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This model is also referred to as a hybrid RANS/LES modelin that it combines the advantages of RANS and LES methodologiesand can resolve the flow at less computational cost as well as withhigh accuracy [20]. Since its inception the method has been appliedto a range of configurations including simple shapes such ascylinders, spheres and aircraft forebodies, in addition to complexgeometries including fighter aircraft [21]. The fluid flow in a hu-manmouth–throat was also studied by DESmodel and good agree-ment with the LES data was obtained [22].It has been proved that eccentric agitation can improve the mix-ing in the laminar, transitional and turbulent flow regime. Theeccentric position of the impeller lead to mixing time reductionwith increasing the impeller eccentricity [23–27]. However, theseinvestigations were all performed by experimental approaches. Inthe present work, an attempt is made to characterize the mixingof an inert scalar in stirred tanks equipped with four pitched-bladeturbines using the DES model. To the best of the authors’ knowl-edge, no comprehensive investigation of mixing characteristicsand assessment of mixing time has been performed by this modelso far. The authors’ previous work [28,29] showed that DES modelcan predict the hydrodynamics of stirred tanks as accurate as LESdoes, can provide considerably improved predictions of the meanand turbulent quantities compared with those obtained by theRANS approach. This suggests that a better prediction of mixingtime by DES model should be expected.The paper is organized as follows: Section 2 presents the stirredsystem geometry and the operating conditions. In Section 3, the governing equations, computational grid and numerical solutionmethod are described. In the following Section 4, the mixing pat-terns are discussed and the mixing times are compared with thePLIF result. Finally, the main conclusions are summarized inSection 5.2. Stirred tank configurationConfigurations of the investigated stirred tank and the impelleris given in Fig. 1. The system is an unbaffled, flat-bottomed, cylin-drical tank (of diameter T = 0.15 m) agitated by a down-pumpingpitched-blade turbine with four blades (named as PBT-4), eachangled at 30 to the horizontal and attached to a hub that ismounted on the impeller shaft with a diameter d = 0. 008 m. Thethickness and width of the impeller blade are tb = 0. 001 m andwb = 0. 01 m, respectively. The tankwas filledwithwater to a heightof H = T. In the numerical simulations, the fluid is assumed to beincompressible with a density of q =1 103kg m 3and a dynamicviscosity of l =1 10 3Pa s. As for the concentric configuration,the shaft of the impeller was concentric with the axis of the tank.For the eccentric agitation, the impeller was positioned at threeoff-axis locations, i.e. at e =2E/T = 0.2, 0.3 and 0.5 fromthe tank axis.For all cases, the diameter of the impeller is D = T/3 and the impelleroff-bottom clearance is C = T/3. The impeller rotates clockwise (asviewed from the above) with a speed of N =5s 1, which corre-sponds to a Reynolds number of Re ¼ qND2l ¼ 1:25 104.3. Methodology3.1. Mathematical formulation of DES modelThe turbulent fluid flow in the stirred tank is predicted by theDES model, which is formulated by replacing the distance functiond in the one-equation Spalart–Allmaras (S–A) model [30] with amodified distance function:~ d ¼ minfd; CDESDgð1Þwhere CDES = 0.65 is the model empirical constant and D is the larg-est dimension of the grid cell in question. This modification of theS–A model changes the interpretation of the model substantially.In regions close to the wall, where d < CDESD, it behaves as a RANSmodel. Away from the wall, where d > CDESD, it behaves in a Smago-rinsky-like manner and is changed to the LES model. The governingequation of DES model can be given as follows:@~ m@tþhuii@~ m@xi¼ Cb1ð1 ft2Þe S~ mþ 1r@@xiðmþ ~ mÞ@~ m@xi þ Cb2@~ m@xi 2() Cw1fw Cb1c2ft2 ~ md 2ð2Þwhere e S is modified vorticity, m is molecular viscosity, ~ m is modifiedturbulent viscosity that linked to the turbulent viscosity mt and awall function fv1 by~ m ¼ mt=fv1; f v1 ¼ v3v3 þ C3v1; v ¼~ mmð3ÞThe modified vorticity e S is defined in terms of the magnitude ofvorticity S in the following equation:e S ¼ S þ~ mc2d2 fv2; f v2 ¼ 1 v1 þ vfv1; f v3 ¼ ð1 þ vfv1Þð1 fv2Þvð4ÞThe function ft2 and fw in Eq. (2) is defined as:ft2 ¼ Ct1 expð Ct2v2Þð5Þ DES model is proposed based on the one-equation S–A model.Since then, some variants, such as the DES model based on theSST k–x and Realizable k–e model, were proposed by [31,32],respectively. No matter what kind of RANS model was used, theprinciple was the same. In this paper, DES model proposed bySpalart et al. [33], which is referred to as the standard edition,was adopted. The closure coefficients in the governing equationof DES model are given as follows: r = 2/3, c = 0.41, Cb1 = 0.1335,Cb2 = 0.622, Cw1 = Cb1/k2+(1 + Cb2)/r, Cw2 = 0.3, Cw3 =2, Cv1 = 7.1,Ct1 = 1.1 and Ct2 =2.3.2. Simulation of the mixing timeMixing time was predicted by using a virtual scalar tracer andmonitoring the scalar concentration variations with the time. Inthe present work, the origin of coordinate system coincide withthe projective point of the impeller shaft on the bottom plane ofthe stirred tank (see Fig. 1).
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