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    method are described. In the following Section 4, the mixing pat
    terns are discussed and the mixing times are compared with th
    PLIF result. Finally, the main conclusions are summarized in
    Section 5.
    2. Stirred tank configuration
    Configurations of the investigated stirred tank and the impelle
    is given in Fig. 1. The system is an unbaffled, flat-bottomed, cylin
    drical tank (of diameter T = 0.15 m) agitated by a down-pumping
    pitched-blade turbine with four blades (named as PBT-4), each
    angled at 30 to the horizontal and attached to a hub that i
    mounted on the impeller shaft with a diameter d = 0. 008 m. Th
    thickness and width of the impeller blade are tb = 0. 001 m and
    wb = 0. 01 m, respectively. The tankwas filledwithwater to a heigh
    of H = T. In the numerical simulations, the fluid is assumed to b
    incompressible with a density of q =1  103
    kg m3
    and a dynami
    viscosity of l =1  103
    Pa 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 thre
    off-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 impelle
    off-bottom clearance is C = T/3. The impeller rotates clockwise (a
    viewed from the above) with a speed of N =5s
    1
    , which corre
    sponds to a Reynolds number of Re ¼ qND2
    l ¼ 1:25  104
    .
    3. Methodology
    3.1. Mathematical formulation of DES model
    The turbulent fluid flow in the stirred tank is predicted by th
    DES model, which is formulated by replacing the distance function
    d in the one-equation Spalart–Allmaras (S–A) model [30] with
    modified distance function:
    ~ d ¼ minfd; CDESDgð1
    where CDES = 0.65 is the model empirical constant and D is the larg
    est dimension of the grid cell in question. This modification of th
    S–A model changes the interpretation of the model substantially
    In regions close to the wall, where d < CDESD, it behaves as a RAN
    model. Away from the wall, where d > CDESD, it behaves in a Smago
    rinsky-like manner and is changed to the LES model. The governin
    equation of DES model can be given as follows:DES model is proposed based on the one-equation S–A model.
    Since then, some variants, such as the DES model based on the
    SST k–x and Realizable k–e model, were proposed by [31,32],
    respectively. No matter what kind of RANS model was used, the
    principle was the same. In this paper, DES model proposed by
    Spalart et al. [33], which is referred to as the standard edition,
    was adopted. The closure coefficients in the governing equation
    of 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 time
    Mixing time was predicted by using a virtual scalar tracer and
    monitoring the scalar concentration variations with the time. In
    the present work, the origin of coordinate system coincide with
    the projective point of the impeller shaft on the bottom plane of
    the stirred tank (see Fig. 1). As shown in Fig. 2, the scalar tracer
    was feeded from the top surface at a point, which is located in
    the plane y = 5 mm, with a distance of 3 mm to the stirred tank
    wall. The concentration of the tracer was initialized as 1 in the
    feeding region, and in the rest region as 0. A total of 15 monitoring
    points in the same plane were selected in regions of different agi-
    tation intensity, and the axial and radial positions were (z
    =z/
    T = 0.2,0.5,0.9) and (x
    =2x0
    /T = ±0.9, ±0.5,0, where x0
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