A GNF framework for turbulent flow models of drag reducing fluids and proposal for a k ε type closure

Transcription

1 J. Non-Newtonian Flid Mech A GNF framewor for trblent flow models of drag redcing flids and proposal for a ε type closre F. T. Pinho Centro de Estdos de Fenómenos de Transporte, DEMEGI, Facldade de Engenharia, Universidade do Porto, Ra Dr. Roberto Frias, Porto, Portgal Received 1 Jly 2002; received in revised form 23 Jne 2003 Abstract Based on a generalised Newtonian flid GNF model, modified to accont for strain-thicening of the extensional viscosity, this paper derives transport eqations for mass, momentm, Reynolds stresses, trblent inetic energy and its rate of dissipation. An analysis of order of magnitde identifies the relevant new terms and sggestions are made to model those terms needed to ensre closre in the perspective of a low Reynolds nmber ε model. Specifically, a closed model for the time-average viscosity is proposed that taes into accont its non-linearity and dependence on the second and third invariants of the flctating rate of deformation tensor. The trblence model is qalitatively shown to increase the rate of decay of trblent inetic energy in isotropic grid trblence for certain rheological conditions. The performance of the trblence model in a pipe flow is assessed in a companion paper by Crz and Pinho [J. Non-Newtonian Flid Mech., in press] Elsevier B.V. All rights reserved. Keywords: Single-point closres; Drag redcing flids; Trblent flow models; Modified generalised Newtonian flid 1. Introdction Since 1948, when Toms [2] first reported the existence of drag redction in trblent pipe flow of non-newtonian flids, many researchers have dedicated their time at nderstanding the behavior of viscoelastic flids nder trblent flow conditions. Sch efforts clminated in the mid 1970s in a fair amont of phenomenological nderstanding as is well docmented in the reviews of Hoyt [3] and Vir [4]. For the next 20 years, research efforts were aimed at a more in-depth physical nderstanding of the details of sch wall-dominated flows, made possible by developments in optical diagnostic techniqes. Examples of detailed investigations with non-intrsive optical techniqes are the earlier wors of Achia and Thompson [5], Reischman and Tiederman [6], those of Lchi and Tiederman [7,8] and Pinho and Tel.: ; fax: address: fpinho@fe.p.pt F.T. Pinho /$ see front matter 2003 Elsevier B.V. All rights reserved. doi: /s

2 150 F.T. Pinho / J. Non-Newtonian Flid Mech Whitelaw [9] in the 1980s and more recently of Warholic et al. [10], Pereira and Pinho [11] and Escdier et al. [12], amongst others. Note that Warholic et al. [10] investigated heterogenos drag redction, rather than the homogeneos form of concern here and in the other wors listed, bt some of their findings are eqally relevant. In 1995 Gyr and Bewersdorff [13] reviewed critically the existent nowledge not only for polymer soltions bt also for srfactants and fibre sspensions. A major finding of Tiederman s and Hanratty s experimental investigations on channel flows was the dramatic redction of the Reynolds shear stress. The ensing shear stress deficit mst be acconted for by the appearance of an extra elastic shear stress which has not yet been directly measred. This finding sggests a new trblence dynamics involving, amongst others, the copling between flctations of elastic stresses and flctating velocity gradients. This was confirmed by the DNS investigation of Massah and Hanratty [14] with the FENE-P model who fond that the added polymer stresses flctate and interact with the trblence and the mean flow. Under certain conditions the effect is similar to an increased dissipation bearing some resemblance to the effect of anisotropic viscosity on stresses, and contribting to drag redction. In fact, when decopling the shear stress into viscos, inertial and polymer components, the latter can assme both positive and negative vales, i.e. it can act as a sorce as well as a sin of trblent inetic energy see also the recent experimental reslts of Ptasinsi et al. [15]. In spite of all the efforts there is still no clear explanation for the relationship between the observed flow characteristics and the rheology of the viscoelastic flids. However, there are strong indications for a relationship between drag redction and extensional viscosity, bt so far there has been no definite proof of this speclation, one reason being the difficlties in measring the extensional viscosity of dilte and semi-dilte soltions reqired to attain trblent flows. The availability of increased compter power has offered an important research alternative by providing the means for direct nmerical simlations DNS. Starting with Massah et al. [16]. DNS have provided sefl information on the effect of specific flid properties on trblent flow characteristics and is presently the most powerfl method available for probing the physics of trblent flow. For viscoelastic flids there is, however, one important difficlty relative to DNS with Newtonian flids: a priori there is no certainty to what is the correct rheological constittive eqation for a given flid. Even so, DNS is providing sefl insight that will enable researchers to select adeqate constittive eqations and, more important, to develop sefl and more accrate single-point closres for classical or newly developed trblence models. The initial DNS investigations were not self-consistent as they only solved the constittive eqation, sally the FENE-P model, for fixed Newtonian inematics. This strategy, adopted by Hanratty and co-worers [14,16], is not able to predict drag redction bt gave insight onto the evoltion of the moleclar configrations and the corresponding flid properties, with the trblence dynamics. One of their main findings were the large moleclar extensions in the viscos sblayer bt no significant moleclar extensions in the bffer layer. Since in a laminar Coette flow, the molecles were also significantly extended bt no drag redction was fond, it was conclded that althogh moleclar extension affected the shear and elongational viscosities, drag was only a fnction of the shear stress which is not affected by the extensional viscosity in laminar flow. However, extensional viscosity interferes with the dynamics of trblence and conseqently the shear stress in trblent flow is affected. Progress in compter technology enabled the first self-consistent DNS, bt the complexity of viscoelastic differential constittive eqations led researchers to adopt simpler rheological eqations aimed at assessing the effects of particlar rheological properties. This was done by Orlandi [17] and Den Toonder et al. [18,19], who adopted viscos constittive eqations to mimic the effects of polymers in DNS. In Orlandi the anisotropy of a polymer viscos contribtion to normal stresses was related to the magnitdes

3 F.T. Pinho / J. Non-Newtonian Flid Mech of strain and rotation rate tensors to mimic the effect of moleclar extension and this was fond to be sfficient to create drag redction. Den Toonder et al. [18] investigated two constittive models: in model 1, an isotropic normal stress polymer contribtion was only sensitive to stretching of the molecles and, in model 2, the extensional viscosity increased with both stretching and compression of molecles. Only the latter model lead to drag redction. In Den Toonder et al. [19] an anisotropic viscos model and an anisotropic simple viscoelastic model, both made especially sensitive to elongational effects, were investigated. In this wor the anisotropic viscos model was able to predict higher drag redctions than the viscoelastic model. More recently, self-consistent DNS investigations of trblent channel flow with viscoelastic constittive eqations derived from inetic FENE-P and networ Gieses theories have been carried ot by Sreshmar et al. [20] and Dimitropolos et al. [21,22]. These wors were able to predict drag redction and have shown qalitative agreement with experimental findings. In fact, the simlations with the FENE-P and Gieses models showed similar amonts of drag redction when their parameters were chosen to match the platea extensional viscosity [22]. Dimitropolos et al. [22] have also provided bdgets of trblence inetic energy, Reynolds stresses and vorticity and conseqently it is a major reference for the development of single-point trblence closres. Drag redction was fond to be directly related to the extensibility of the polymer chains and that a pre-reqisite for drag redction is a sfficiently enhanced elongational viscosity in agreement with the findings from varios other sorces. The recent experiments of Ptasinsi et al. [15] also provided bdgets of mean energy and of trblent inetic energy, the reslts of which confirm some findings by DNS. As in Newtonian trblent flow, DNS is a powerfl research tool bt reqires enormos compting resorces and is not practical for calclations of indstrial flows. For this prpose, single-point trblence closres are reqired. In contrast to experimental wor, progress in single-point trblence models for viscoelastic flow predictions has been rather slow. After an initial effort by varios research grops in the late 1970s [23 26] very few developments have taen place henceforth. In some of these initial investigations, the ε trblence model for Newtonian flids was sed with very specific modifications in wall fnctions standard model or damping fnctions low Reynolds nmber model. In Hassid and Poreh [24] a one-eqation model was sggested, bt in general these were rapidly being discarded for better models de to difficlties in defining the appropriate length scale. In Poreh and co-worers [25,26] the same modified version of the low Reynolds nmber ε model of Jones and Lander [27] was sed. However, this was not sfficient for predicting drag redction and the coefficient A of the Van Driest type of damping fnction for eddy viscosity had to be determined from the reslts of drag redction at a particlar Reynolds nmber for a specific flid and pipe geometry to enable the accrate calclation of the flow characteristics. A similar strategy was adopted by Drst and Rastogi [23]. In essence, these modifications were nable to deal with drag redction with generality, bt showed that adeqate modifications of the law of the wall or of damping fnctions wold lead to correct predictions. This deficiency stems from the lac of connection between trblence model and flid rheology. A soltion to this shortcoming was attempted as early as 1973 by Mizshina et al. [28]: the Van Driest damping factor in a zero eqation trblence model, was modified to accont for the viscoelasticity of the flid by incorporating Rose s relaxation time [29] to be determined from experiments. This approach enabled predictions of several flow conditions with the same flid, a sitation that not even the later models of Hassid and co-worers [24 26] and Drst and Rastogi [23] cold achieve. More recent excrsions into the same sbject were also of limited application: Politis [30], Crz et al. [31] and Malin [32,33] independently dedced the relevant ε eqations for trblent flows of inelastic

4 152 F.T. Pinho / J. Non-Newtonian Flid Mech power law flids bt the intense drag redction of elastic origin cold not be predicted. However, some of their ideas are prsed in the present wor and its contination [1]. Developments of eddy viscosity models have also been attempted. Poreh and Dimant [34] developed a model based on Van Driest s mixing length with a variable damping parameter to represent the effect of drag redcing additives, and more complex expressions were derived by Edwards and Smith [35] and recently by Azoz and Shirazi [36]. In the latter, the prpose was to predict trblent flow of polymer soltions in annli bt again the model reqired previos measrements in pipe flow with the same flids. There is clearly the need for frther progress in trblence modelling for viscoelastic flids, in particlar taing into accont the extensional behavior of the drag redcing flids. This paper constittes the first step of a wor aimed at developing and testing trblence models for drag redcing flids obeying a specific constittive eqation. Specifically, a generalised Newtonian flid model, that was modified to mimic some effects of the extensional viscosity, was chosen as constittive eqation. Instead of adopting immediately a complex differential constittive eqation, for which modelling wold be more difficlt, a generalised Newtonian flid modified to mimic extensional viscosity enhancement effects will lead to a trblence model with many similarities to a Newtonian trblence model. Trblence models for Newtonian flids do not sffer from the difficlty of choosing a constittive eqation and yet they often rely on damping fnctions and other approaches to compensate for inadeqate or incomplete physics modelling. There is no reason to assme that it will not be so for viscoelastic flids. If a more complex, bt faithfl, viscoelastic constittive eqation is adopted, it is more liely that there will be more ad hoc assmptions and simplifications than if a simpler Newtonian-lie rheological expression is sed. This will provide s with the experience for more elaborate trblent closres in the ftre. This first paper is rather general and is aimed at deriving the transport eqations needed in single-point first- and second-order trblence models. First, the constittive model adopted and its modifications are explained and the corresponding time-averaged conservation eqations of mass, momentm, trblent inetic energy and its rate of dissipation are presented their derivation is presented in Appendix A. An analysis of order of magnitde is then carried ot on those eqations to identify relevant new terms and proposals are made for closre especially regarding a low Reynolds nmber ε model. At the end the new ε model is tested qalitatively in isotropic decay of grid generated trblence. In a second companion paper [1] the ε model is frther developed for wall flows and its capacities are investigated via predictions of pipe flow and comparisons with experimental data from the literatre. 2. A constittive eqation The most important rheological property of polymer soltions that mst be taen into accont is the viscometric viscosity. It can be constant, as with Newtonian flids, bt most often exhibits some degree of shear-thinning. A generalised Newtonian flid model GNF is adeqate to predict this flid property accrately. For a long time there was controversy as to what rheological properties cased drag redction and this was discssed in Oliveira and Pinho [37]. There, the relevance of extensional viscosity was made clear and the recent DNS wors of Dimitropolos et al. [21,22] and De Angelis et al. [38] have extensively confirmed it. Experimentally, Escdier et al. [12] were probably the first to measre drag redction and detailed velocity profiles with polymer soltions for which they provided elongational viscosity data measred

5 F.T. Pinho / J. Non-Newtonian Flid Mech with an opposed-jet nozzle rheometer. Their measrements with different types of polymers showed shear-thinning of the viscometric viscosity and confirmed the strain-thicening of the Troton ratio, relevant rheological featres for drag redction [37]. As mentioned in the introdction, the constittive eqation adopted is simple: the GNF model modified to mimic extensional viscosity strain-thicening. This was the model sed by Oliveira and Pinho [37], here with small modifications. The elongational viscosity is introdced into the GNF constittive eqation by maing it a fnction of the strain rate ε as explained in more detail by Oliveira and Pinho [37]. The GNF flid, with dependence on the shear rate γ and strain rate ε, is written as σ = 2µS, 1 where µ is the viscosity fnction and S is the rate of deformation tensor defined by S 1 i + j. 2 x j x i The viscosity fnction µ = µ γ, ε, depends on γ and ε which are related to the invariants of S in the following way: γ = 4II S = 2S ij S ij, ε = 6 det S tr S 2 = 2trS3 tr S 2 = 2S is j S ij. 5 S ij S ij An algebraic form for the viscosity fnction 3 can be a Bird Carrea type of eqation µ = µ 0 [1 + λ s γ 2 ] n 1/2 [1 + λ e ε 2 ] p 1/2, 6 bt, for simplicity in the derivation of the trblence model, a power law-based eqation is preferred. Ths, the viscosity eqation adopted henceforth is µ = K v [ γ 2 ] n 1/2 K e [ ε 2 ] p 1/2, 7 where some constraints to the varios parameters are imposed later. The model is the prodct of a shear rate dependent term with a strain rate dependent term. Both terms do not have to be dimensionally identical bt their prodct mst be a viscosity and Eq. 7 mst also obey some limiting conditions and agree with rheological measrements. The specific meanings of the varios model parameters are specified in Crz and Pinho [1], and at this stage it is only important to consider K v, K e, n and p as nown flid properties. In any case, the model pretends to represent a behavior where there is shear-thinning de to dependence on γ n <1, p = 1 and strain-thicening de to the dependence on ε n = 1, p>1 as is setched in Fig. 1. For this GNF constittive eqation it is now necessary to dedce the corresponding conservation eqations for trblent flow, bearing in mind that there are flctations in the viscosity becase of its non-linear dependence on the flow inematics. Althogh specific viscosity expressions are presented in Eqs. 6 and 7, the wor in Section 3, and in Appendix A, is totally independent of the adopted eqation for µ. A specific viscosity model is only reqired from Section 4 onwards, bt even then part of this wor remains general and independent of specific forms adopted for µ

6 154 F.T. Pinho / J. Non-Newtonian Flid Mech µ increasing p n=1, p>1 n<1, p=1 decreasing n.. Fig. 1. Schematic representation of the effect of n and p on the variation of the viscosity with the shear and strain rates in log log coordinates. 3. Conservation eqations Modern developments of one-point closres for trblence are sally based on first- or second-order models. Since an objective of this wor is to establish a framewor for developing a specific type of cople trblence rheology closre, conservation eqations for mass, momentm, trblent inetic energy, its rate of dissipation and for the Reynolds stress tensor need to be derived for these flids of variable viscosity, where the first major conceptal difference relative to a Newtonian flid is the existence of flctations of viscosity. The derivation of all transport eqations is tedios and has similarities to that for Newtonian flids. In order not to brea the logical seqence of the text, sch wor is left to Appendix A and here only the final forms of the varios eqations are presented. Throghot the paper the Reynolds decomposition is sed and the average of the flctating qantities, inclding the viscosity, is zero. Capital letters or an overbar designate average vales, small letters or a prime designate flctating qantities, the exception being the average pressre represented by p. The Reynolds-averaged momentm eqation for a GNF flid is ρ U i t + ρu U i = p x i + 2 µs i + 2µ s i ρ i. 8 Relative to the momentm eqation for a Newtonian flid there is a new diffsive term 2µ s i and the classical term 2 µs i is modified. Both need to be evalated later for closre. As will be seen in Section 4, µ also depends on s ij and S ij althogh at high Reynolds nmbers the dependence on s ij is more important. In recent experimental and DNS investigations of trblent dct flows of viscoelastic flids sing the FENE-P model [14,15,21,22] amongst others, the total stress is written as the sm of the solvent, the polymer and the Reynolds shear stress tensors: τ ij = τ ij,s + τ ij,p ρ i j. It is important to nderstand that τ ij,p acconts not only for an elastic contribtion bt also for a possibly large viscos contribtion.

7 F.T. Pinho / J. Non-Newtonian Flid Mech In or formlation, however, the separation of effects is different becase both 2 µs ij and 2µ s ij inclde prely viscos and extensional contribtions of the polymer soltion notice that µ s ij 0 for an inelastic shear-thinning flid. Ths, τ ij,s is totally inclded in 2 µs ij bt so is also part of τ ij,p. The Reynolds stress transport eqation is given by ρ D i j Dt U i U j + ρ j + ρ i = ρ i j x i p j + 2 µ i j + µ i j + µ 2µ i j + µ i j + µ i 2 U j + j µ p x i j j + µ Ui + U x i x i j j + p + i + µ 2 i j x i x j + i x j + i x i x j µ + i 2 s ij + µ 2 i j 2 U + µ i j Uj + U x j, 9 and contraction of indices gives the transport eqation for the trblent inetic energy 2 ii /2: ρ D Dt = jp [ ] 1 x j x j 2 ρ i j i 2 µ i s ij 2µ i S ij 2µ i s ij 2 µs 2 ij 2µ s 2 ij 2µ s ij S ij ρ i j S ij. 10 Eqs. 9 and 10 inclde varios non-newtonian terms, bt even some of the Newtonian terms are presented in an nsal way. The reader can recover the classical eqation fond in many papers and textboos bearing in mind properties de to the symmetry of the rate of deformation tensor S ij see also Appendix A The rate of dissipation of trblent inetic energy For Newtonian flids the average rate of dissipation ε of trblent inetic energy, per nit of mass, is defined as ρε n 2µs 2 ij, 11 where the sbscript n is sed here to distingish it from the rate of dissipation of for GNF flids. Eq. 11 is the time-average of the instantaneos rate of dissipation defined by ρˆε n 2µs 2 ij. 12 By analogy, one can define an instantaneos rate of dissipation for non-newtonian flids obeying the GNF model by sing the instantaneos viscosity: ρˆε 2 ˆµs 2 ij. 13

8 156 F.T. Pinho / J. Non-Newtonian Flid Mech Time-averaging eqation 13 provides the average rate of dissipation for the GNF flid ρε = 2 ˆµs 2 ij = 2 µ + µ s 2 ij = 2 µs2 ij + 2µ s 2 ij. 14 These definitions of instantaneos and average rates of dissipation are identical to those sed by Politis [30] in his derivation of a ε model of trblence for prely viscos shear-thinning flids. The definition of the rate of dissipation of trblent inetic energy also deserves a comment in light of the literatre involving the FENE-P or similar models. Several wors reviewed in Section 1 reported the existence of a deficit in Reynolds shear stress and the conseqent existence of a polymer stress, here called τij P. The flctations of τp ij contribte to increase or decrease trblence via the term τ P ij s ij which can tae positive or negative vales, respectively [10]. Also, in a viscoelastic formlation the transport eqations of and i j contain a term for the interaction between the elastic stress and the rate of strain as shown in the DNS investigation for a FENE-P flid of Dimitropolos et al. [22] their ε v ij term. Their reslts do confirm that ε v ij in combination to the interaction between τ ij P and i acts as a trblence prodction term near the wall and as dissipation elsewhere. By defining the total stress tensor as τ ij = 2 µs ij 2µ s ij ρ i j and the rate of dissipation as in Eq. 14, its two terms in Eq. 10 already inclde sch interaction between flctating stress and flctating shear rate. ε is defined as a single qantity bt still it is the sm of a positive definite term 2 µs ij 2 with a term that can be either a sorce or a sin of dissipation 2µ sij 2. So, in a limiting sitation of a 1D shear flow, where the Reynolds shear stresses are fond to be negligible de to the presence of additives, cf. [10], nder the classical eqilibrim condition the definition of Eq. 14 reslts in negligible prodction of trblence and negligible dissipation. However, for the flow to be trblent there mst be prodction of trblence, as well as dissipation, and so what is really happening is 2µ sij 2 = 2 µs ij 2. Note that recently Ptasinsi et al. [15] showed that in pipe flow the Reynolds stress stays definitely non-zero even at maximm drag redction The transport eqation for the rate of dissipation of trblent inetic energy ρ ε t {{ Ia ε + ρu = 2 U i x {{ x Ib [ ] [ ] µ + µ i 2 U µ + µ x m x m x i i m x m {{{{ IIa IIb [ ] 2 2 U i µ + µ x i 2 µ + µ m x i i m x m x {{{{ m IId IIc µ + µ [ ] i i 2 ν + ν x m x [ ] p i {{ m x i x m x {{ m IIIa IIIb + µ + [ ] µ 2 i i 2 µ + µ 2 2 i 2 i ρ x m x {{ m ρ x x m x {{ m IV V

9 F.T. Pinho / J. Non-Newtonian Flid Mech [ ] + 4 ν + ν 2 µ + µ i s i + 4 ν + ν x m x µ + µ i s i {{ m x x m x {{ m VI VII + 2 ν + ν µ + [ ] µ i i + 4S i ν + ν x m x m x i 2 µ {{ x m x m {{ VIII IX [ ] [ ] + 4 S i ν + ν x i µ U i ν + ν m x m x i µ x m x m {{{{ X XI [ ] [ ] U i ν + ν x m x µ i 4ν x i µ s i m x m x m {{{{ XII XIIIa [ ] 2ν i µ 2 x m x i + 2µ [ ] m x i i x m x m {{{{ XIIIb XIV + µ + µ i i t x m x {{ m XVa µ + µ + U i x {{ XVb x m i x m. 15 The transport eqation for the time-average rate of dissipation ε of trblent inetic energy of a GNF flid is given by Eq. 15. In Eq. 15 the alternative definitions of the average and instantaneos rates of dissipation are ρε µ + µ i /x m i /x m and ρˆε µ + µ i /x m i /x m, respectively. These two definitions of rate of dissipation are eqivalent to those of Eqs. 13 and 14 only nder the assmption of homogeneos trblence. For convenience, in writing down Eq. 15 se was made of the inematic viscosity ν µ/ρ. Eq. 15 incldes two types of terms: those having similarities with terms fond in the dissipation eqation for a Newtonian flid originate from the inertial, pressre and one of the viscos terms in the momentm eqation, and inclde both the average and flctating viscosities. The second set of terms is new and involves the flctating viscosity µ. Often, their physical meaning can be easily identified de to similarities with terms involving the average viscosity. 4. The time-average moleclar viscosity In trblent flow, the moleclar viscosity of a variable-viscosity flid depends on the flctating rates of shear and strain, a major difference relative to a Newtonian flid. Hence, a major contribtion of this wor is the relationship between the time-average moleclar viscosity and trblent qantities to ensre closre of the set of eqations. In order to arrive at sch relationship it is now necessary to adopt a specific form for the viscosity fnction and here Eq. 7 is sed.

10 158 F.T. Pinho / J. Non-Newtonian Flid Mech In terms of instantaneos vales the viscosity is ˆµ = K v [ ˆ γ 2 ] n 1/2 K e [ˆ ε 2 ] p 1/2, with ˆ γ and ˆ ε following from Eqs. 4 and 5. The maximm vale of ˆ ε was estimated by Oliveira and Pinho [37] to be ˆ ε max = 2 3ŜijŜij For a high Reynolds nmber trblent flow Tennees and Lmley [39] have shown that Ŝ ij Ŝ ij = sij s ij, ths Oliveira and Pinho [37] conclded that 2 ˆ ε max = 3 s ijs ij. 18 Typical vales of ˆ ε being smaller, let s assme that in general ˆ ε = sij s ij A ε, 19 where the vale of A ε is to be fond from experimental data see [1] bt mst be higher than 3/2. Bac-sbstitting these definitions into the viscosity model and simplifying gives ˆµ = K vk e A p 1 2 n 1/2 [sij 2 ]n+p 2/2. ε This expression can now be sed to calclate the instantaneos rate of dissipation in Eq. 13: ρˆε = 2 n+1/2 K vk e A p 1 [sij 2 ]n+p/2. ε Eqs. 20 and 21 are combined to eliminate sij 2 and yielding a relation between the instantaneos viscosity and rate of dissipation [ ] Kv K 2/n+p e ˆµ = A p p/n+p ρˆε n+p 2/n+p. 22 ε Introdcing parameters m n + p 2 n + p, 23a and [ ] Kv K 1 m e B A p 1 2 n 1 mn+1/2 ρ m, 23b ε for compactness, Eq. 22 assmes the simple form ˆµ = Bˆε m. 24

11 F.T. Pinho / J. Non-Newtonian Flid Mech The average viscosity and the average rate of dissipation are determined sing their probability distribtion fnctions. By definition, the time-average viscosity µ = 0 Bˆε m Pˆε dˆε. Since the instantaneos viscosity is always a positive qantity, the instantaneos rate of dissipation ˆε is positive-definite and is associated with small scale motion, assmed here to be locally isotropic at high Reynolds nmber flows. Ths, as explained by Monin and Yaglom [40], ˆε follows a log-normal distribtion { 1 Pˆε = ˆεσ 2π exp 1 ln ˆε M 2, 26 2 σ with M and σ standing for the mean and standard deviation of ˆε. Following Monin and Yaglom [40] pp , { ˆε m 0 ˆεσ 2π exp 1 ln ˆε M 2 dˆε = exp mm + m2 σ 2 2 σ 2 so, the average viscosity comes ot as µ = B expmm m2 σ 2. 25, 27 The average rate of dissipation is also obtained from the probability distribtion fnction of ˆε { 1 ε = ˆεPˆε dˆε = ˆε 0 0 ˆεσ 2π exp 1 ln ˆε M 2 dˆε = exp M + σ σ 2 Now, µ and ε can be related to each other by solving Eq. 28 to get e M and sbstitting it bac into Eq. 29. The reslt is µ = Bε m e mσ2 m 1/2, with m and B given by Eqs. 23a and 23b. In Eq. 30, σ 2 is the variance of the distribtion of ln ˆε which is given by L σ 2 = A 1 + A 2 ln, 31 η with L representing an external trblence length scale, sch as a large eddy scale, and η an appropriately defined internal length scale sch as the Kolmogorov microscale [40]. A 1 is a parameter depending on the trblence macrostrctre and A 2 is a niversal constant. The extensive discssion of these qantities, in pp of Monin and Yaglom [40], sggests that A 2 = , althogh the most appropriate vale seems to be between 0.4 and 0.5, ths 0.45 is considered. A 1 depends on the form of the space regions and there is very little information concerning it p. 634 in [40], so it is assmed to be zero. Hence, σ 2 is finally given by σ 2 = A 2 ln L η

12 160 F.T. Pinho / J. Non-Newtonian Flid Mech The Kolmogorov scale is [ ν 3 ] 1/4 [ µ 3 ] 1/4 η =, 33 ε ρ 3 ε where the average moleclar viscosity was sed. The large length scale for the energy containing eddies is given by the inviscid estimate proportional to 1.5 /ε. A commonly sed eqation for L derived from inviscid theory argments is adopted here as given by Yonis [41] L 2C0.75 µ 1.5, 34 ε where C µ is a niversal constant that sally assmes the nmerical vale of 0.09 for Newtonian flids. Finally, combining Eqs provides the final form of an explicit expression for µ µ = C µ ρ 3mm 1A 2/8+3mm 1A 2 2 4mm 1A 2/8+3mm 1A 2 6mm 1A 2/8+3mm 1A 2 ε [8 3m 1A 2]m/8+3mm 1A 2 B 8/8+3mm 1A 2 35 with m and B defined above. In the limiting case of a Newtonian flid n = p = l a constant average moleclar viscosity of µ = K v K e is recovered. With this relationship the fll closre of the set of transport eqations can be ensred provided the new non-newtonian terms of the conservation eqations are adeqately modelled. This is carried ot in the next sections, bt only in the context of a first-order ε trblence closre. In the companion paper [1] the final details of the model are derived and the model is sed to mae predictions and comparisons with data from the literatre. The developments for more elaborate first-order and second-order models are left for the ftre. 5. Order of magnitde analysis An order of magnitde analysis is carried ot for all the transport eqations to help in their simplification. There will be a preoccpation to identify similarities with the corresponding eqations for Newtonian flids and, becase of the lac of experience and nowledge on trblence modelling for viscoelastic flids, the nmber of modifications is ept to a minimm except when based on solid argments. To perform the order of magnitde analysis, the following scales are sed: L represents a large length scale of the energy containing eddies, U is the velocity scale of mean flow, is the velocity scale of flctations and l is the length scale associated with small flctations and its gradients, which is related to the Kolmogorov scale. Note also that the inviscid estimate of the rate of dissipation is sed ε = 3 /L and that both the instantaneos moleclar viscosity ˆµ and the moleclar viscosity flctations µ are needed. This analysis starts with an estimate of these two viscos qantities, bt henceforth their inematic eqivalents ˆν and ν ν µ/ρ are sed instead. The ratio of instantaneos to average moleclar viscosities is needed and is estimated as [ mσ 2 1 m ˆν ν = ˆε m ε m exp[mσ 2 m 1/2] exp 2 ], 36

13 F.T. Pinho / J. Non-Newtonian Flid Mech becase ε is determined from the pdf of ˆε and m Eq. 23 is a coefficient of order 1 it depends on n and p where sally n 1 and p 1. Upon sbstittion of σ 2 Eq. 32 ˆν L 0.225m1 m ν. 37 η For convenience a 0.225m1 m is defined. Considering typical vales of n and p, m = 0 for Newtonian flids and for non-newtonian flids for which the shear-thinning exponent n differs from 1 by exactly the same amont that the strain-thicening p differs from 1 i.e. when p = 1 n. For other realistic vales of n and p [1] the exponent a stays in the range limited by 0.05 and Finally, considering the Kolmogorov scale Eq. 33 and L from Eq. 34: ˆν ν 0.33a and, by definition, L 3a/4 L 3a/4, 38 ν ν L 3a/4 1. ν ν 39 ν The se of the difference in estimating ν is advantageos, becase it will prodce a nll contribtion for flids having a constant viscosity Momentm eqation The momentm eqation Eq. 8 has a modified diffsive term that can now be calclated since there is a relationship for µ and a new diffsive term 2µ s i that mst be evalated. The relevance of this new term is assessed by comparison with the modified diffsion as 2µ s ij ν /l 2 µs ij ν U/L = [ L 3a/4 1] ν L U l, 40 so in principle µ s ij cannot be neglected becase l/l /U in fact, l/l L/ ν 3/4. Under certain conditions, however, it is possible to neglect µ s ij as shown by Oliveira and Pinho [37]: basically, in 2D flows of shear rate γ independent viscosity flids the average strain rate ε is zero, hence the flctating viscosity is an even fnction of ε. Ths, any correlation of µ with other approximately Gassian-distribted variable related to the small scale of trblence will vanish becase the odd moments of the normal distribtion are zero [42]. So, no terms are dropped and the final form of the time-average momentm eqation is that of Eq The transport eqations for i j and The transport eqation for the Reynolds stress i j Eq. 9 contains several new moleclar diffsion or dissipative terms. Their orders of magnitde are compared with that of the Newtonian-lie dissipative term 2 µ i / j / which, note, is not eqal to ε ij de to the new definition of ε. The otcome of this analysis is contained in Table 1. At first sight, it is obvios that only the third and forth terms in

14 162 F.T. Pinho / J. Non-Newtonian Flid Mech Table 1 Order of magnitde relative to Newtonian dissipative term Term Order Nmerical µ i j + j + i L 3/ x i x j ν µ L 3/4 s ij ν µ 2 i j 2µ x i j L 3a/4 1 ν 1 L 3a/4 j + i 1 x i x j ν 1 [ 2 U µ i 2 U j + µ x j L 3a/4 U L 3/2 i 1] ν ν µ Ui j + U µ Uj + i + U [ L 3a/4 U L 3/4 1] x i x j ν ν µ i j + µ the table shold be retained in a high Reynolds nmber formlation. Possibly, in a low Reynolds nmber formlation some of the other terms may be retained, bt that reqires a more extensive analysis left for the ftre as the present paper concentrates on modelling only a ε closre. Anyway, note also that the first term does not reqire modelling. In conclsion, the simplified transport eqation of i j is ρ D i j Dt U i U j + ρ j + ρ i = ρ i j x i p j + j p x i + p j + µ 2 i j 2µ x i j + µ i j + µ + i x i x j j + i x i x j + µ 2 i j 2 µ i j. 41 The transport eqation of Eq. 10, obtained by contraction of the indices of the eqation of i j, also contains new dissipative and diffsive terms and, as with the Reynolds stress eqation, comparing their orders of magnitde with the order of magnitde of the dissipation term allows s to simplify it. The dissipative terms 2µ sij 2 and 2 µs2 ij are lmped together to define the dissipation ε cf. Eq. 14. The other dissipative term 2µ s ij S ij is neglected in comparison to ε as in the Reynolds stress eqation. Terms 2 µ i s ij and 2µ i s ij are components of the moleclar diffsion of which can be recast as 2 µ + µ i s ij. Here, the diffsivity coefficient is the average moleclar viscosity pls a contribtion from the flctating moleclar viscosity. This second contribtion taes on positive and negative vales and so it shold be smaller than the former. The diffsion of is sally very small, except at low Reynolds nmbers and in the vicinity of walls, so the contribtion from the flctating viscosity is in principle smaller and neglected by comparison, at least ntil frther research shows that it shold be ept,

15 F.T. Pinho / J. Non-Newtonian Flid Mech i.e. as a first approximation one has 2 µ i s ij + 2µ i s ij 2 µ i s ij. 42 Therefore, the simplified version of the transport eqation of is given by ρ D Dt = x j 5.3. Transport eqation for ε [ i p + 12 ρ i j i 2µ i s ij ] ρε ρ i j S ij. 43 For the ε eqation Eq. 15 the estimated order of magnitde of its varios terms are smmarised in Table 2. Using terms Ia + Ib as reference, and considering l to be identical to the Kolmogorov length scale leading to l/l L/ ν 3/4, one ends p with the relative reslts of Table 3 which gives a better idea of the relevance of the varios terms. To help in this determination, nmerical vales are given to a and to the two Reynolds nmbers. For a there are three typical vales of 0, 0.05 and +0.05, bt in terms of order of magnitde a = 0.05 and are eqivalent. For the Reynolds nmbers, a bl Reynolds nmber Re U of abot 50,000 is considered and a trblence intensity of abot 10% gives Re = However, before proceeding, and to help in the critical assessment of the relevance of the varios terms of the ε eqation, the eqivalent Newtonian eqation is presented in its general form [43]: ρ ε t {{ Ia +ρu i ε x i {{ Ib = µ 2 ε 2µ i U i 2µ j j U i i 2 U i 2µ {{ 2µ i i x j x j x i x IV {{{{ j x j x m x {{{{ m IIa IIb IIc IId [ ] [ ] µ i i 2 µ p 2 µ2 2 i 2 i. 44 x m x m ρ x m x m ρ x m x m {{{{{{ V IIIa IIIb In Eq. 44 the varios terms are identified by the same codes sed in the non-newtonian eqation 15 to facility comparisons, bt cation mst be exercised becase the presence of moleclar viscosity flctations can change their physical meaning as will be seen. Using now the nmerical vales of the previos paragraph, the nmerical estimate of the relative order of magnitde of the terms in Table 3 leads to Table 4 and the following conclsions are drawn: i For Newtonian flids the prely non-newtonian terms VI XV vanish. ii The terms that are common to the Newtonian and non-newtonian eqations terms I V, most of which have been modified, have the same order of magnitde regardless of the vale of parameter a. The exceptions are the diffsive terms trblent diffsion: IIIa + IIIb; moleclar diffsion: IV which have been significantly modified and enhanced by the viscosity flctations. These terms were named by analogy to the Newtonian eqation and considering the physics of the latter. iii Of the new terms of the non-newtonian eqation, terms IX XIV are irrelevant in comparison with terms VI + VII + VIII + XV. Terms IX XII originate in the diffsive term 2 µs ij of the momentm eqation and terms XIII and XIV come from the other diffsive term 2µ s ij. These two sets of

16 164 F.T. Pinho / J. Non-Newtonian Flid Mech Table 2 Estimated order of magnitde of terms of ε Eq. 15 Term Ia + Ib IIa + IIb IIc IId IIIa + IIIb IV V VI + VII + VIII IX X + XI XII XIIIa + XIIIb XIV XVa + XVb Order U 3 L 2 ˆν U2 Ll 2 ˆν U2 L 2 l ˆν 3 l 3 3 l 2 2 l 2 ˆν 2 2 l 4 ν L + ν l ν 2 ˆνν 2 l 4 l 2 ˆνν U Ll 3 ˆνν U L 2 l 2 ˆνν U L 3 l 2 2 ν Ll 3 ν 3 Ll 2 ν U2 l 3 + ν2 L + ν ν 2 l 2 terms have a negligible inflence except perhaps in the perspective of a low Reynolds nmber flow and wall proximity. In this case terms IX and XIV loo more important than the others bt more detailed investigations are reqired to ascertain which shold be ept nder those conditions. Note that, with viscoelastic drag redcing flids, a low Reynolds nmber formlation is essential becase there is no niversal law of the wall in contrast to what happens with Newtonian flids. Since many of the argments sed in this order of magnitde analysis were formlated on the basis of high Reynolds nmber flow, some of the neglected terms may need to be re-evalated close to walls. At this stage of nowledge on modelling trblent viscoelastic flow we opted to neglect them. iv Term XV is the most important of all terms as its order of magnitde is the highest. This does not mae complete sense as the term shold be balanced by at least another one. It is an indication that

17 F.T. Pinho / J. Non-Newtonian Flid Mech Table 3 Estimated order of magnitde of terms of ε Eq. 15 relative to terms Ia + Ib a Term Order IIa + IIb IIc IId Re 2+3a/4 Re 3a 1/4 IIIa + IIIb Re 3/2 IV Re 1/2 V VI + VII + VIII IX X + XI XII Re 9+3a/4 Re 1 U Re 1 U [1 + Re3+3a/4 Re 3/4 ] Re 1 U Re 4+3a/2 Re 1 U {1 + Re3/2 [Re3a/4 Re 8+3a/4 Re 1 U [Re3a/4 1] Re 1+3a/4 [Re 3a/4 1] Re 3a 2/4 [Re 3a/4 1] Re 3a 5/4 [Re 3a/4 1] XIIIa + XIIIb Re 5/4 Re 1 U [Re3a/4 1] 2 XIV Re 3/2 Re 1 U [Re3a/4 1] XVa + XVb Re 5/4 [Re3a/4 1] a Re U is based on U and Re is based on. In both cases the length scale is L and ν is sed. 1] 2 + Re 3/2 [Re3a/4 1] perhaps the viscosity gradient shold not be scaled with the length l as it was, bt with L. Inany case, the term is important and hence it is ept. At this stage, and since the trblent flows of viscoelastic soltions are almost always associated with Reynolds nmbers well below those of Newtonian flids, and given the need for a low Reynolds nmber Table 4 Nmerical estimate of the order of magnitde of terms of Eq. 15 relative to terms Ia + Ib Term Order IIa + IIb 100 IIc 0.1 IId 1000 IIIa + IIIb a = 0 1, a IV a = , a V 100 VI + VII + VIII a = 0 0, a IX a = 0 0, a 0 1 X + XI a = 0 0, a XII a = 0 0, a XIIIa + XIIIb a = 0 0, a XIV a = 0 0, a 0 1 XVa + XVb a = 0 0, a

18 166 F.T. Pinho / J. Non-Newtonian Flid Mech formlation, the ε transport eqation is [ ] [ ] ρ ε t + ρu ε = 2 U i µ + µ x i 2 U µ + µ x m x m x i i m x m [ ] 2 2 U i x m µ + µ i x m 2 µ + µ i x m i x m µ + µ [ ] i i 2 ν + ν x m x [ ] p i m x i x m x m + µ + [ ] µ 2 i i 2 µ + µ 2 2 i 2 i ρ x m x m ρ x m x m [ ] + 4 ν + ν 2 µ + µ i s i + 4 ν + ν x m x µ + µ i s i m x m x m + 2 ν + ν µ + [ ] µ i i + 4S i ν + ν x m x m x i 2 µ x m x m [ ] [ ] 4ν i µ s i 2ν x m x m x i µ 2 x m x i + 2µ [ ] m x i i x m x m + µ + µ t i i µ + µ + U i x m x m x m i x m. 45 This eqation will be frther simplified when discssing its modelling in the perspective of a ε closre, in the next section. 6. Modelling the transport eqations for a ε closre The analysis of order of magnitde of the previos section has shown what terms of the transport eqations mst be retained and which to neglect nder some assmptions. The relevant terms are of two types: those that can be directly evalated, sch as the moleclar diffsion or the prodction of trblence in the eqation, and those that mst be modelled, sch as the trblent diffsion Momentm eqation The aim is to solve the momentm eqation Eq. 8 which possesses three terms that reqire modelling: the mean moleclar stress 2 µs ij, the moleclar trblent stress 2µ s ij and the Reynolds stress i j. For the mean moleclar term, closre is ensred by the expression for the average viscosity Eq. 35. The moleclar-trblent stress is a new term copling the flctations in viscosity and in the rate of strain tensor. It is a relevant term for shear-thinning flids, that can only be neglected in non- or wealy shear-thinning flids in 2D mean flows, sch as in bondary layers, jets or pipe flows. As mentioned in Section 3, 2µ s ij brings into the momentm eqation of the polymer soltion both viscos and elastic

19 F.T. Pinho / J. Non-Newtonian Flid Mech contribtions from the polymer molecles. However, 2 µs ij also inclde viscos and elastic contribtions from the polymer molecles in addition to he viscos contribtion from the Newtonian solvent. It is difficlt to ascertain how the polymer contribtions are split between both terms, bt the fact that 2µ s ij becomes negligible nder certain conditions sggests that 2 µs ij taes in a significant amont of the effect. At present 2µ s ij is dropped and its effect is basically taen by 2 µs ij and a new damping fnction f v to be introdced in the follow-p paper [1]. In the near ftre, this mst be improved by an adeqate modelling of 2µ s ij. Finally, the Reynolds stress reqires modelling and henceforth in this wor a first-order trblence closre of the ε type is the choice. The selection of sch model may seem too simplistic an approach bt the trth is that there is no single-point trblence model for drag redcing flids that combine the effects of trblence and non-linear rheology. Therefore, a ε formlation seems adeqate as a starting point for semi-qantitative predictions meaning that the trends will be captred, a significant drag redction will also obtained bt predictions will not always match experimental reslts. For the Reynolds stress, the Bossinesq approximation or gradient hypothesis is sed i j = ν T Ui x j + U j x i 2 3 δ ij, 46 with the trblent eddy viscosity given by the Prandtl Kolmogorov eqation 2 ν T = C µ f µ ε. 47 In Eq. 47 there is a damping fnction f µ that is needed for low Reynolds nmber models, bt is eqal to 1 in high Reynolds nmber formlations and away from the wall. This, and other damping fnctions appearing later, accont for physical inadeqacies in modelling [44]. Natrally, since the flids involved are now non-newtonian, the damping fnction f µ, parameter C µ and other parameters and damping fnctions fond in Newtonian models mst be evalated differently. This is done by Crz and Pinho [1] and is partially inspired by the wor of Crz et al. [31] for inelastic power law flids Transport eqation for In Eq. 43 only the terms within the sqare bracets need modelling. There is one term de to the interaction of velocity and pressre flctations pressre diffsion and a second term de to the interaction of velocity flctations with Reynolds stresses, both groped together nder the name of trblent diffsion. The third term is the moleclar diffsion of trblent inetic energy differing from the Newtonian definition becase it involves the average moleclar viscosity. Pressre diffsion has a small contribtion in Newtonian flows and is not expected to behave differently with the non-newtonian flids. In fact, the DNS simlations of Dimitropolos et al. [22] and De Angelis et al. [38] show that viscoelasticity decreases pressre diffsion. Hence, the sal approach in its modelling is sed: pressre diffsion is lmped with trblent diffsion and modelled as a symmetric term or, eqivalently it is neglected. Since trblent diffsion is independent of the constittive eqation, it is modelled exactly as for Newtonian flids sing the classical gradient model j p i j i = ρ ν T, 48 σ x j

20 168 F.T. Pinho / J. Non-Newtonian Flid Mech where σ is an empirical coefficient called the trblent Prandtl nmber. The final form of the transport eqation of is ρ D Dt = [ ρ ν T + µ ] ρ i j S ij ρε. 49 x j σ x j x j 6.3. Transport eqation for ε The eqation with the most sbstantial nmber of modifications is the transport eqation for the rate of dissipation of trblent inetic energy. Unfortnately, there is still no viscoelastic DNS wor with a bdget of ε to gide modelling. The dissipation eqation concerns physical processes in the dissipative range bt for its modelling with Newtonian flids ε is viewed rather as an energy flow rate in the energy cascade, i.e. a large scale motion qantity. Conseqently, the modelled transport eqation of ε for Newtonian flids is basically empirical and the same approach is sed in this wor. Therefore, inspection of the varios terms of the exact eqation of ε basically serves the prpose of identifying similarities with the corresponding Newtonian eqation and the physical mechanisms involved in order to help in their modelling. Terms IIa, IIb and IIc correspond to the prodction of ε, becase they qantify the interaction between ε and the mean flow gradient, and now they also inclde the effect of variable viscosity. Terms IId in Eqs. 15 and 44 pertain to the generation of vorticity flctations which are often inclded as part of the destrction of ε [45]. In the Newtonian eqation, term IIc is sally neglected in comparison with term IId with the argment that the flctations of velocity and its gradient in IIc are less well correlated than the gradient qantities in IId [45]. The same approach is adopted here and it is worth remembering that in the presence of drag redcing flids the correlations between trblent qantities sally decrease in comparison with their Newtonian eqivalents. This was seen to be the case in the Reynolds stress and bdgets obtained by Dimitropolos et al. [22] and De Angelis et al. [38] with DNS and experimentally in several wors [13], so there is no reason to believe that it can not be so also for the ε eqation. Regarding terms IIa and IIb, nder the assmption of isotropic dissipation small scales they are neglected here. This is one of the isses that may have to be reviewed in the ftre given the tendency for flid elasticity to accentate anisotropy of trblence large scales. µ i j = 0, when i = j and l m, i j and l = m. 50 x l x m For Newtonian flids, Hanjalic and Lander [46] modelled the prodction of ε terms IIa + IIb by not assming isotropy of dissipation bt then modelled the generation of vorticity flctations IId and the destrction of ε V in a different way so that, the sm IIa + IIb + IIc + V gives rise to two terms in the modelled eqation that exactly match the modelling of other athors who assmed isotropy of dissipation Eq. 50. Therefore, defining the prodction of ε as P ε IIa + IIb + IIc + IId, this is modelled as P ε ρf 1 C ε1 ε i j U i x j = ρf 1 C ε1 ε ν T Ui x j 2 ε = ρf 1 C ε1 C 2 µf µ ε 2 Ui. 51 This model of prodction atomatically considers the modifications de to the variable moleclar viscosity in the modified definition of average rate of dissipation. The damping fnction f 1 acconts for x j

21 F.T. Pinho / J. Non-Newtonian Flid Mech modelling inadeqacies at low Reynolds nmbers and near walls may need to be modified to accont for non-newtonian effects. The next two terms in Eqs. 15 and 44 terms IIIa + IIIb mst be analysed catiosly. In the Newtonian eqation 44 IIIa represents the trblent diffsion of dissipation by the velocity flctations, whereas the role of pressre flctations on trblent diffsion is acconted for by term IIIb. For the non-newtonian flids it is convenient to frther maniplate these terms. Term IIIa is separated into two sbterms as follows: µ + µ [ ] i i x m x {{ m IIIa = ρ ˆε {{ IIIa1 µ + µ + 2 i x {{ m IIIa2. 52 The first term on the right-hand side is the classical trblent diffsion and the second term is prely non-newtonian, having the same order of magnitde of the whole term IIIa in Table 2. This is very important becase the order of magnitde of term IIIa for non-newtonian flids was 100 times larger than for Newtonian flids, i.e. the magnitde of IIIa1 is qite small in comparison to that of term IIIa2. Modelling of IIIa2 is discssed below in Eq. 59. Term IIIb can also be split as ν + ν [ ] p i = 2 [ ν + ν x m x m x ] i p + 2 p i µ + µ, 53 x m x m x m x m x i {{ IIIb {{ IIIb1 {{ IIIb2 with term IIIb1 acconting for diffsion by pressre flctations and the second term associated with viscosity variations. The latter is again the predominant contribtion and is the main responsible for the order of magnitde of the whole term IIIb. However, at the moment term IIIb2 is neglected for lac of nowledge on how to model it. Politis [30] has considered this term to have a negligible inflence and assmed that its magnitde was smaller and similar to that of or term IIIb1. To estimate the order of magnitde of term IIIb2, he sed L as the length scale for the gradient of pressre flctations we sed l. In the absence of more information, we consider the view of Politis [30] and so the effect of copling pressre gradient with viscosity gradient flctations is assmed inclded in the modelling of term IIIb1. In conclsion, terms IIIa1 + IIIb1 + IIIb2 are modelled together and as part of trblent diffsion, bt in the ftre term IIIb2 may have to be specified separately and in a different way. Finally, there is an extra non-newtonian trblent diffsion term de to viscosity flctations term XIV in Eq. 15. As seen above, term XIV is small compared with terms IIIa1 + IIIb1 and so, together these three contribtions, in addition to term IIIb2, will be denoted as D ε. In the analysis of Politis [30] for prely viscos flids term XIV was also shown to be negligible. In a classical ε closre, the trblent diffsion D ε is often modelled with a gradient transport hypothesis: the argment is that, for a continm flow, the time and length scales of the flctations and of the moleclar processes are different by many orders of magnitde, bt the time and length scales of the mean and flctating flows are of similar magnitdes. For the ε closre it is frther assmed that the trblence is isotropic leading to D ε = ρc 2 ε ε x i ε x i = x i ρ ν T ε. 54 σ ε x i