Reynolds Averaged Simulation on Turbulent Drag ...

Proceedings of the ASME 2014 4th Joint US-European Fluids Engineering Division Summer Meeting FEDSM2014 August 3-7, 2014, Chicago, Illinois, USA

FEDSM2014-21327

REYNOLDS-AVERAGED SIMULATION ON TURBULENT DRAG-REDUCING FLOWS OF VISCOELASTIC FLUID BASED ON USER-DEFINED FUNCTION IN FLUENT PACKAGE Zhi-Ying Zheng, Feng-Chen Li†, Qian Li School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

ABSTRACT A new numerical simulation methodology for turbulent flows of viscoelastic fluid was developed for engineering application purpose based on commercial computational fluid dynamics code FLUENT package. An in-house subroutine was established and embedded into FLUENT code through userdefined function functionalization. In order to benchmark this methodology, numerical simulations on turbulent channel flows of viscoelastic fluid are conducted under different cases with drag reduction rates varied from low level to high level. FENE-P (finitely extensive nonlinear elastic-Peterlin) constitutive model is used to describe the viscoelastic effect of viscoelastic fluid flow. The turbulent model is developed in the 2 framework of k    v  f model, for which the elliptic

been conducted to understand the mechanism of turbulent DR. However, the up-to-date experimental techniques cannot yet obtain the key information for studying the mechanism of turbulent DR, such as the molecular or micelle conformation tensor and elastic stress field in viscoelastic fluid flows. The development of computer technology prompts researchers to pay more attentions to numerical methods. Among the numerical simulation methods, direct numerical simulation (DNS) can obtain comprehensive turbulence information till the smallest dissipation scale in the turbulent flow field and thus has been widely adopted to study the mechanisms of turbulent DR by simulating viscoelastic fluid flows, such as channel flows [2, 3] and decaying homogeneous isotropic turbulence [4]. But the multi-scale characteristic of turbulence makes DNS calculation considerably expensive with extremely high demand for computer’s memory and time consumption, limiting Reynolds number of feasible flows for DNS to be small or moderate, which cannot satisfy the need of engineering practice. Compared with DNS, commercial computational fluid dynamics (CFD) software using RANS (Reynolds averaged Navier-Stokes) method such as FLUENT can provide economic, mature, reliable, and robust numerical simulation algorithms for turbulent flows, which is suitable for engineering applications. And the functionalization of userdefined function (UDF) provided by some commercial CFD packages can overcome the shortcoming that there is no constitutive model for viscoelastic fluid in those commercial CFD codes for turbulent flow simulations. Thus, we have been then motivated to develop a new numerical simulation methodology for viscoelastic fluid flow based on the CFD code FLUENT package. For the next step of research that an inhouse subroutine embedded into FLUENT code through UDF functionalization has been benchmarked with simulations on laminar flows of viscoelastic fluid [5], the extension of

relaxation model is modified to account for the Reynolds stress equilibrium established by the presence of elasticity in the fluid. The numerical simulation results, including velocity profiles, turbulent flow characteristics, elastic stress and conformation field, show good agreements with published DNS results, which validates the newly established method on turbulent flows of viscoelastic fluid based on FLUENT software platform for engineering applications. INTRODUCTION Friction drag of turbulent flows of viscoelastic fluid can be significantly reduced compared with that of Newtonian fluid with the drag reduction (DR) rate up to 80%. Since Toms reported the turbulent drag-reducing effect in polymer solution flow for the first time [1], researchers have recognized great potentials of this phenomenon in energy saving for industrial application systems, and a large amount of experimental and numerical studies on viscoelastic fluid turbulent flows have 

†Corresponding author

1

Copyright © 2014 by ASME

methodology to simulating turbulent flows of viscoelastic fluid is presented in this paper. In the past decade, several attempts have been made to simulate turbulent flows of viscoelastic fluid based on RANS modeling method. Leighton et al. [6] first proposed a turbulent model for viscoelastic fluid flows based on FENE-P (finitely extensive nonlinear elastic-Peterlin) constitutive equation model. In their work, a Reynolds-stress transport equation model was presented based on an extended set of RANS equations combining with polymer stresses specified in terms of the mean polymer conformation tensor, and closures for unknown correlations were also developed. Their approach was validated in turbulent channel flow and showed encouraging results. However, the extreme complexity of 6 partial differential equations for Reynolds stresses and 6 more for polymer stresses in the set of governing equations with unclosed correlations in each equation limited its application in tackling practical engineering problems. In another attempt, Pinho et al. [7] developed a new RANS model for polymer turbulent drag-reducing flows by utilizing FENE-P dumbbell model based on a single-point k   turbulent model, and presented closures for turbulent correlations induced by viscoelasticity. But this model was only valid for low DR regime. An improved model [8] was subsequently developed by providing new closures for the Reynolds-averaged nonlinear term of the polymer conformation equation and the eddy viscosity, and the inclusion of direct viscoelastic contributions into the transport equations for turbulent kinetic energy (k) and its dissipation rate. The existing closures for the viscoelastic stress work model in [7] were also improved. This improved model was extended to moderate DR regime by validation against DNS results. However, the fact that the inherent turbulence isotropy of k   model can’t simultaneously predict mean velocity, turbulent kinetic energy and its dissipation rate as a result of the increase of turbulence anisotropy with DR confines the simulations on polymer turbulent flows at high DR level [9]. 2 Recently, Iaccarino et al. [10] presented a k    v  f

obtained for many quantities, such as DR rates, mean velocity profiles, polymer stress, turbulent kinetic energy and its dissipation rate, the predictions of viscoelastic eddy viscosity, polymer mean shear stress, the budgets of turbulent kinetic energy and the evolution equation for conformation tensor didn’t consist with DNS results well. Therefore, Masoudian et 2 al. [9] proposed a new improved k    v  f model with more accurate predictions by improving the existing closure for viscoelastic eddy viscosity, and developing new closures for nonlinear fluctuating terms in FENE-P constitutive equation and polymer stress work terms in the transport equations of k 2 and v . In this paper, an in-house subroutine modeling turbulent 2 drag-reducing flow of viscoelastic fluid by k    v  f

model introduced by Masoudian et al. [9] with several modifications is developed and embedded into the FLUENT code through utilizing the functionalization of UDF. Afterwards, this approach is validated against DNS results for fully developed turbulent channel flows of viscoelastic fluid over the whole range of DR. NOMENCLATURE a1, a2, a3 turbulent model constants turbulent model constants C1, C2 turbulent model constants C  , C C 1 , C 2

turbulent model constants

Cij CL CTij

p

components of conformation tensor turbulent model constant contribution to the advective transport of the conformation tensor by the fluctuating velocity field (1/s) viscoelastic contribution to the transport equation of  (m2/s4) turbulence energy redistribution process (1/s) Peterlin function height of upstream channel (m) turbulent kinetic energy (m2/s2) maximum extensibility of polymer molecule turbulent length scale (m) mean flow distortion term contained with the Oldroyd derivative of Cij (1/s) interactions between the fluctuating components of the conformation tensor and of the velocity gradient tensor (1/s) pressure (Pa)

Pk

turbulence production ( kg

Qp Re

viscoelastic stress work (m2/s3) Reynolds number

S

strain rate magnitude (1/s)

Ep f f(r) h k L Lt Mij

model for turbulent wall-bounded flows of viscoelastic fluid, which can reproduce the level of DR over the whole range. This model was an extension of k   model and included two additional equations – a transport equation for wallnormal fluctuating velocity variance ( v2 ) which dominates the scaling of turbulent viscosity near the wall and an elliptic relaxation equation for turbulence energy redistribution process (f) – to represent the turbulence anisotropy caused by the existence of solid walls without damping functions. A pseudo viscosity was introduced to model the polymer stresses directly with the combination of the mean velocity gradients, and the increased wall damping of the turbulent fluctuations was obtained by modifying the pressure-strain redistribution term in the equation of f. Although accurate predictions were

NLTij

2

m  s  ) 3


Sij Tt ui

v

strain rate tensor (1/s) turbulent time scale (s) velocity components (m/s) 2

ui

Wi

wall-normal fluctuating velocity variance (m2/s2) Weissenberg number

xi(x, y, z)

Cartesian coordinates

u j

p

s

 ij



x j

p

s



xi

p

 ij



x j

 ij

(2)

x j

equals 1000 kg/m3, p is the pressure,  ij (i = 1, 2, 3; j = 1, 2, s

3) is the viscous stress caused by the solvent defined as:

 ui

s

 ij   s 

polymer stress tensor (Pa)



 x j

stress tensor caused by Newtonian solvent

  xi 

u j

(3)

where  s is the Newtonian solvent viscosity and constantly

(Pa) t

ui

where ui is the velocity,  is the fluid density and constantly

Greek Symbols Kroneker symbol  ij

 ij

(1)

0

xi

p

equals 0.001 Pa  s . And  ij is the elastic stress induced by

 ij

Reynolds stress tensor (Pa)

  

p

turbulent model constant solvent viscosity ratio dissipation rate of turbulent kinetic energy (m2/s3) viscoelastic turbulent transport (m2/s3)

k ,     p

relaxation time (s) density (kg/m3) coefficient of artificial diffusion term dynamic viscosity of viscoelastic polymer

s

( Pa  s ) dynamic viscosity of solvent ( Pa  s )

t

turbulent (or eddy) viscosity ( Pa  s )

where L is the maximum extensibility of polymer molecule. cij in Eq. (4) is the conformation tensor of polymer molecule, for i  j , there exists cij  c ji , and its corresponding transport

t , p

viscoelastic eddy viscosity ( Pa  s )

equation is:

s

the elasticity, whose formulation is based on FENE-P model: p p (4)  ij  n ( x, t )  f  r  cij   ij   where n( x, t ) is the unit concentration and equals 1 when the concentration of polymer is homogeneous (assumed in this paper),  p is the viscosity of viscoelastic polymer,  is the

turbulent Prandtl numbers for k and 

relaxation time of viscoelastic fluid,  ij is the Kroneker symbol, f  r  is the Peterlin function described as: 2

f r 

2

kinetic viscosity of solvent (m /s)

uk

Superscripts and/or subscripts i, j, k, m, q Cartesian components (from 1 to 3) p polymer s Newtonian solvent t turbulent

cij xk

 cik

u j xk

 ckj

L 3

(5)

2

L  ckk

ui 1   f  r   ik ckj   ij  xk  

Reynolds averaging procedure Considering ui  U i  ui and

cij  Cij  cij ,

(6)

and

conducting several substitutions and Reynolds averaging for governing equations, we can obtain: U i 0 (7) xi

NUMERICAL SIMULATION PROCEDURES In order to facilitate the clear expression, in the following lower-case letters denote instantaneous quantities, upper-case letters or overbars denote Reynolds-averaged quantities and primes denote ﬂuctuating quantities.





 U i     uiu j  ijp (8)    s   x j x j  x j  p C  U j cij  u j U i  u   U k ij   Cik  Ckj   cik  ckj i    ij   uk xk  xk xk  xk  xk xk  p

U j

Governing equations In the present work steady flows are resolved. Thus for incompressible flow of viscoelastic fluid, the continuity and momentum equations in the Cartesian coordinate system are as follows:

U i P    x j xi x j

(9)

3


 ijp =

 p  L2  3  p L2  3 Cij   ij  + cij  2 2   L  Ckk   L   Ckk +ckk 

the equation of f. Their expressions are as follows and need closure. u  (20)  p   ijp i x j

(10)

where   uiu j is the Reynolds stress tensor and according to Boussinesq hypothesis it can be expressed as: 2 (11)   uiu j = ijt =2t Sij  k ij 3 herein t is eddy viscosity, k is turbulent kinetic energy and Sij is strain rate tensor defined as: 1  U U j Sij =  i  xi 2  x j

  

Qp  E p  2 s

C  70

,

Development of closures Except for the first term on the left hand side of Eq. (9), other terms respectively represent the mean flow distortion term contained with the Oldroyd derivative of Cij, the contribution to the advective transport of the conformation tensor by the fluctuating velocity field and the interactions between the fluctuating components of the conformation tensor and of the velocity gradient tensor, which are denoted by Masoudian et al. [9] as: U j U i (24) M ij  Cik  Ckj xk x k

CTij  uk

cij xk

u j

(25)

ui (26) xk xk According to the analysis of Masoudian et al. [9], the first term on the left hand side of Eq. (9) vanishes for fully developed channel flow, and the polymer stress can be computed by the extension of the polymer molecule via Ckk, thus only the extension of the polymer molecule was taken into consideration rather than all of the components in conformation tensor as Iaccarino et al. did in [10], which can be obtained as the trace of Reynolds-averaged conformation by combining Eq. (9) and Eq. (10).  1 L2  3 (27) Ckk   0 M kk  NLTkk  CTkk   3  2  L  Ckk  Note that the second term on the right hand side of Eq. (10) can be neglected in the light of the comparison with the DNS data conducted by Masoudian et al. [9]. NLTij  cik

Lt and Tt are turbulent length scale and time scale, respectively. Their definitions are: 14 k3 2   s3   (18) , C    Lt  CL max        k   (19) Tt  max  , 6 s      is kinetic viscosity of Newtonian solvent defined

E p and  yyp

 ui     L2  3 L2  3  Cij   (22) cqq   2 2   L  3 xk x j  xk  L  Ckk L  cmm   2

C 1  1.4 1  0.045 k v2  , C 2  1.9 ,   C  0.22 , CL  0.23 ,  k  1.0 ,    1.3 .

P 5 v 2 k 2 3  v 2 k 2 f   pyy (16)   C1  1  C2 k  x j x j k Tt Tt

as  s  s  .  p , Qp ,

p

The turbulent model constants in the above equations are the same as those in Newtonian form: C1  1.4 , C2  0.3 ,

where  is the dissipation rate of turbulent kinetic energy, Pk is turbulence production defined as: (17) Pk  2 t Sij Sij

where  s

(21)

the wall-normal direction. The eddy viscosity is denoted as: (23) t   C v2Tt

 v 2       v 2    kf  6  v 2    s  t     p , yy   Q p , yy x j k x j   k  x j  (15) f  L2t

x j

The subscript yy in  p , yy , Qp, yy and  yyp means the part in

(12)

Turbulence model 2 The k    v  f model was proposed by Durbin [11] and subsequent researches have been performed to improve this model for the enhancement of numerical stability, resulting in several versions. The version adopted in this paper is derived from the code-friendly one introduced by Lien and Durbin [12]. Moreover, additional terms appear in the model equations for the presence of viscoelasticity.  t  k  k   U j  Pk      s     p   Qp (13)   k  x j  x j x j  t     C 1 Pk  C 2    U j     s     E p (14)  Tt    x j  x j x j 

U j

 ijp ui

are the terms

introduced due to the existence of viscoelasticity, and respectively, represent the viscoelastic turbulent transport, the viscoelastic stress work, the viscoelastic contribution to the transport equation of  and the viscoelastic contribution to

4

 ckj


Comparison of the first three terms on the left hand side of Eq. (27) by Masoudian et al. [9] indicated that the term of CTkk can be neglected. And the closures for the left two terms were developed by Masoudian et al. [9] as: 2 L2  C t , p  dU  (28) M kk  2 2 kk   L  3  p  dy  NLTkk  a2 M kk

where t , p

t s   p

where the expressions for t , p , M kk , NLTkk ,  p , E p and

 p, yy are given in Eqs. (28) - (30) and Eqs. (32) - (34). The present model is embedded into the FLUENT code through UDF functionality based on the low Reynolds number k   model provided by FLUENT software. In UDF, the general user-defined scalar transport equation is shown in Eq. (42), in which from left to right there are four terms to be customized: transient term, convection term, diffusion term and source term, respectively.      (42)    ui     S t xi  xi  where  is an arbitrary scalar,  is diffusion coefficient and S is the source term. The transient term is omitted due to the consideration of steady flow mentioned above, and each term in the governing and model equations can be categorized into the left three terms in Eq. (42). Afterwards, the customizations of these terms are realized by writing corresponding functions with C language and then connecting the functions with interfaces through the corresponding settings in the code. The accuracy of the present model is assessed by simulating a fully developed turbulent channel flow of viscoelastic fluid with computational domain size of Lx  Ly  Lz  10h  2h  5h , in which Lx, Ly, and Lz represent the sizes in the streamwise, wall-normal and spanwise directions, respectively; h is the half-height of the channel set to be 0.1 m. The flows are characterized by Reynolds number and Weissenberg number Re  U h   s   p 

(29)

is viscoelastic eddy viscosity introduced by

Iaccarino et al. [10] to model the elastic stress in Eq. (8), and Masoudian et al. [9] developed the closure for it as: L2  C L2  3 t , p   p 2 kk  a1 L2 Wi 2 t (30) L 3 L  Ckk And then elastic stress in Eq. (8) is denoted as: (31)  ijp =2t , p Sij Following the analysis of Masoudian et al. [9], the terms of Qp , Q p , yy and  yyp can be neglected, and the left terms introduced in the presence of viscoelasticity were developed as:  L2  3 p  p 2 NLTkk (32) 2  L  Cmm C 1 p (33) Ep  Tt

L2  3 kf (34) L2  Ckk The coefficients that appear in the above equations are: a1=0.02, a2=0.16, a3=0.15.

 p, yy  a1a3 L

Wi  U2

Summary of the model and implementation details Based on the closures in the above subsections, the final form of the governing and model equations are given below. U i 0 (35) xi  U i    s  t  t , p   x j   t  k  k   U j  Pk      s     p   k  x j  x j x j 

U j

U i P    x j xi x j

P 5 v 2 k 2 f 2 3  v 2 k   C1  1  C2 k  k x j x j Tt Tt

M kk  NLTkk 

 1 L2  3 Ckk   0 3 2   L  Ckk 

  p  (both are based on friction velocity

 defined as    s



s

  p  with constant value of 0.9

for each case simulated in this paper. And the parameters of all cases are the same as those of DNS provided by Masoudian et al. [9], which are shown in Table I. In order to enhance the numerical stability, an artificial diffusion term ( 2 Ckk ) is

(36)

introduced in Eq. (41) with the value of  U h ~ O(102 ) which is also shown in Table I.  is the coefficient of artificial diffusion term. Table I Parameters for each case of simulated channel flow  Case DR(%) L2  Re Wi

(37)

 C 1 Pk  C 2     Tt x j x j

f  L2t

s

U ). Another rheological parameter is solvent viscosity ratio

 t      s      E p (38)    x j    v 2       v 2  U j   kf  6  v2    s  t     p , yy (39) x j k x j   k  x j 

U j



395 900 18 25 0.9 8×10-6 100 0.9 395 900 37 8×10-6 100 0.9 395 3600 51 1×10-5 -5 395 100 0.9 14400 63 1×10 Non-uniform collocated mesh is adopted in the computational domain with 64 grids in each direction. The grid size along the direction normal to the top and bottom walls is gradually increased with a constant factor of 1.08912, A B C D

(40) (41)

5


and the meshes along the streamwise and spanwise directions are uniform. Double precision is employed for the calculations in this paper. SIMPLE scheme is adopted to solve the coupling of velocity and pressure; standard scheme is utilized to discretize the pressure equation; QUICK scheme is used to discretize the equations of momentum, turbulent kinetic energy, dissipation rate of turbulent kinetic energy, wallnormal fluctuating velocity variance, turbulence energy redistribution process and polymer extension. The periodic boundary conditions in the streamwise and spanwise directions with constant pressure difference in the streamwise direction and no-slip boundary condition for the top and bottom walls are used in the simulation, and boundary conditions for the turbulent scalars at the walls are the same as those of Newtonian form [12], while the same Dirichlet boundary condition for Ckk as Iaccarino et al. introduced in [10] is adopted in this paper.

near wall region possesses large polymer stress, which consists with the distribution of polymer elongation.

RESULTS AND DISCUSSIONS For the facilitation of comparison with DNS data provided by Masoudian et al. [9], all the predicted results are given in the dimensionless form which are normalized by the friction velocity ( U ) and viscous length ( xi  xi U   s   p  ). 

The predictions of k  and v2 for different cases are shown in Fig. 1. The normalized turbulent kinetic energy is underestimated for all cases, and the discrepancy increases with DR, which is probably caused by the shortcoming of FENE-P model as Ptasinski et al. analyzed in [2]. However, the upward shift of the peak location of k+ with increasing DR and 

the satisfactory agreement with DNS data for v2 are captured. The predicted normalized dissipation rates of turbulent kinetic energy accompanied by DNS data for different cases are depicted in Fig. 2. The predictions agree well with DNS data near and away from the wall for LDR (case A) and MDR (case B), while for HDR (case D), the prediction is consistent with DNS data far from the wall, but overestimated near the wall. Moreover, the peak location of   shifts away from the wall as DR increases. The polymer extension normalized by polymer maximum extensibility predicted by the present model for LDR (case A) and HDR (case D) are shown and compared with DNS data in Fig. 3. The maximum elongation of polymer molecule is limited to about 40% of the maximum extensibility for both cases, and the region of large extension is confined to be close to the wall. Besides, the predictions comply with DNS data, as shown in Fig. 3. In Fig. 4 the predicted normalized polymer stresses for the cases with different DR are presented, as well as the comparison with DNS data. The predictions are in accord with DNS data with only a small difference for HDR (case D). The

Fig. 1 Comparison between predictions (lines) and DNS data 

(symbols) of normalized turbulent kinetic energy and v2 for (a) case A (LDR), (b) case B (MDR), and (c) case D (HDR)

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underestimated for other cases with increasing difference as DR increases. Besides, the profiles in the viscous sublayer for all cases comply with the linear distribution, and the mean velocity of viscoelastic fluid far from the wall increases with increasing DR by comparing with that of Newtonian fluid. The upward shift of the log-law layer indicates the thickening of viscous sublayer, which is explained by Lumley [13] as the drag-reducing mechanism, and the slope of the profile in the log-law layer remains the same with that of Newtonian fluid for LDR (case A) and increases with DR, indicating the occupation of the central region of the channel by the buffer layer for HDR (case D). These observations are all in qualitative agreement with DNS data.

Fig. 2 Comparison between predictions (lines) and DNS data (symbols) of normalized dissipation rate of turbulent kinetic energy for case A (LDR), case B (MDR), and case D (HDR)

Fig. 4 Comparison between predictions (lines) and DNS data (symbols) of normalized polymer stresses for case A (LDR), case B (MDR), and case D (HDR) CONCLUSIONS An in-house subroutine modeling turbulent drag-reducing flow of viscoelastic fluid described by FENE-P model is developed and embedded into the FLUENT code through utilizing the functionalization of UDF. This approach is based

Fig. 3 Comparison between predictions (lines) and DNS data (symbols) of polymer extension normalized by polymer maximum extensibility for cases A and D Figure 5 illustrates the comparison between the predictions and DNS data of normalized shear stresses, including Reynolds shear stress, polymer shear stress and the stress caused by Newtonian solvent, for different cases. The predicted shear stresses show both qualitative and quantitative agreement with DNS data for LDR (case A) and MDR (case B), while for HDR (case D), the results of the present model predict the polymer shear stress well, but overestimate Reynolds shear stress. Moreover, with the increase of DR, Reynolds shear stress decreases and polymer shear stress increases to be comparable to Reynolds shear stress, which conforms to DNS results. The predictions and DNS results for normalized mean velocity profile of viscoelastic fluid are shown in Fig. 6, as well as the predicted results of Newtonian fluid. The predictions are in consistence with DNS data well for case A, but

2 on the k    v  f model introduced by Masoudian et al. [9] with several additional modifications, and validated against DNS results for fully developed turbulent channel flows of viscoelastic fluid over the whole range of DR. The predictions of the present model agree well with DNS data in terms of velocity profiles, turbulence physical quantities, polymer elastic stress and polymer extension. For the next step, the present model will be improved by modifying the model constants and improving the closures for better accuracy, especially for the discrepancies of the predictions on velocity profile and turbulent kinetic energy.

ACKNOWLEDGMENTS This work is supported by National Natural Science Foundation of China (51276046), Foundation for Innovative

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Fig. 6 Normalized mean velocity profiles in wall coordinates for Newtonian and viscoelastic fluids with DNS data Research Groups of the National Natural Science Foundation of China (51121004), Specialized Research Fund for the Doctoral Program of Higher Education of China (20112302110020). The authors would like to thank the enthusiastic help of all members of Complex Flow and Heat Transfer Laboratory of Harbin Institute of Technology. REFERENCES [1] Toms, B. A., 1948, “Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers”, In Proceedings of the 1st International Congress on Rheology, pp. 135-141. [2] Ptasinski, P. K., Boersma, B. J., Nieuwstadt, F. T. M., Hulsen, M. A., Van den Brule, B. H. A. A. and Hunt, J. C. R., 2003, “Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms”, Journal of Fluid Mechanics, 490(1), pp. 251-291. [3] Dimitropoulos, C. D., Dubief, Y., Shaqfeh, E. S., Moin, P. and Lele, S. K., 2005, “Direct numerical simulation of polymer-induced drag reduction in turbulent boundary layer flow”, Physics of Fluids, 17, 011705. [4] Cai, W. H., Li, F. C. and Zhang, H. N., 2010, “DNS study of decaying homogeneous isotropic turbulence with polymer additives”, Journal of Fluid Mechanics, 665, pp. 334-356. [5] Zheng, Z. Y., Li, F. C. and Yang, J. C., 2013, “Modeling Asymmetric Flow of Viscoelastic Fluid in Symmetric Planar Sudden Expansion Geometry Based on UserDefined Function in FLUENT CFD Package”, Advances in Mechanical Engineering, 2013, 795937. [6] Leighton, R., Walker, D. T., Stephens, T. and Garwood G., 2003, “Reynolds stress modeling for drag reducing viscoelastic flows”, in: 2003 Joint ASME/JSME Fluids Engineering Symposium on Friction Drag Reduction, Honolulu, Hawaii, USA.

Fig. 5 Comparison between predictions (lines) and DNS data (symbols) of normalized shear stresses for (a) case A (LDR), (b) case B (MDR), and (c) case D (HDR)

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[7] Pinho, F. T., Li, C. F., Younis, B. A. and Sureshkumar, R., 2008, “A low Reynolds number turbulence closure for viscoelastic fluids”, Journal of Non-Newtonian Fluid Mechanics, 154(2), pp. 89-108. [8] Resende, P. R., Kim, K., Younis, B. A., Sureshkumar, R. and Pinho, F. T., 2011, “A FENE-P k-ε turbulence model for low and intermediate regimes of polymer-induced drag reduction”, Journal of Non-Newtonian Fluid Mechanics, 166(12), pp. 639-660. [9] Masoudian, M., Kim, K., Pinho, F. T. and Sureshkumar, R., 2013, “A viscoelastic turbulent flow model valid up to the maximum drag reduction limit”, Journal of NonNewtonian Fluid Mechanics, 202, pp. 99-111.

[10] Iaccarino, G., Shaqfeh, E. S. and Dubief, Y., 2010, “Reynolds-averaged modeling of polymer drag reduction in turbulent flows”, Journal of Non-Newtonian Fluid Mechanics, 165(7), pp. 376-384. [11] Durbin, P. A., 1991, “Near-wall turbulence closure modeling without “damping functions” ”， Theoretical and Computational Fluid Dynamics, 3(1), pp. 1-13. [12] Lien, F. S. and Durbin, P. A., 1996, “Non-linear k-ε-v2 modeling with application to high-lift”, in: Proceedings of the Summer Program 1996, Stanford University, pp. 5-26. [13] Lumley, J. L., 1969, “Drag reduction by additives”, Annual Review of Fluid Mechanics, 1, pp. 367384.

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