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As shown in Fig. 2, the scalar tracerwas feeded from the top surface at a point, which is located inthe plane y = 5 mm, with a distance of 3 mm to the stirred tankwall. The concentration of the tracer was initialized as 1 in thefeeding region, and in the rest region as 0. A total of 15 monitoringpoints 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= x–E),respectively. The tracer feeding and monitoring points used inthe numerical simulations match exactly those used in the exper-iments for the purpose of validation.In the simulation of a passive scalar transport, the instanta-neous velocity affects the scalar evolution, while the passive scalarhas no effect on the flow characteristics, which means there is onlyone way interaction with the flow field. The scalar field is mathe-matically decoupled from the dynamical equations that governthe flow field and the solution of the scalar field. Therefore, a fullydeveloped turbulent flow was used to solve the transient passivescalar transport equation.The mixing process was simulated by solving the scalar trans-port Eq. (7) to follow the transient concentration variations ofthe scalar tracer:@ðq cÞ@tþ r ðq u cÞ¼ r ½ðCl þ Ct Þgradðq cÞ þ Sc ð7Þ where c is the scalar variable, u is the velocity vector, Sc is the scalarsource term, Cl is themolecular diffusivity of the scalar and Ct is theeddy diffusivity which is related to the eddy viscosity by the follow-ing relationship:Ct ¼ ltrtð8Þwhere rt is the turbulent Schmidt number and is used to define theratio of eddy viscosity to eddy diffusivity, which is in the range0.7–0.9 depending on the type of flow. For the turbulent flow in stir-red tanks, it was taken as 0.7 in the present study.3.3. Computational grid and modeling methodComputational model was solved using the commercial CFDsolver Fluent 6.3. Considering the unsteady nature of the flow,the whole stirred tank was simulated. The determination of thegrid resolution is a critical point for numerical simulation. In thepresent work, the grids were prepared by the pre-processor Gam-bit 2.3 according to the guide of Spalart [34,35] and a non-uni-formly distributed hybrid unstructured mesh consist of about500,000 nodes was used in total. Much attention had been takento put more mesh points in the regions of high gradient aroundthe blades and discharge region, where about 220,000 nodes wereemployed. Along the impeller width, 25 nodes were assigned withthe minimal grid length equals to 0.4 mm, which equals to 0.008D.A similar grid resolution (970, 997 cells for a stirred tank with andiameter T = 0.3 m and Re = 4.17 104) was employed by Zadghaf-fari et al. [17] in their LES study of the turbulent flow and mixing ina stirred tank driven by a Rushton turbine, and satisfactory resultswere obtained. The grid used used here is also finer than the locallyrefined grid (0.023D) used by Revstedt et al. [36], who reported agood LES prediction of the turbulent flow. This implies that the gridresolution used in the present work is adequate to resolve the tur-bulent flow accurately.
The pressure-based Navier–Stokes algorithm was used for thesolution of the model with implicit solver formulation where theabsolute velocity formulation was adopted. To ensure smoothand better convergence, initially the k–e computation was per-formed until the steady state flow field is obtained. Subsequently,the result of the steady-state computation was used as the initialsolution to perform the unsteady DES computation. For modelingthe impeller rotation, the multiple reference frame (MRF) methodwas used for the k–e computation. Then it was switched to the fullytransient sliding mesh (SM) method for the DES computation. Inthis method, the rotation of the impeller is explicitly taken into ac-count and two fluid zones are defined: an inner rotating cylindricalvolume centered on the impeller (rotor region) and an outer sta-tionary zone containing the rest of the tank (stator region). In thepresent study, the boundary of the rotor region was positioned atr = 0.075 m and 0.03 m < z < 0.07 m (where z is the axial distancefrom the bottom of the tank). For more details about this method,the readers are referred to the literature [37] and [38].The initial condition for each simulation was that of still liquid.A flat liquid surface was assumed at the liquid surface by setting allthe shear stress equal to zero. No-slip boundary conditions wereused at the impeller blades, the shaft and the tank walls. For thesteady-state k–e simulation, the standard wall functions were usedto solve the near-wall flow. The SIMPLE algorithm was performedto couple velocities and pressure terms. The continuity equation,momentum conservation equation and k–e equation were all dis-cretised using the second order upwind scheme to obtain a high-precision result. For the DES computation, The PISO discretizationscheme was adopted to couple velocities and pressure terms. ThePRESTO algorithm was adopted for the spatial discretization ofthe pressure item because it is optimized for swirling and rotating flow. The second order implicit scheme was used for time discret-ization. The bounded central differencing scheme was adopted forspatial discretization of momentum and the modified turbulentviscosity equation. This scheme blends the pure central differenc-ing scheme with first- and second-order upwind schemes, andcan reduce the unphysical oscillations in the solution field inducedby the central differencing scheme. The momentum conservationequation and concentration equation were all discretized usingthe second order upwind scheme to obtain a high-precision result.The time step and number of iterations are crucial to the tran-sient DESmodeling. The time stepmust be small enough to capturethe flow features induced by the motion of the impeller. Further-more, it also must be considered with the grid to ensure a stableand converged solution. A time step of Dt =1 10 3s was adoptedhere, i.e. each complete impeller revolution requires 200 iterations.Given the time step and the mesh used in the present work, thecourant number is less than two. Within each time step, a maxi-mum of 40 iterations were performed and the solution was consid-ered to be fully converged when the normalized residuals of allvariables were less than 1 10 4. All the simulations were con-ducted using the commercial CFD software Fluent 6.3 (FluentInc., USA) on a HP workstation XW6200 with 4 GB memory. Withthe adopted spatial and temporal discretization, it took about30 CPU hours to simulate 1 s of actual flow time. spatial and temporal evolution of the scalar concentration andaccordingly give the mixing characteristics during the mixingstages at various positions. From these plots we can see that DEScan capture the fully transient mixing process in the stirred tank.For concentric agitation, the concentration field in the stirred tankis almost symmetric. Under eccentric agitation configuration, thesymmetry of the stirred tank is broken, and a faster mixing processis achieved. Analysis of time evolution of scalar concentration forother cases, i.e., e = 0.2, e = 0.3, also indicates that the DES modelcan capture the transient mixing process in the stirred tank. Forreason of simplicity, these concentration contour plots are not pre-sented here.4.2.
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