) [7]). In the second-ordermodelling, a system of statistically averaged equa-tions is closed at the second moment so that all thethird-moment terms appearing in the transport equa-tions for second-moment terms, and one auxiliaryequation which determines the length scale, must bemodelled.

Second-order models certainly have an advantageover lower-order models for flows where the transportof second-moment terms, such as the Reynolds stressand heat flux, play important roles. This is because thephysics of the transport of second moment terms canbe built into the second-order models. Since, in mostturbulent flows the turbulent velocity fluctuations areanisotropic, and, differ in magnitude depending ontheir orientation. The stress transport models have anadvantage over the two-equation transport models foraccurate prediction of turbulence anisotropy.

Axisymmetric turbulence is the simplest type

of turbulence where redistribution of energy, relatedto pressure-strain and anisotropic dissipation, exists.The lowest level of turbulence models at which sucheffects enter explicitly is that in which transport equa-tions are formulated for the individual Reynolds stresscomponents. To study these effects experimentally,highly anisotropic axisymmetric turbulence was gen-erated in an axi-symmetric duct using medium to highcontraction (depending on inlet/outlet area ratio) inthe wind-tunnel at LSTM Erlangen. This experimen-tal work is reported in detail in the dissertation of Er-tunc (2007) [1].

The theory of axisymmetric turbulence was firstanalysed by Batchelor [8] followed by Chandrasekhar(1950) [9] and recently by Lindborg (1995) [10].Lindborg used a representation that makes use of the“cylindrical” symmetry properties and considerablysimplified the earlier theory which used a represen-tation more suited to the isotropic “spherical” case.The historical development of the theory of turbulencepassing a contraction appears to have begun with theideas of Prandtl (1930) [11] and was extended by Tay-lor (1935a) [12], who addressed already, at that time,well known effect that a contraction reduces the longi-tudinal fluctuations. Experimental studies of axisym-metric strained turbulence began with the measure-

1st WSEAS International Conference on MARITIME and NAVAL SCIENCE and ENGINEERING (MN'08) Malta, September 11-13, 2008

ISSN:1790-2769 135 ISBN: 978-960-474-004-8

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ments of Uberoi (1956) [13], (1957) [14], who inves-tigated the influence of different axisymmetric con-traction shapes. Warhaft (1980) [15] was able to in-corporate thermal fluctuations in grid-generated tur-bulence in order to examine the effect of an axisym-metric strain on passive scalar fluctuations.

In the absence of any driving mechanism,anisotropic turbulent flows tend to return to anisotropic state the state of least order. This nonlin-ear process is often called return-to- isotropy. In 1951,Rotta [16] proposed a turbulence model for the return-to-isotropy term assuming a simple relationship be-tween the rate of return to isotropy and the degreeof anisotropy of turbulence. It is a linear model inthe sense that the rate of return to isotropy is lin-early proportional to the degree of anisotropy. Rotta’smodel has been used in most of the second-order tur-bulence models, and has proved to predict many shearflows quite well (Launder et al. (1975) [2]; Reynolds(1976) [3]; Lumley & Newman (1977) [17]; Lum-ley (1978) [4]). In fact, there have been claims fromexperimentalists that Rotta’s linear model does notaccount for the correct energy redistribution amongthe components in homogeneous shear flows (Cham-pagne, Harris & Corrsin 1970 [18]; Harris, Graham &Corrsin 1977 [19]).

There have also been disagreements on the pro-portionality constant in Rotta’s model, different val-ues were used by different models depending on theturbulence structure and the Reynolds number of aparticular flow to be modelled. Lumley & Newman(1977) [17] and Lumley (1978) [4] developed tur-bulence models for the return to isotropy, which arefunctions of both the anisotropy of turbulenceaij andthe turbulent Reynolds numberReλ.

aij =uiuj

q2−

1

3δij , q2 = uiui,

II = aijaji III = aijajkaki (1)

Most of the earlier studies were focused on the be-haviour of the measured single point Reynolds stresscomponents to analyse the ‘return-to-isotropy’ prob-lem. This approach was extended to include two-pointmeasurements by Johansson et. al (1998) [20]. Tobetter understand the energy redistribution process Jo-hansson et. al isolated the two parts of the “return-to-isotropy” mechanism representing the viscous andpressure related effects. These two parts are extremelydifficult to measure separately in a wind tunnel be-cause of the several different velocity derivative mo-ments contained in the total dissipation rate tensorand because of the difficulty of measuring the pres-sure fluctuations. This split can, however, easily beperformed on data from direct numerical simulations

(DNS) where one can solve the Poisson equation forthe fluctuating pressure, see e.g. Lee & Reynolds(1985) [21], Mansour, Kim & Moin (1988) [22]. Thetask is much more difficult in a physical experiment.

The main objectives of the experiments of Er-tunc [1] were:

• To provide reliable measurements of symmetri-cal distortions of isotropic and axisymmetric tur-bulence.

• To provide reliable measurements of decayingisotropic and anisotropic axisymmetric turbu-lence.

• To evaluate available models of axisymmetricturbulence.

The experimental results of Ertunc [1] were used todevelop a new turbulence model for the return-to-isotropy term. The effect of the turbulent Reynoldsnumber on the rate of return to isotropy was also in-vestigated and the results incorporated in the proposedmodel.

Using the theory of barycentric coordinates ofBanerjee et. al. (2007) [23] we shall provide a rationaldescription of the partition process, satisfying the lim-iting states of turbulence and the assumed behavior forlarge Reynolds number and small anisotropy. Fromthis we will derive a fully closed model for axisym-metric strained turbulence. The theory of barycen-tric coordinates used for the development of turbu-lence closures provides exact results only in ratherrare circumstances. In more general situations onlyapproximate results can be obtained. The errors re-sulting from simple analytic approximations of the un-known correlation functions needed in applications ofinvariant functionsCa

1c, Ca2c can be estimated from the

exact expression for the turbulent effective viscositydeduced by Banerjee et. al. [24], to show that theyare smaller than about30%. Comparison of the theo-retical results with direct numerical simulations DNSactually show considerably better agreement. In thederivations to follow the equality sign is used only forthe exact relations, and the symbols∼= and≈ (in thatorder) imply increasing uncertainty of the results.

1.1 Partition of the Stress Dissipation in Ax-isymmetric turbulence

Let the turbulence be statistically axisymmetric. Wemay thenwrite following the terminology, introducedby Banerjee et al. (2007) [23].

aij = Ca1ca1c +Ca

2ca2c (2)

eij = Ce1ca1c + Ce

2ca2c (3)

1st WSEAS International Conference on MARITIME and NAVAL SCIENCE and ENGINEERING (MN'08) Malta, September 11-13, 2008

ISSN:1790-2769 136 ISBN: 978-960-474-004-8

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whereaij is the anisotropic Reynolds stress tensor andeij is the anisotropic dissipation tensor.Ca

1c, Ca2c, C

e1c

andCe2c are scalar functions of the eigenvalues ofaij

and eij . a1c and a2c are unit vectors constructed insuch a way that (2) and (3) are invariant under rotationabout a particular axis.

For axisymmetric expansion Ca2c = 0,

Ce2c = 0

aij = Ca1ca1c (4)

eij = C1ce a1c (5)

Since the eigenvectors composing thea1c are unit vec-tors. Solving (4) and (5) the following relationshipexactly holds in axisymmetric expansion.

eij =Ce

1c

Ca1c

aij (6)

eij = Aaij (7)

For axisymmetric contraction Ca1c = 0, Ce

1c = 0

aij = Ca2ca2c (8)

eij = Ce2ca2c (9)

Since the eigenvectors composing thea2c are unit vec-tors. Solving (8) and (9) the following relationshipexactly holds in axisymmetric contraction.

eij =Ce

2c

Ca2c

a2c (10)

eij = Aaij (11)

As expected, there is very good agreement betweenthis analytical results (6) and (10) and DNS resultsof J. Kim, P. Moin and R. Moser (1999) [25] (seefigure 1). This linear alignment between the orderedanisotropy tensoraij and the ordered dissipation ten-sor eij .

1.2 Construction of the Invariant FunctionA

We now use the theory developed in Baner-jee(2007) [23] to determine the scalar functionA forthe limiting states of turbulence.

For axisymmetric contraction Ca1c = 0

Ca2c ∈ [0, 1] (12)

For axisymmetric expansion Ca2c = 0

Ca1c ∈ [0, 1] (13)

while in two-component turbulence,

Ca2c + Ca

1c = 1 (14)

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

eij

Figure 1: Cross-plot for the constitutive modelling re-lationship for the turbulent dissipation anisotropy ten-soreij using equations (6) and (10). Data is extractedfrom theexperiments of J. Kim, P. Moin and R. Moser(1999) [25] for the fully developed turbulent channelflow Reτ = 395.

Plotted in the Ca1c, C

a2c andCa

3c plane, the relations(12) and(13) yields left-running and right-runningbranches of the barycentric map. As shown in Baner-jee (2007) [23] the barycentric map contains all phys-ically realizable states of turbulence, (see figure 2)which shows the values ofCa

1c, Ca2c andCa

iso andAat the three vertices. The orign corresponds to threedimensional isotropic turbulence at whichCa

iso = 1.

2 comp 1 comp

3 comp

C2c

=1 C1c

=1

C3c

=1

C1c

=0

C2c

=0

C1c

+ C2c

=1

A ≃

1 Reλ → 00 Reλ → ∞

Figure 2: Barycentric map based on scalar metricswhich are functions of eigenvalues of the second-order stress tensor describing turbulence. At theisotropic point with metricC3c = 1 A has two val-ues based onReλ, at the two-component point hasC2c = 1 A = 1 and theone-component point hasC1c = 1, A = 1

For two-component isotropic turbulence

Ca2c = Ce

2c (15)

For onecomponent state

Ca1c = Ce

1c (16)

1st WSEAS International Conference on MARITIME and NAVAL SCIENCE and ENGINEERING (MN'08) Malta, September 11-13, 2008

ISSN:1790-2769 137 ISBN: 978-960-474-004-8

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and therefore for these casesA is given by

A2c = 1 (17)

A1c = 1 (18)

For very small Reynolds number, the dissipation andthe energy-containing ranges of the spectrum overlap,with little seperation between corresponding lengthscales, and therefore we may assume

A → 1, Ca3c → 1, Reλ → 0 (19)

and this relation (19) holds for arbitrary anisotropy.HereReλ is the turbulent Reynolds number:

Reλ =qλ

ν(20)

based on the Taylor microscaleλ. The microscale isstrictly defined in terms of the mean square derivativeof the streamwise velocity fluctuation but for practicalpurposes usually defined so that

ǫ = 5νq2

λ2(21)

Since the limits (17), (18) are fixed, the only logicalway to approach the hypothesis of locally isotropicturbulence in the sense suggested by Kolmogorov [26]is to assume

A → 0, Ca3c → 1, Reλ → ∞ (22)

For vanishing anisotropy of turbulence and largeReynolds number,ǫ may be related to the integrallength scaleLf of the energy-containing range (Kol-mogorov[26]) by

ǫ ∼= 0.192q3

LfRlambda >> 1 (23)

In this case of isotropic turbulence we see that the inte-gral length scale is directly proportional toq3/ǫ ≡ L.At very low Reynolds numbers, e.g, in the “final de-cay period” of classical grid generated turbulence, therelationship betweenλ andLf can be derived analyt-ically to yield (Hinze [27], pp 210)

Lf =1

2

π

2

1/2

λ Rlambda << 1 (24)

Using it in the expression forǫ (21) can be rewrittenas

ǫ = 1.959νq2

λ2Rlambda << 1 (25)

Following the suggestion of Rotta [16] we combine(23) and (25) to obtain an interpolation formula forǫvalid for low and large Reynolds numbers:

ǫ ∼= 1.959νq2

λ2ǫ ∼= 0.192

q3

Lf(26)

With (21) and (26), we are in a position to correlatethe length scale ratioλ/Lf in terms of the Reynoldsnumber

λ

Lf

∼= −0.049Reλ +1

2(0.009604Re2λ + 10.208)1/2

(27)which attains a maximum value of 1.597, say 1.6,whenReλ → 0 and vanishes forReλ → ∞. It issuitable to normalize the above relation and introducethe function

W =1

1.597

λ

Lf(28)

which can be used to match the limiting properties ofthe turbulence for very low and very large Reynoldsnumbers. We can use the above relationships in con-junction with the weighting technique based on (28)to interpolate between the two quasi-asymptotic lim-its (19) and (22) suggested forA:

A → (1−W)(A)Reλ→∞+W(A)Reλ→0, Ca3c → 1

(29)This simple proposal appears to be sufficient to en-sure (19) and (22) and is consistent with the conceptof asymptotic invariance (Tennekes and Lumley [28],p. 6) which broadly states that the overall behavior ofturbulent flows, outside the viscous wall region or sub-layer is almost independent of the fluid viscosity. Theconcept of asymptotic invariance is well supported byexperimental observations Uberoi [14], Comte-Bellotand Corrsin [29], which show that the direct effects ofviscosity, away from viscous wall regions, decreaseas the Reynolds number increases. More general andtherefore more complicated formulations than (29) arepossible but would require additional constraints orinput data. It is not clear if something better than (19)and (22) can be formulated rationally. Using it, wefind

A → W Ca3c → 1 (30)

The two exact limiting values ofA, (17) and (18),can be matched to (30) utilizing properties of the two-dimensionality parameterJ introduced in the work ofBanerjee et al. (2007) [23]:

J = 1 − (Ca1c + Ca

2c) (31)

J = 0 indicates two-component turbulence andJ =1 corresponds to isotropic turbulence. Thus we may

1st WSEAS International Conference on MARITIME and NAVAL SCIENCE and ENGINEERING (MN'08) Malta, September 11-13, 2008

ISSN:1790-2769 138 ISBN: 978-960-474-004-8

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write an approximate expression forA in thecase ofinitially isotropic turbulence submitted to axisymmet-ric expansion as

A ∼= (1 − J )(A)1c + J (A)iso Ca1c ∈ [0, 1] (32)

and also for turbulence developing in axisymmetriccontraction:

A ∼= (1 − J )(A)2c + J (A)iso Ca2c ∈ [0, 1] (33)

Taking axisymmetry into (12), (13), (30), (32), and(33)

A ≃

1 + 1 − Ca

1c (W − 1) , Ca1c ∈ [0, 1]

1 + 1 − Ca2c (W − 1) , Ca

2c ∈ [0, 1](34)

The forms deduced forA imply that the anisotropyof eij will persist at all Reynolds numbers. Thisconclusion, reached only from kinematical consider-ations. Using Rogallo data [30], distributions ofAwere calculated for different Reynolds numbersRetfor turbulence developing in axisymmetric contrac-tion (C2c

a ∈ [0, 1]) and in axisymmetric expansion(C1c

a ∈ [0, 1]). It can be seen thatA rises fasterwith increasing anisotropy in axisymmetric contrac-tion than in axisymmetric expansion (see figure(3)).

Figure 3: Distribution of the invariant functionA inaxisymmetric strained turbulence. Symbols are theDNS results of Rogallo [30] and Lee [21]. The linescorrespond to the suggested forms (34) forA; [−−] ,C2c = 1; [−] , C1c = 1.

1.3 The Dissipation Rate Equation

We shall now demonstrate how the influence ofanisotropy of turbulence penetrates into the dynam-ics of ǫ. For axisymmetric homogeneous turbulence

the dissipation rate equation reads

∂ǫ

∂t= −2ν

∂ui

∂xl

∂ui

∂xl

∂U i

∂xk︸ ︷︷ ︸P 1

epsilon

−2ν∂us

∂xi

∂us

∂xk

∂U i

∂xk︸ ︷︷ ︸P 2

epsilon

−2ν∂ui

∂xl

∂ui

∂xl

∂ui

∂xk︸ ︷︷ ︸P 4

epsilon

−2ν2 ∂2ui

∂xl∂xn

∂2ui

∂xl∂xn︸ ︷︷ ︸γ

(35)

The timederivative on the left-hand side of (35) is theusual substantial derivative following the mean mo-tion of the fluid. The first two terms on the right-handside,P 1

epsilon andP 2epsilon, represent the production of

ǫ by the mean velocity gradients. These terms involvethe following tensors:

Pik = −2ν∂ui

∂xl

∂ui

∂xl(36)

Rik = −2ν∂us

∂xi

∂us

∂xk(37)

which have the same trace:

Pss = Rss = 2ǫ (38)

Given the axisymmetry of the whole flow, (36) and(37) must both be axisymmetric. Since the tensors(36), (37) are positive semi-definite and can be writ-ten following the same analytical path (2 and 3) asPik andRik, they can be represented following themethodology of Banerjee et al. (2007) [23] as

Pik = Cep1cD1c + Cep

2cD2c + Cep3cD3c (39)

Rik = Cer1cD1c + Cer

2cD2c + Cer3cD3c (40)

where

D1c =

1 0 0

0 0 00 0 0

(41)

D2c =

1 0 0

0 1 00 0 0

(42)

D3c =

1 0 0

0 1 00 0 1

. (43)

For axisymmetric contractionCep1c = 0 andCer

1c = 0

Pik = Cep2cD2c + Cep

3cD3c (44)

Rik = Cer2cD2c + Cer

3cD3c (45)

1st WSEAS International Conference on MARITIME and NAVAL SCIENCE and ENGINEERING (MN'08) Malta, September 11-13, 2008

ISSN:1790-2769 139 ISBN: 978-960-474-004-8

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Since the eigenvectors forming basis inD2c are thesame forPik and Rik, substituting the value ofD2c

from (45) in (44) the following relationship is ob-tained

Pik = (Cep3c −

Cep2cC

er3c

Cer2c

)D3c +Cep

2c

Cer2c

Rik (46)

Contracting the above equation, and taking into ac-count (46), we get

2ǫ = 3(Cep3c −

Cep2cC

er3c

Cer2c

) +Cep

2c

Cer2c

2ǫ (47)

By comparing left-hand and right-hand sides of (47 ),we find

Cep3c =

Cep2cC

er3c

Cer2c

Cep2c = Cer

2c (48)

For axisymmetric expansionCep2c = 0 andCer

2c = 0

Pik = Cep1cD1c + Cep

3cD3c (49)

Rik = Cer1cD1c + Cer

3cD3c (50)

Since the eigenvectors forming basis inD1c are thesame forPik and Rik, substituting the value ofD1c

from (50) in (49) the following relationship is ob-tained

Pik = (Cep3c −

Cep1cC

er3c

Cer1c

)D3c +Cep

1c

Cer1c

Rik (51)

Contracting the above equation, and taking into ac-count (51), we get

2ǫ = 3(Cep3c −

Cep1cC

er3c

Cer1c

) +Cep

1c

Cer1c

2ǫ (52)

By comparing left-hand and right-hand sides of (52),we find

Cep3c =

Cep1cC

er3c

Cer1c

Cep1c = Cer

1c (53)

and thereforePik = Rik (54)

This result implies that in axisymmetric turbulence,the two production termsP 1

ǫ andP 2ǫ of the dissipation

rate equation are identical:

P 1ǫ = P 2

ǫ (55)

The cross-plot of P 1ǫ versusP 2

ǫ shown in Fig. 3confirms that the derived relationship (55) and thatit holds in axisymmetric turbulence. However, the

Figure 4: Cross-plot ofP 2ǫ as a function of P 1

ǫ inaxisymmetric strained turbulence from the databaseof Rogallo [30]. The slope of the straight line fittedthrough the numerical data is 1.12. The dashed linecorresponds to the exact analytical solution given by(55).

slope of the straight line betweenP 1ǫ versusP 2

ǫ fit-ted through the data of Rogallo [30] was 1.12 (seefigure(4)), and we must conclude that the accuracy ofthese early DNS results is not as high as we wouldwish.

Using the defination ofǫij andPik we may writeP 1

ǫ as

P 1ǫ = −ǫik

∂U i

∂xk(56)

Using (11)

P 1ǫ = −Aǫuiuk

k∂U i

∂xk−

2

3ǫ(1 −A)

∂U i

∂xkδik︸ ︷︷ ︸

=0

for incompressible flow(57)

Thus, for incompressible flow, the source term of thedissipation rate equation is

P 1ǫ + P 2

ǫ = −2Aǫuiuk

k

∂U i

∂xk(58)

To formulate a turbulence closure for the differencebetween the turbulent production of dissipation ratedue to vortex stretching,P 4

ǫ , and the viscous destruc-tion,γ, wecan use a result that holds in grid-generatedturbulence to get

P 4ǫ − γ ∼= −ψ

ǫ2

k(59)

Using Loitsianski invariant Hinze [27] , p. 260; to es-timateψ in the limitReλ → 0, and Saffman invariantin thelimit Reλ → ∞, we obtain Jovanovic et al. 34)

ψ ≃

1.4 Reλ → 01.8 Reλ → ∞

Ca3c → 1. (60)

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Application of the weighting technique based on thelengthscale ratio used in the derivation of (28) yields

ψ → 1.8 − 0.4W, Ca3c → 1 (61)

To complete the closure of theǫ equation, we mustestimate the influence of anisotropy onψ. This canbe done by interpolation if values ofψ are availableat the left and right upper vertices of the barycen-tric map fig. (2), which correspond, respectively, totwo-component isotropic turbulence and to the one-component state. Let us denote the values ofψ atthese two points by

ψ2c = a (62)

ψ2c = b (63)

Following the methodology introduced above inBanerjee et al. (2007) [23] forCa

3c → 1.

ψ ≃

[9bCa

1c] + 1 −Ca1c (1.8 − 0.4W)Ca

1c ∈ [0, 1][9aCa

2c] + 1 − Ca2c (1.8 − 0.4W)Ca

2c ∈ [0, 1](64)

To determineψ in two-component isotropic turbu-lence, we used the results fork, ǫ, uiuj and turbu-lent microscales obtained by Rogallo for axisymmet-ric contraction. From the numerical data of Rogallo∂ǫ∂t , P

1ǫ andP 2

ǫ were computed, and the decay termP 4

ǫ − γ was deduced from the balance of (35) by dif-ference. Values ofψ computed from eight differentfields, of Rogallo [30]. These results are somewhatscattered owing to numerical difficulties in evaluatingthe time derivative ofǫ. The results indicate that theψincreases continuously with increasing anisotropy andattains a maximum value of about

a = 2.5 (65)

near the isotropic two-component limit.Lumley and Newman [17] provided information

about the form ofψ in unstrained one-component tur-bulence. They argued that nonlinear terms in the equa-tion for the two-point correlation vanish in this flow sothatψ can be calculated exactly, to yield

b = 1.4 (66)

the same result as in the final stage of decay ofisotropic turbulence. Lumley and Newman [17]equatedcomponentalityand dimensionalityof one-component turbulence, implying that it is also one-dimensional turbulence. This assumption is logicaland has been proved to exist all along the two com-ponent limit of the barycentric map, but it was arguedby Jovanovic et al. [31], that it is not permitted by thedynamic equations for the instantaneous fluctuations

for any finite values of the latter. This fact, is a mat-ter of controversy in the turbulence literature. There-fore, if true, the assumption implies that turbulencemust vanish completely in the one-component, one-dimensional state, giving:

b = 0 (67)

DNS results for wall-bounded and free-shear flowssupport for Lumley and Newmans [17] assumptionand the theoritical claim of Banerjee [24] of the equal-ity of componentality and dimensionality near the 1-Climit.

1.4 Closure for rapid pressure-strain term

Most of the popularly used models have utilized thesuggestions of Rotta [16] in developing models forthis correlation. Rotta [16] showed that the fourthrank tensorMijkl has the following properties whichare exact.

Mijkl = Mjikl, (68)

Miikl = 0, (69)

Mijjl = 2uiul (70)

1. The first condition (68) implements the symme-try in i andj;

2. The second condition (69) preserves the trace ofthe anisotropy tensor is equal to zero;

3. The third condition (70) normalizesMijkl.

These equations are very helpful for constructingmodels for theMijkl tensor. They have been used inall conventional turbulence models and it is thoughtnormally that a model which doesnot satisfy the aboveconditions has little hope of success in general appli-cations. In the current work, the models which havebeen cited most frequently in turbulence literature forthe rapid and slow terms of the pressure-strain corre-lations are used to gain theoretical knowledge aboutthe background of the work.

Using kinematic constraints alone, the fourth or-der tensorMijkl for the fast part ofΠr

ij can be evalu-ated exactly for initially isotropic turbulence exposedto rapid distortion due to Crow (1968) [32] to yield

M0ijkl = Aδijδkl +Bδikδjl + Cδilδjk (71)

Using (68) gives B=C, the second condition (69) gives

Aδiiδkl +B(δikδil + δilδik) = 0 (72)

Sinceδij is an identity matrix andδii = 3 andδikδil =δkl. Thusthe above equation (72) reduces to(3A +

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2B)δkl = 0, or 3A = −2B. From the normalizationcondition (70), it can be written as

Mijjl = 2k(bil +2

3δil) (73)

Πrij = 2prsij = 2

∂Uk

∂xl(Miljk +Mjlik) (74)

Expression (71) is a first term expansion inbil, thusonly theδil term of equation (73) contributes at thisorder, using the (72) and putting it in (74) one canderive the Crow constraint (75).

Πrij =

2

5q2Sij (75)

Daij

Dt= P a

ij +1

K(Πr

ij + Πsij) −

ǫ

K(eij − aij) +Da

ij

(76)From theequation for the anisotropy tensoraij in ho-mogeneous turbulence (76). Near the wall, where thetwo-component limit is approached, the proper kine-matic condition can be stated aseij = aij. This islooked as the strong form of realizability as the sourceterm in the anisotropy equation (76) arising from dis-sipation− ǫ

K (eij −aij) will then vanish everywhere atthe two-component limit. The asymptotic behaviourof Πij as ∂aij

∂t → 0 for the case of initially isotropicturbulence subjected to axisymmetric deformation:

(Πij)2c → aijPss +1

3Pssδij − Pij C2c = 1 (77)

(Πij)1c → aijPss +1

3Pssδij − Pij C1c = 1 (78)

This behavior suggests thatΠij vanishes in two-component isotropic turbulence and also in the one-component state:

(Πij)2c = 0 (79)

(Πij)1c = 0 (80)

We can exactly define the asymptotic behavior ofΠij

near all three vertices of the barycentric map. UsingRapid Distortion limit the fast pressure-strain can beformulated as:

Πrij →

3

5(1

3Pssδij − Pij) C3c = 1 (81)

The above result (81) suggest an approximation forthe fast pressure-strain termΠr

ij as:

Πrij ≈ aijPss + F

(1

3Pssδij − Pij

)(82)

whereF = 1 C3c = 0F = 3

5C3c = 1

(83)

1.5 Closure for slow pressure-strain term

Closure for the slow part ofΠij can be formulated us-ing thetransport equation foraij in decaying homoge-neous axisymmetric turbulence without mean velocitygradients such that the equation (76) becomes:

2k∂aij

∂t= Πs

ij + 2ǫaij − 2Aǫaij (84)

ForC3c → 1, i.e around the isotropic limit

Πsij = 2(A)isoǫaij −Dǫaij (85)

where D is the new scalar function. In the final pe-riod of decay of grid turbulence, which corresponds toReλ and negligible nonlinear terms, intercomponentenergy transfer should be negligibleas is confirmed bythe experiments Batchelor [33] and we may write

D → 2, C3c → 1 Reλ → 0 (86)

Datafor the decay of slightly anisotropic grid turbu-lence of Comte-Bellot and Corrsin [29] yield the be-havior ofD at moderate and high Reynolds numbers:

D → 2.5, C3c → 1 Reλ → ∞ (87)

Matching (86) and (87)

D → 2.5 − 0.5W, C3c → 1 (88)

yields

Πsij = (2.5W − 2.5)ǫaij C3c → 1 (89)

Thus, the form suggested forΠsij is obtained by in-

terpolating between the data represented by (89) andthe exact results (79) and (80) for the limiting statesof turbulence.

Πsij = Caij C3c → 1 (90)

where

C = (1 − C1c)(2.5W − 2.5)ǫ C1c ∈ [0, 1]C = (1 − C2c)(2.5W − 2.5)ǫ C2c ∈ [0, 1]

(91)

1.6 Comparison of Predictions With the Ex-perimental Data

We have calculated the axisymmetric flows measuredby Ertunc (2007) [1], using his specified initial val-ues of the Reynolds stress componentsuiuj and ofthe dissipation rateǫ0.

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1.6.1 Reliability of the Experimental Data

Typical turbulence levels measured in grid-generatedturbulence are of the order of 0.32%, so the back-ground turbulence level in the test rig must be lessthan, say, 0.05% so that it can be reliably subtracted.In addition to this, the height of the test sectionDmust be much larger than the length scale of the en-ergy containing eddies,l say, which is of the sameorder as the mesh size of the gridM

D

l≈D

M>> 1 (92)

Measurements of grid-generated turbulence made byComte-Bellot and Corrsin [29] are the only onesreported in the literature which satisfy these re-quirements (92). Another useful class of flows fortesting turbulence closures is the passage of grid-generated turbulence through expanding or contract-ing and preferably axisymmetric ducts. All experi-ments on these flows Uberoi [13] and Tan-atichat [34]were carried out in small medium-quality test rigswith background turbulence levels of0.2% or more,and with D/l too small to ensure full homogene-ity. The major purpose of these experiments was todemonstrate the ability of the rapid distortion theory,developed by Batchelor and Proudman [35] and Rib-ner and Tucker [36], to predict the effect of contrac-tion on turbulence quantities. Even for application tothe larger energy-containing eddies alone, this theory,which neglects the viscosity and the nonlinear termsin the equations, requires (Batchelor [8])

l

L≈

l

D>>

q

U(93)

whereL is the duct length, which is usually not muchlarger thanD. In view of (92), this inequality im-plies that it is not possible to reach the rapid distortionlimit and satisfy constraints for good flow homogene-ity. Therefore, the observed deviations of the mea-surements from rapid distortion theory reported in theliterature are not surprising.

A serious problem, arising from inadequate flowquality, occurs in the interpretation of measurementsin axisymmetric contractions. When nearly isotropicgrid turbulence is distorted by contraction, the two lat-eral components of turbulent intensity increase whilethe streamwise component decreases, and the grid-induced contribution to the streamwise componentmay fall below the background turbulence level. Thebackground disturbances originate in part from pres-sure fluctuations generated by the fan blades and bythe turbulent boundary layers on the test section walls.Both of these influences mainly contribute to the lon-gitudinal intensity component of the free stream, and

the effect of a contraction is unlikely to be the sameas the effect on true turbulence. Thus, the existingexperimental data for the streamwise intensity com-ponent in contracting ducts are uncertain and shouldnot be used to validate predictions of turbulence mod-els. The normal component data should be largely freefrom these objections.

1.6.2 Decay of Homogeneous Isotropic Turbu-lence

The decay of homogeneous isotropic turbulence is thesimplest problem for numerical prediction. The basicexperiment is due to Comte-Bellot and Corrsin [29]reffered as (0371), who carried out the most completeexperimental investigations of grid turbulence at mod-erate and high Reynolds numbers. Figure (5, (right))shows that the numerical prediction of kinetic energyq2 match with the experiments closely.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

k

Figure 5: Sketch of the experimental setup (0371)(left), Experimental data [] and model predictions[−] for the turbulent kinetic energyk (right)

∂uiuj

∂t= Pij + aijPss + F

(1

3Pssδij − Pij

)

+ (C − 2Aǫ)aij −2

3ǫδij (94)

The dissipation equation can be written as

∂ǫ

∂t= −2A

ǫuiuk

k

∂U i

∂xk− ψ

ǫ2

k(95)

Using (94) and (95), the histories of the energy com-ponents measured downstream of an axisymmetriccontraction by Ertunc [1] were calculated.

1.6.3 Solution Procedure

According to the property of homogenity all spatialgradients∂(·)/∂xi represented by the diffusion termsDij in the transport equations for the Reynolds stresstensor (see equation (96))and the termsDǫ in thetransport equation for the dissipation rate are equalto zero. In order to start the computational algorithm

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one has to provide the following necessary informa-tion: the fluid and flow properties – the kinematicviscosity ν and the mean velocity gradient structure∂Ui(t)/∂xj in terms of the strainSij and the rotationΩij tensors. Since the experimental data of homoge-neous axisymmetric turbulence is subjected to irrota-tional strain, (see Ertunc (2007) [1]) so the rotationtermΩij is not considered in the following work. Thenumerical simulation is divided into three parts:

1. Firstly the set of mathematical equations are de-rived from the standard RST equations (96), bysubstituting the appropriate closure models.

2. These equations are written with the aid of carte-sian tensor notation, which uses Einstein summa-tion convention.

3. The problem is then solved by using software al-gebra program MAPLE, to generate appropriatematrices to be implemented in Fotran 90 programwritten by the author.

In addition to this, the corresponding initial con-ditions should be specified – values for the Reynoldsstress tensoruiuj(t0) and the turbulent dissipationrate ǫ(t0) in the initial time statet0. Generally, theinitial values of the Reynolds stress tensor can alwaysbe taken directly from the experimental database. Theinitial turbulent dissipation rateǫ(t0) (in the case ifit is not measured and reported) can be evaluated us-ing the following balance equation for the turbulentkinetic energyk(t0) at the timet0:

∂k

∂t

∣∣∣∣t0

≃ P (t0) − ǫ(t0). (96)

The turbulent kinetic energy gradient can be ap-proximated by the second-order accuracy scheme:∂k/∂t|t0 ≃ (k(t1) − k(t−1))/(t1 − t−1). Assumingthe upstream steadiness of the flowk(t−1) ≃ k(t0)and uniformity of the time step sizest−1−t0 = t0−t1one has∂k/∂t|t0 ≃ 0.5(k(t1) − k(t0))/(t1 − t0).Therefore, the computational expression for the ini-tial value of the turbulent dissipation rateǫ(t0) can bederived directly from equation (97):

ǫ(t0) ≃ P (t0) −k(t1) − k(t0)

2(t1 − t0). (97)

1.6.4 Decay of Anisotropic Axisymmetric Turbu-lence

This turbulent flow case was generated by the strain-ing of grid-generated turbulence through a3.69 : 1contraction and relaxing it downstream of the contrac-tion in a constant diameter duct. Measurements for

this turbulent case are comprehensively analyzed inthe Section5.4 of the dissertation of Ertunc [1]. Theq2 prediction of the banerjee model based on barycen-tric coordinates is shown in fig. (6), (left) and thea11 prediction in fig. (6), (right). The comparisonof prediction of Reynolds stressesuu shown in fig.(7), (left) andvv in fig. (7), (right). It is found thatthe quasi-linear models of Banerjee (2007) [37]) per-forms quite satisfactory in the prediction of Reynoldsstresses. The sudden increase inuu in the experimen-tal data set might be attributed to the end effect, whichis not well reproduced by the current model.

0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.80.0165

0.017

0.0175

0.018

0.0185

0.019

0.0195

0.02

0.0205

x+

q2

banerjeeexp

0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8−0.224

−0.222

−0.22

−0.218

−0.216

−0.214

−0.212

x+

b 11

banerjeeexp

Figure 6: Prediction of the kinetic energyq2 and thestreamwise anisotropya11 for anisotropic grid decay(case AXC-AGD).

0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.81.85

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25x 10

−3

x+

uu

banerjeeexp

0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.87.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9

9.2x 10

−3

x+

vv

banerjeeexp

Figure 7: Prediction of the streamwiseuu and nor-mal stressesvv for anisotropic grid decay (case AXC-AGD).

1.6.5 Axisymmetric Contraction

The three selected data sets with 1.27:1 (case AXC-c1.27), 3.69:1 (case AXC-c3.69) and 14.75:1 (caseAXC-c14.75) contractions correspond to slow, mod-erately rapid and rapid axisymmetric contraction(strain) cases respectively. This selection is expectedto show the reaction of different models to differentlevels of mean velocity gradients. For the slow ax-isymmetric contraction (case AXC-c1.27), the bestpredictions ofq2 (see, (fig. 8), (left)) anda11 (see,(fig. 8), (right)) is obtained by the current model. Thepredictions of the Reynolds stressesuu (see, (fig. 9),(left)) andvv (see, (fig. 9), (right)) is very good forthe current model. For the axisymmetric contrac-tion (case AXC-c3.69), satisfactory prediction ofq2

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0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

x+

q2

banerjeeexp

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

x+

b 11

banerjeeexp

Figure 8: Prediction of the kinetic energyq2 and thestreamwise anisotropya11 for axisymmetric contrac-tion (case AXC-c1.27).

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.752

3

4

5

6

7

8

9

10

11x 10

−3

x+

uu

banerjeeexp

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.752.5

3

3.5

4

4.5

5

5.5

6

6.5

7x 10

−3

x+

vv

banerjeeexp

Figure 9: Prediction of the streamwiseuu and normalstressesvv for axisymmetric contraction (case AXC-c1.27).

(see, (fig. 10), (left)) anda11 (see, (fig. 10), (right))is obtained by the current model. The predictions ofthe Reynolds stressesuu (see, (fig. 11), (left)) andvv (see, (fig. 11), (right)) is also good for the currentmodel. For the axisymmetric contraction (case AXC-

0.45 0.5 0.55 0.6 0.65 0.7 0.750.017

0.018

0.019

0.02

0.021

0.022

0.023

0.024

0.025

0.026

x+

q2

banerjeeexp

0.45 0.5 0.55 0.6 0.65 0.7 0.75−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

x+

b 11

banerjeeexp

Figure 10: Prediction of the kinetic energyq2 and thestreamwise anisotropya11 for axisymmetric contrac-tion (case AXC-c3.69).

c14.75), the predictions ofq2 (see, (fig. 12), (left))anda11 (see, (fig. 12), (right)) is under-predicted bythe current model. The predictions of the Reynoldsstressesuu (see, (fig. 13), (left)) andvv (see, (fig.13), (right)) is under-predicted by the current model.

1.6.6 Axisymmetric Expansion

All the axisymmetric expansion cases of Ertunc(2007) [1] are examples of slow straining. For the cur-rent computation the case (case AXE-c0.72) is used.Figure (14), (left), shows the kinetic energy predic-

0.45 0.5 0.55 0.6 0.65 0.7 0.751

2

3

4

5

6

7

8

9

10x 10

−3

x+

uu

banerjeeexp

0.45 0.5 0.55 0.6 0.65 0.7 0.756

7

8

9

10

11

12x 10

−3

x+

vv

banerjeeexp

Figure 11: Prediction of the streamwiseuu and nor-mal stressesvv for axisymmetric contraction (caseAXC-c3.69).

0.5 0.55 0.6 0.65 0.7 0.750.005

0.01

0.015

0.02

0.025

0.03

x+

q2

banerjeeexp

0.5 0.55 0.6 0.65 0.7 0.75−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

x+

b 11

banerjeeexp

Figure 12: Prediction of the kinetic energyq2 and thestreamwise anisotropya11 for axisymmetric contrac-tion (case AXC-c14.75).

0.5 0.55 0.6 0.65 0.7 0.750

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

−3

x+

uu

banerjeeexp

0.5 0.55 0.6 0.65 0.7 0.750

0.005

0.01

0.015

x+

vv

banerjeeexp

Figure 13: Prediction of the streamwiseuu and nor-mal stressesvv for axisymmetric contraction (caseAXC-c14.75)

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tions of the current model, which matches with theexperimental data very well. Since the kinetic energyequation is insenstive to the pressure-strain closure, sothe Reynolds stress closures perform very well. Theanisotropy prediction (see fig. (14), (right)) of thequasi-linear models of Banerjee (2007) [37] is alsosatisfactory. The predictions of the Reynolds stressesuu (see, (fig. 15), (left)) andvv (see, (fig. 15), (right))by thecurrent model is very good.

0.4 0.5 0.6 0.7 0.8 0.9 10.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

x+

q2

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

x+

b 11

banerjeeexp

Figure 14: Prediction of the kinetic energyq2 andstreamwise anisotropya11 for axisymmetric expan-sion (case AXE-c0.72).

0.4 0.5 0.6 0.7 0.8 0.9 10.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

x+

uu

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 12

4

6

8

10

12

14x 10

−3

x+

vv

banerjeeexp

Figure 15: Prediction of the streamwiseuu and nor-mal stressesvv for axisymmetric expansion (caseAXE-c0.72).

1.6.7 Decay of Grid-Generated Turbulence

The measurements for these experiments were per-formed in the wind tunnel, discussed in detail in thedissertation of Ertunc (2007) [1] (see Chapt. 4 and5). All the turbulence models predicts the kinetic en-ergyq2 very well (see fig. (16), (18), (20), (22), (24),(26)), (left)). A detailed computation is made forUgrid = 3, 5, 6, 8, 10, 12m/s. The prediction of thestreamwise anisotropya11 (see fig. (16), (18), (20),(22), (24), (26)), (right)) is not satisfactory, since theanisotropya11 is calculated by a local algebraic rela-tion which cannot express the memory effect. The de-velopment of the streamwise stressuu (see fig. (17),(19), (21), (23), (25), (27)), (left)) and normal stressvv (see fig. (17), (19), (21), (23), (25), (27)), (right))is captured very well by the current model.

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

x+

q2

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.065

0.07

0.075

0.08

0.085

0.09

0.095

x+

b 11

banerjeeexp

Figure 16: Prediction of the kinetic energyq2 andstreamwise anisotropya11 for grid decay (case ofAXC-GDu3).

0.4 0.5 0.6 0.7 0.8 0.9 1 1.11

2

3

4

5

6

7

8x 10

−3

x+

uu

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3

x+

vv

banerjeeexp

Figure 17: Prediction of the streamwiseuu and nor-mal stressesvv for grid decay (case of AXC-GDu3).

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

x+

q2

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.06

0.065

0.07

0.075

0.08

0.085

x+

b 11

banerjeeexp

Figure 18: Prediction of the kinetic energyq2 andstreamwise anisotropya11 for grid decay (case ofAXC-GDu5).

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

x+

uu

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.12

4

6

8

10

12

14x 10

−3

x+

vv

banerjeeexp

Figure 19: Prediction of the streamwiseuu and nor-mal stressesvv for grid decay (case of AXC-GDu5).

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0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.01

0.02

0.03

0.04

0.05

0.06

0.07

x+

q2

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

x+b 11

banerjeeexp

Figure 20: Prediction of the kinetic energyq2 andstreamwise anisotropya11 for grid decay (case ofAXC-GDu6).

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.005

0.01

0.015

0.02

0.025

0.03

x+

uu

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

x+

vv

banerjeeexp

Figure 21: Prediction of the streamwiseuu and nor-mal stressesvv for grid decay (case of AXC-GDu6).

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x+

q2

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

x+

b 11

banerjeeexp

Figure 22: Prediction of the kinetic energyq2 andstreamwise anisotropya11 for grid decay (case ofAXC-GDu8).

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

x+

uu

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

x+

vv

banerjeeexp

Figure 23: Prediction of the streamwiseuu and nor-mal stressesvv for grid decay (case of AXC-GDu8).

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

x+

q2

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.055

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

x+

b 11

banerjeeexp

Figure 24: Prediction of the kinetic energyq2 andstreamwise anisotropya11 for grid decay (case ofAXC-GDu10).

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

x+

uu

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.01

0.02

0.03

0.04

0.05

0.06

0.07

x+

vv

banerjeeexp

Figure 25: Prediction of the streamwiseuu and nor-mal stressesvv for grid decay (case of AXC-GDu10).

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x+

q2

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

0.105

x+

b 11

banerjeeexp

Figure 26: Prediction of the kinetic energyq2 andstreamwise anisotropya11 for grid decay (case ofAXC-GDu12)

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.02

0.04

0.06

0.08

0.1

0.12

0.14

x+

uu

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

x+

vv

banerjeeexp

Figure 27: Prediction of the streamwiseuu and nor-mal stressesvv for grid decay (case of AXC-GDu12)

1st WSEAS International Conference on MARITIME and NAVAL SCIENCE and ENGINEERING (MN'08) Malta, September 11-13, 2008

ISSN:1790-2769 147 ISBN: 978-960-474-004-8

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1.6.8 Succesive Axisymmetric Strain

The measurements for these experiments were per-formed in the wind tunnel, discussed in the disserta-tion of Ertunc (2007) [1]. The prediction ofq2 (see,(fig. 28), (left)) anda11 (see, (fig. 28), (right)) is ob-tained quite good. The predictions of the Reynoldsstressesuu (see, (fig. 29), (left)) andvv (see, (fig. 29),(right)) is quite good, but the model, overpredicts theuu andvv development near the downstream regionx+ = 0.6.

0.4 0.5 0.6 0.7 0.8 0.9 10.015

0.02

0.025

0.03

0.035

0.04

0.045

x+

q2

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 1−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

x+

b 11

banerjeeexp

Figure 28: Prediction of the kinetic energyq2 and thestreamwise anisotropya11 for succesive strain (caseof ASS).

0.4 0.5 0.6 0.7 0.8 0.9 10.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

x+

uu

banerjeeexp

0.4 0.5 0.6 0.7 0.8 0.9 10.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

0.016

x+

vv

banerjeeexp

Figure 29: Prediction of the streamwiseuu and nor-mal stressesvv for succesive strain (case of ASS).

2 Conclusions

1. Analysis of axisymmetric strained turbulencedata of Ertunc (2007) [1] using the theory ofbarycentric coordinates shows that in such tur-bulence all second-order correlations are alignedwith each other. Kinematic metricsC1c, C2c canbe used to asses the flow axi-symmetry and ob-tain analytical closure for the production terms ofthe dissipation rate equation. All these findingsare compared with the available results of directnumerical simulations of Rogallo (1981) [30].

2. Complete turbulence closure was formulated forthe dissipation and pressure-strain correlationsby assuming that these vary monotonically be-tween the limiting states of turbulence at the ver-tices of the barycentric map. This assumption is

supported qualitatively by the simulation results.A suitable procedure is suggested for matchingthe behavior of turbulence around its limitingstates and the asymptotic trends for very largeand very low Reynolds numbers.

3. Analytical treatment of axisymmetric turbulenceshows that the dynamics of the dissipation cor-relations are more complicated than those of thepressure-strain correlations, since the behavior oflatter can be defined mathematically at the ver-tices of the anisotropy invariant map. Closuresfor the partition of the dissipation tensor and forthe “slow” part of the pressure-strain correlationsindicate that the turbulence functions depend onthe turbulent Reynolds numberRλ.

4. Predictions of axisymmetric flows by the im-proved turbulence closure, which accounts forthe effects of anisotropy of turbulence, agree wellwith the experimental data.

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

[2] B. E. Launder, G. J. Reece, and W. Rodi.Progress in the development of a Reynolds stressturbulence closure.Journal of Fluid Mechanics,68:537–566, 1975.

[3] W. C. Reynolds. Computation of turbulent flows.American Institute of Aeronautics and Astronau-tics, 74:556–567, 1974.

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

[5] Cebeci P. Bradshaw and Whitelaw. Engineer-ing calculation methods of turbulent flows.Aka-demic Publishers, page 331, 1981.

[6] C. G. Speziale. Analytical methods for the de-velopment of Reynolds stress closures in tur-bulence. Annual Review of Fluid Mechanics,23:107–157, 1991.

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

[8] G. K. Batchelor.The theory of homogeneous tur-bulence. Cambrige University Press, 1982.

[9] S. R. Chandrasekhar. The theory of axisymmet-ric turbulence. 1950.

[10] E. Lindborg. Kinematics of axisymmetric ho-mogenous turbulence. 1995.

[11] L. Prandtl. Uber ein neues Formelsystemfur die ausgebildete Turbulenz.Nachrichtender Akademie der Wissenschaften zu Gottingen,Mathematisch-Physikalische Klasse, 6:6–19,1945.

[12] G.I Taylor. Statistical theory of turbulence.Proc.Roy. Soc. London, Part1-4:421–478, 1935.

[13] M. S. Uberoi. Effect of wind tunnel contractionon free stream turbulence.Journal of Aero Sci-ence, 1956.

[14] M. S. Uberoi. Equipartition of energy and localisotropy in turbulent flows.Journal of AppliedPhysics, 1957.

[15] Z. Warhaft. An experimental study of the effectof uniform strain on thermal fluctuations in grid-generated turbulence. 1980.

[16] J. C. Rotta. Statistische Theorie nichthomogenerTurbulenz.Zeitschrift fur Physik, 129:547–572,1951.

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

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[19] V Harris, G. Graham, and S Corrsin. Further ex-periments in nearly homogenous turbulent shearflow. Journal of Fluid Mechanics, 1977.

[20] J Johansson, A.V. & Sjogren. Measurement andmodelling of homogeneous axisymmetric turbu-lence. Journal of Fluid Mechanics, 374:59–90,1998.

[21] 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.

[22] 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.

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

[24] F. Durst S. Banerjee, O. Ertunc. Kinematic re-lationships between second-order tensors in tur-bulence.To be submitted, 2007.

[25] R. Moser, J. Kim, and N. Mansour. Direct nu-merical simulation of turbulent channel flow upto Reτ =590. Physics of Fluids, 11(4):943–945,1999.

[26] A. N. Kolmogorov. Equations of turbulentmotion of an incompressible fluid. IzvestiyaAkademii Nauk SSSR, 6:56–58, 1942.

[27] J. O. Hinze. Turbulence. McGraw-Hill, NewYork, 1975.

[28] J. L. Lumley and H. Tennekes.A first coursein turbulence. MIT Press, Massachussets andLondon, 1972.

[29] G. Comte-Bellot and S. Corrsin. Simple eule-rian time correlations of full and narrow-bandvelocity signals in grid-generated isotropic tur-bulence. Journal of Fluid Mechanics, 48:273–337, 1971.

[30] R. S. Rogallo. Numerical experiments homo-geneous turbulence.NASA Technical Memoran-dum 81315, 1981.

[31] 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.

[32] S. C. Crow. Viscoelastic properties of finegrained incompressible turbulence.Journal ofFluid Mechanics, 33, 1968.

[33] A.A Batchelor, G.K. & Townshend. Decay ofisotropic turbulence in the initial period.Proc.Roy. Soc. London, A193:539–558, 1948.

[34] J. Tan-atchichat.Effects of axisymmetric con-traction on turbulence of various scales. PhDthesis, Illinois Institute of Technology, 1980.

[35] G. K. Batchelor and Proudman. The effect ofrapid distortion of a fluid in turbulent motion.Quart. J Mech. Appl. Math 7, 7:83–103, 1954.

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