Assessment of standard k–ε, RSM and LES turbulence models in a baffled stirred vessel agitated by...

Chemical Engineering Science 63 (2008) 5468 -- 5495

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Chemical Engineering Science

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Assessment of standard k–�, RSMand LES turbulencemodels in a baffled stirred vesselagitated by various impeller designs

B.N. Murthy, J.B. Joshi∗

Department of Chemical Engineering, Institute of Chemical Technology, University of Mumbai, Matunga, Mumbai 400 019, India

A R T I C L E I N F O A B S T R A C T

Article history:Received 25 September 2007Received in revised form 22 May 2008Accepted 24 June 2008Available online 2 July 2008

Keywords:Stirred vesselCFDDisc turbinePitched blade turbineHydrofoilRANSLESLDA

In the present work, laser-doppler anemometry measurements as well as CFD simulations have beenperformed for the flow generated by various impellers, namely disc turbine (DT), a variety of pitchedblade down flow turbine impellers varying in blade angle (Standard PBTD60, 45 and 30) and hydrofoil(HF) impeller. The tank was fully baffled, and the flow regime was turbulent. The objective of the presentwork was to carry out a detailed investigation of the predictive capabilities of the various turbulencemodels, i.e. the standard k– � model, Reynolds-stress transport model (RSTM) and large eddy simulations(LES). In case of LES, effect of subgrid scales on the resolved scales has been modeled by dynamic oneequation subgrid-scale model. The simulated values of the mean axial, radial and tangential velocitiesalong with the turbulent kinetic energy have been compared with the measured LDA data. It has beenidentified that the present SGS LES model performs well for predicting all the flow variables. Whereas,RSM and standard k– � model underpredict the turbulent kinetic energy profiles significantly in theimpeller region. RSM can capture well all the mean flow characteristics and the standard k– � model failsto simulate the mean flow associated with the strong swirl. Energy content of the precessional vortex hasbeen quantified for all the five impeller designs. Intermediate frequencies inbetween the mean circulationand the precession instability have been identified having a non-dimensional frequency of 0.04 to 0.07for all the impeller designs under consideration.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Stirred vessels are extensively used in the chemical process in-dustries over a wide range of applications. In majority of cases, theflow field in baffled stirred vessels is highly turbulent hence it isthree-dimensional, complex and chaotic in nature. During the last25 years, there have been continuous efforts on understanding theseflows using both sophisticated experimental and computational fluiddynamics tools.

In view of the above, in the past, the flow generated by disc tur-bine (DT) and pitched blade turbines (PBT) have been subjected todetailed experimental and computational studies. The initial objec-tives were preliminary such as the estimation of gross flow param-eters and the average flow field. However, during the past 10 years,the objectives have become deeper such as the characterization ofturbulent flow, flow instabilities, etc., and tailoring the impeller de-sign so as to get the desired flow field. Majority of the work hasbeen devoted to the modeling of the mean and the turbulence flow

* Corresponding author. Tel.: +912224145616; fax: +912224145614.E-mail address: [email protected] (J.B. Joshi).

0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2008.06.019

field and its comparison with the experimental flow measurements.However, the comparison of predictive capabilities of various turbu-lence models has been preliminary even for the flow generated byconventional impeller designs like DT and PBT. It is well known thatthe k– � model assumes the isotropy for turbulence, and therefore,anisotropic models such as Reynolds stress model (RSM) and thelarge eddy simulation (LES) model are being recommended for thesimulation of complex three-dimensional flows. However, RSM hasgot shortcomings like, non-universal model parameters, numericaldifficulties and is computationally expensive by an order of magni-tude as compared to the k– � model. Further, the RSM model doesnot capture the time dependent nature of the flow. This limitation isovercome by the LES approach. During the last few years, the abil-ity to resolve all but the smallest turbulence scales using LES hasbecome more viable. LES can potentially produce more accurate re-sults by modeling only the smallest scales, which tend to be moreisotropic, while fully resolving the turbulence at the larger scales.

From a practical point of view, the use of LES or DNS as designtools is far from easy due to the high computational costs associatedwith them. For these reasons, it is envisioned that the RANS equationsassociated with turbulence modeling will be the main CFD tool usedby the practitioners and part of the research community, at least in

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B.N. Murthy, J.B. Joshi / Chemical Engineering Science 63 (2008) 5468 -- 5495 5469

the near future. As a result of this trend, there is a need to improvethe accuracy and reliability of the solutions of turbulent flow fieldsobtained from the RANS equations. In light of this discussion, theRANS models can be improved by selecting the model parameterson the basis of the understanding of mean and the turbulent flowfields. For this purpose, it is desirable that, an extensive benchmarkdatabase is made available with equally extensive exercise of thecomparative performance of these turbulencemodels. Therefore, oneof themain objectives of the presentwork is to present a comparativestudy of k– �, RSM and LES with inhouse LDA measurements.

2. Previous work

As regards to standard k– � modeling, extensive work (Ranadeet al., 1992; Jenne and Reuss, 1999; Brucato et al., 1998; Jaworski andZakrzewska, 2001; Joshi and Patwardhan, 1999; Nere et al., 2001,2003; Aubin et al., 2004) has been reported in the published litera-ture. Hence, the present work reviews the past computational workon the flow generated by DT and PBT using RSM and LES models.

2.1. Reynolds stress modeling

There have been two attempts partially focused on the RSM forthe flow generated by Rushton turbine in a baffled stirred vessel.

Table 1Previous work on LES of the stirred tank

Authors, Impeller-SGSmodel, Grid size

System investigated Flowvariable

Region of comparison Remarks

Eggels (1996), Adaptive force fieldtechnique to model the actionof impeller, Standard Smagorinskymodel, 1.73 × 106, 13.8 × 106

Rushtine turbine, T=0.48m, H=T , C/T = 0.33, D/T = 0.33, N =250 rpm, Re = 107, 000

〈u1〉 0.144 < z/w<0.176 Qualitative flow was presented. Comparison was givenonly in the impeller region. Tangential velocity and turbu-lent kinetic energy comparison was missing. Energy bal-ance was missing.

u1′ 0.37 < r/R <0.780 < r/R <1z = 0

〈u2〉 0 < z/T <0u2′ r/R = 0.34

Revstedt et al. (1998), Impellermotion was modeled by specify-ing momentum source term


〈u1〉 z/T = 0.33 Qualitative flow features were presented. Flow and pump-ing numbers have been in good agreement with exper-imentally measured values. Mean axial velocity was inagreement with the experimental data all over the do-main. The mean radial velocity and turbulent kinetic en-ergy were compared only in the impeller region. Spectralanalysis was reported.

0 < r/R <1〈u2〉 0.25 < z/T <0.75

0 < r/R <1No sgs model, 32,768, 262,144 K z/T = 0.33

0 < r/R <1

Derksen and Van den Akker(1999), Adaptive force fieldtechnique to model the action ofimpeller, Standard Smagorinskysgs model, 6 × 106


〈u1〉 0.144 < z/w<0.176 Detailed information was given for phase resolved veloc-ities, kinetic energy and dissipation rate distribution allover the domain. The vortex paths, both above and belowthe impeller disk, were predicted correctly. Axial velocitycomparison was not given. However, comparison was re-stricted only to the impeller region. Power number waspredicted well.

r/R = 0.5, 0.65〈u3〉 0.144 < z/w<0.176

r/R = 0.5, 0.65K 0.144 < z/w<0.176

r/R = 0.5, 0.65

Bakker et al. (2000), Sliding meshtechnique, standard smagorinsky'ssgs model, 5,27,000 for PBTD7,63,000 for DT

PBTD: T = 0.292m, D/T =0.35 C/T = 0.46, N = 60 rpm, DT:T = 0.202m, C/T = D/T = 1

3 , N =290 rpm

Realistic qualitative flow patterns have been presented andcompared with flow field snapshots of PIV. Quantitativecomparison was missing.

Revstedt et al. (2000), Impellermotion was modeled by specify-ing momentum source term,Scale similarity sgs model, Scalartransport was studied

6RT, 6SRGT: multiple impellersystem, T = 0.8m, H = 1.5 T,inter impeller clearance=0.25T,bottom clearance = 0.5 T. Re =100, 000

〈u3〉 0 < r/R <1.0 This was the first study which employed multiple impellersystem. This study gave more importance to prediction ofglobal parameters (Np and NQP). The mean axial velocityand turbulent kinetic energy were predicted well. Spectralanalysis has been carried out for the radial velocity com-ponent. LES scalar mixing studies have been performed.All the flow variables were not compared with experi-mental data.

z/T = 0k 0 < r/R <1.0

z/T = 0

Derksen (2001), Adaptive forcefield technique in Lattice Boltz-mann frame work. StandardSmagorinsky sgs and structuredfunction sgs models, 1803, 2403

and 3603

4 bladed PBTD, Impeller-vesselgeometry similar to the one in-vestigated by Schafer et al.(1998), Re = 7300

〈u3〉 z/T = 0.145, 0.276 Spatial resolution has significant impact on the overallaverage flow field results. The impact of a slightly differ-ent subgrid-scale (SGS) model (a structure function modelinstead of a Smagorinsky model) was studied. No signifi-cant effect change in the overall and phase-averaged flowfield results due to change in subgrid scale model. Onlythe formation and strength of the tip vortex differed forthe two SGS approaches (vortex formed in the structurefunction simulations was found to be weaker). Validationwas made only with in the impeller region.

0 < r/R <1.0〈u2〉 z/T = 0.145, 0.276

0 < r/R <1.0k z/T = 0.145, 0.276

0 < r/R <1.0

First one, Bakker and Van den Akker (1998) employed the simpli-fied RSM, i.e. algebraic stress model (ASM) using the IBC methodto model the flows produced by a Rushton turbine. Their objectivewas to improve the predictive capabilities of CFD modeling by ac-counting anisotropy using less computational intensive ASM model(simplified RSM). Their study concluded that the results predicted bythe ASM compare better with the experimental data than those pre-dicted by the standard k– � model. Oshinowo et al. (2000) performedthe CFD study using different turbulence models like, k– �, RNG k– �and RSM for the prediction of tangential velocity distribution in abaffled vessel using multiple reference frame (MRF) model. The tan-gential velocity distribution above the impeller has been correctlypredicted. They attributed the occurrence of the counter-intuitivereverse swirl in the simulation to poor convergence and coarse griddensity. It may be pointed out, however, that both the aforesaid in-vestigations have shown comparisons in only a small region whilemost of the vessel region remained unexplored. More importantly,both the studies have not presented the predictive capability of CFDmodels for the turbulent kinetic energy and the turbulent energydissipation rate.

2.2. Large eddy simulations

Table 1 summarizes the previous work on the LES of the flowpattern generated in a stirred vessel. It gives details with regard to

5470 B.N. Murthy, J.B. Joshi / Chemical Engineering Science 63 (2008) 5468 -- 5495

Table 1 (Continued.)

Authors, Impeller-SGS model, Gridsize

System investigated Flowvariable

Region ofcomparison

Remarks

Revstedt et al. (2002), Impeller mo-tion was modeled by specifying mo-mentum source term, Scalesimilarity sgs models

6RT, 6SRGT: multiple impellersystem, T = 0.8m, H = 1.5 T,inter impeller clearance=0.25T,bottom clearance = 0.5 T. Re =100, 000

〈u1〉 0 < r/R <1 They presented results both in a fixed and moving frameof reference. There was a slight over prediction of themean tangential velocity component due to insufficientgrid resolution. Comparison was made between Rushtonand Scaba turbines using their predictions. Finally, thisstudy explained the high levels of turbulence of flow gen-erated by Rushton than Scaba turbine with the help ofspectral density of radial fluctuating component obtainedfrom LES.

z/T = 0.33〈u3〉 0 < r/R <1

z/T = 0.33k 0 < r/R <1

z/T = 0.33

Roussinova et al. (2003), Slidingmesh technique, StandardSmagorinsky sgs model, 500,000

PBTD454, T = 0.24, D/T = 0.5,W/D=0.2, dh/D=0.22, h/D=0.20,C/T = 0.5, N = 200, Re = 48, 000

〈u1〉 z/T = 0.5 This study compared predicted values of the radial profileof the mean axial velocity. It focused on the identificationof low frequency macro instabilities. Their predicted valueof frequency of macro instability was compared well withLDA measurement.0 < r/R <1

Yeoh et al. (2004), Sliding deform-ing mesh technique, StandardSmagorinsky sgs model, 490,000

DT6, T=0.10, D/T=0.33, C=T/3,H=T , N=36.08 rps, Re=40, 000

〈u1〉 0 < r/R <1 The predictions were compared with LDA data of meanand rms velocities and energy dissipation. Both standardk– � and LES models predicted the mean flow field well.LES out performed standard k– � with regard to the kpredictions. The power number obtained from integrationof the energy dissipation was predicted to within 15% ofthe measured value. This study also made the comparisononly in the impeller region.

z/T = 0.33〈u3〉 0 < r/R <1

z/T = 0.33k 0 < r/R <1

z/T = 0.33� 0 < r/R <1

z/T = 0.33

Hartmann et al. (2004a), StandardSmagorinsky and Voke sgs models,Adaptive force field technique

T = 0.15, D = T/3, C = T/3, H =T , baffle clearance=0.017T, N=7 rps, Re = 7300

〈u1〉 0.52 < z/R <0.8 Predictive capabilities of LES and RANS were assessed. Thesimulated phase-averaged flow fields were in good agree-ment with the experimental results. LES overpredictedthe tangential velocity at the center of the impeller tip.Their simulations revealed that Smagorinsky sgs modelperforms better than Voke sgs model especially when itcomes to prediction of trailing vortex pair. It was shownthat LES better predicted turbulent kinetic energy in theimpeller discharge flow where RANS underperforms. Theyfound nearly isotropy in the circulation loops. But in theimpeller stream, the boundary layers, and at the separa-tion points turbulence was found more anisotropic due tohigh shear rate. They did not see relative merits of usingVoke sgs model.

0.366 < r/R <0.64〈u3〉 0.52 < z/R <0.8

0.366 < r/R <0.64k 0.52 < z/R <0.8

0.366 < r/R <0.64

Yeoh et al. (2005), Sliding deform-ing mesh technique, StandardSmagorinsky sgs model, 490,000

DT6, T=0.10, D/T=0.33, C=T/3,H=T , N=36.08 rps, Re=40, 000

This was the first attempt to study the mixing time. LESprovided a very detailed evolution pattern of the concen-tration field in space and time. At the impeller midsectiona substantial amount of scalar is trapped in a region be-tween two vertical baffles. In the impeller region it wasobserved that instantaneous concentrations being as highas three times the equilibrium value. They cited that LEScan be used to identify stagnant zones, mixing inhomo-geneities due to the vessel geometry, and also for optimallocation of feed pipe.

Alcamo et al. (2005), standardSmagorinsky sgs model, slidingmesh technique, 761,760

DT6, unbaffled vessel, T = 0.19,D/T = 0.5, C = T/3, H = T , N =200 rpm, Re= 3× 104, unbaffledvessel

〈u1〉 0.26 < r/R <0.44 The LES predictions were compared with experimentaldata on an unbaffled tank. The simulated values comparedvery well with the measured data as far as tangential ve-locity are concerned. And a fairly good comparison ob-tained for radial velocity profiles as well. The LES couldcaptured the existence of pair of trailing vortices in un-baffled tanks.

0.033 < z/T <0.16〈u3〉 0.27 < z/R <0.105

0.033 < z/R <0.16〈u2〉 0.033 < z/R <0.16

Derksen et al. (2007), standardSmagorinsky and mixed sgs models,adaptive force field technique, 803,1283 and 1603

DT6, non-standard geometry,T = 0.44, H = 1.86T, D = 2

3 T,C=0.0375T, Re=14, 000, 82,000,350,000

There was no significant difference between the perfor-mance of the two sgs models. Further, the particle track-ing technique has been extended for the study of mixingperformance. They reported that after some 5–10 impellerrevolutions uniform conditions have been obtained, exceptfor the bottom region that is characterized by the pres-ence of fresh feed. However, LES predictions have beendiscussed more qualitatively as non-availability of the ex-perimental data for the industrial scale equipment.

the vessel and the impeller geometry and the subgrid scale modelalong with the grid size. The table also gives remarks on the qual-ity of comparison of predicted radial, axial, and tangential velocitycomponents, and turbulent kinetic energy (k) with the experimentaldata.

The very first study of LES for stirred vessel was performed byEggels (1996) using a lattice-Boltzmann discretization scheme andthe adaptive force field technique for the impeller rotation. The sim-ulations have been performed for coarse and fine mesh and the

corresponding uniform grid size of 4mm (1.73 × 106) and 2mm(13.8 × 106), respectively. The main objectives were to make thelattice-Boltzmann scheme based solver more general to simulateflows in real complex geometries and to understand the detailed lo-cal flow field in the vicinity of the impeller which was not possiblewith the RANS based turbulence models. Therefore, this study pre-sented the instantaneous and themean flow field in both vertical andhorizontal planes which would help in initial understanding of thegiven system. However, there were differences in the quantitative

Fig. 1. Impeller designs used for study. (A) DT; (B) PBTD60; (C) PBTD45; (D) PBTD30 and (E) hydrofoil impeller.

predictions of the mean radial and mean axial velocities especiallyin the near impeller region. The discrepancies were due to insuffi-cient statistical sampling time which was limited by the comput-ing time with the fine grid and adaptive force field technique usedfor the impeller modeling. Further, the quantitative comparison hasbeen restricted to a few flow variables (axial and radial velocity) andalso at limited locations (one radial and one axial line) and energybalance was not reported.

Revstedt et al. (1998) investigated both the mean flow and thespectral distribution of flow by power density function at variouslocations in a baffled strirred tank. In this study, the effect of sgsscale motion on the large scale was not taken into account and theprocess of energy dissipation was numerically treated by using thirdorder upwind scheme. The LES could capture the trailing vorticesbehind the blades and their detailed movement away from the im-peller with increasing distance from the blade. These observationswere in close agreement with the expensive and time-consumingexperiment. They demonstrated that the LES could generate instan-taneous flow field like sophisticated flow measuring techniques and

sometimes even better as far as the flow in the impeller region andnear wall regions are concerned. The mean axial velocities have beenbetter predicted. Comparison for the mean radial velocity at only oneaxial location was given (impeller center line), which was very goodat all the locations except 0.5 < x1/D <0.6. They have also obtained afairly good prediction for the turbulent kinetic energy in the impellercenter plane. Acceleration of the fluid at the impeller tip because ofthe fluid entrainment due to the trailing vortex pair was well simu-lated. Prediction of radial and tangential velocity with the angle waspoor. They have suggested that the boundary conditions on the im-peller region should be improved. Their instantaneous data from LESsupported the blade frequency and the existence of Kolmogorov's− 5

3 region in the energy spectrum. However, the detailed comparisonof all the flow variables with the experimental data throughout thevessel has not been reported.

Derksen and Van den Akker (1999) took forward the Eggels (1996)study with a more refined forcing algorithm. The real potential ofLES was explored as the simulations could identify a detailed lo-cal flow in the wakes behind each blade which were found to be

D/T = 1/3H = T

C

T/10

zr

T = 0.30m

Z = 0.010m

Z = 0.044m

Z = 0.082m

Z = 0.10m

Z = 0.118m

Z = 0.154m

Z = 0.244m

Z = 0.190m

D

Fig. 2. Geometrical details of the stirred vessel.

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Fig. 3. Comparison between the simulated and experimental profiles of the dimensionless mean axial velocity for disc turbine at various axial levels. (A) H = 0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

significantly higher than the impeller tip speed (two times tip speed).As far as the quantitative comparison is concerned, good agreementhas been obtained between the experimental data and the simula-tion of the radial profiles of the radial velocity. However, the tan-gential velocities were overpredicted (by maximum 15%) and it wasattributed to a lack of spatial resolution and improper treatment ofthe SGS viscosity. The axial profiles of random (turbulent), coher-ent (pseudoturbulence) and total kinetic energy have been shownto agree with the LDA phase resolved experimental data. Further, itwas observed that the dissipation rate distribution throughout thetank is very inhomogeneous. However, the comparison for axial ve-locity was not presented and all the predictions have been reportedonly in the impeller region.

Revstedt et al. (2000, 2002) have made the first attempt to studythe flow generated in multiple impeller system and the spatial dis-tribution of inert tracer using the LES. The objective of this studywas to investigate the influence of the impeller type (Rushton andScaba) on the flow structure and scalar transportation. This studygave relatively more emphasis to the prediction of global param-eters (NP and NQP) rather than the detailed estimation of meanand turbulent flow field. Further, only axial velocity and turbulentkinetic energy were compared with the experimental data (by con-stant temperature anemometry in the impeller center plane). Inaddition, there were significant discrepancies in the prediction ofturbulent kinetic energy in the impeller region. They attributed it to

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Fig. 4. Comparison between the simulated and experimental profiles of the dimensionless mean radial velocity for disc turbine at various axial levels. (A) H = 0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

the improper treatment of impeller rotation and inadequate meshresolution. However, they have clearly demonstrated the advantagesof LES over RANS using the data generated from LES by carryingout the spectral analysis of both the velocity and scalar flow field. Itrevealed the large spatial inhomogeneities in the concentration fieldsand low-frequency variations seemed to dominate the concentrationfluctuations in the impeller stream. They concluded that, at equalpower input, the center plane velocities and the volumetric flowin the impeller stream did not differ with respect to the impellertype (DT and Scaba). Also, for the Scaba type impeller, the periodicfluctuations associated with the blade passing, were found to be lesspronounced due to the absence of trailing vortices.

Derksen (2001) employed the previously developed lattice-Boltzmann based solver for the flow generated by 4-PBTD45. Thisstudy focused on the influence of the spatial resolution and sgsmodeling on the flow predictions. Furthermore details can be seenin Table 1. It has been observed that the spatial resolution has signif-icant impact on the overall average flow field results. No significanteffect was observed in the overall and phase-averaged flow field re-sults due to a change in the subgrid scale model. Only the formationand strength of the tip vortex differed for the two SGS approaches(vortex formed in the structure function simulations was found tobe weaker). Validation was made only within the impeller region.

For the PBTD45 impeller, Roussinova et al. (2003) performed LESsimulations using sliding mesh technique. The mean axial veloc-ity in the impeller center plane has been in agreement with theexperimentally measured values. The study was extended for theidentification of low frequency macroinstabilities. Their predictedvalue of frequency of macroinstability agreed very well with thoseobtained from LDA measurement. However, the comparison of theLES predicted mean radial, and tangential velocities and the tur-bulent kinetic energy with the experimental data have not beenreported.

Hartmann et al. (2004a) employed LES to investigate the preces-sional vortex phenomenon with the help of LES. This study opted thesame numerical techniques as those of Derksen and Van den Akker(1999). Their LES model could capture the vortical structure movingaround the tank centerline in the same direction as the impeller. Us-ing LES, they also observed that the strength of the vortex below theimpeller to be much stronger and pronounced in size as comparedto that prevailing above the impeller and both vortices move with amutual phase difference which was qualitatively in good agreementwith the experimental data. Their LES model could accurately predictthe characteristic MI (macroinstability) frequency. It could even pre-dict a second frequency peak at f =0.092N at a Reynolds number of12,500. Using LES data, they observed that the flow was dominated

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Fig. 5. Comparison between the simulated and experimental profiles of the dimensionless mean tangential velocity for disc turbine at various axial levels. (A) H=0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

by the low-frequency precessional vortex in the bulk flow region andit was associated with a significant amount of the kinetic energy.

Yeoh et al. (2004) have performed LES using sliding deformingtechnique. The main objective was to assess the capability of the LESmodeling technique, coupled with the sliding deformation method-ology (SDM) and relative performance with RANS approach. RANShave been performed using the standard k– � model. The predictedmean radial and tangential velocities and turbulent kinetic energywere compared with LDA data only in the impeller center plane. Forall the three variables good qualitative and quantitative agreementwas obtained using both the RANS and the LES models. However,LES made superior predictions of the correct radial profiles than thestandard k– � model. With regard to the prediction of the turbulentkinetic energy, LES clearly outperformed the standard k– � model.The power number obtained from integration of the energy dissi-pation was predicted within 15% of the experimentally measuredvalue. Similar to earlier investigations, this work has also presentedthe comparison at only one axial location (impeller center plane).

Hartmann et al. (2004b) made a detailed study to evaluate thepredictive capabilities of LES and RANS. Themain difference betweenthis study and Yeoh et al. (2004) was, this study employed the lattice-Boltzmann solver as against the finite volume based solver. Further,the RANS based simulations were performed with the shear-stress-transport (SST model) in place of standard k– � model. The RANS, LES

predictions have been compared with the LDA measured radial, tan-gential velocities and turbulent kinetic energy in the impeller zone.The radial velocities of both the models were in good agreementwith the experimental data. With regard to the tangential velocity,the overall performance of LES was found to be much superior toRANS simulations. Their simulations revealed that the Smagorinskysgs model performed better than Voke sgs model, especially, as faras the prediction of trailing vortex pair is concerned. It was furthershown that the LES predictions of the turbulent kinetic energy inthe impeller discharge flow were substantially better than the RANSwhich invariably underperformed. Overall they did not see the rel-ative merits of using Voke sgs model. Further, they calculated theenergy dissipation rate by assuming local equilibrium between pro-duction and dissipation at and below subgrid scale level and com-pared with the RANS predictions in the baffle midplane. This studyalso presented comparisons in a limited region. Further, axial veloc-ity comparison has not been given.

Yeoh et al. (2005) extended their previous work to study themixing of inert tracer using LES. This was the first attempt to studythe detailed mixing phenomenon using LES. It was clear from thestudy that the LES provides a very detailed evolution pattern of theconcentration field in space and time. It was at the impeller mid-section that a substantial amount of scalar gets trapped in a regionbetween two vertical baffles. In the impeller region they observed


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Fig. 6. Comparison between the simulated and experimental profiles of the dimensionless turbulent kinetic energy profiles for disc turbine at various axial levels. (A)H = 0.01m; (B) H = 0.044m; (C) H = 0.082m; (D) H = 0.1m; (E) H = 0.118m; (F) H = 0.154m; (G) H = 0.190m and (H) H = 0.244m: �—Experimental; (1) LES model; (2)RSM and (3) k– � model.

instantaneous concentrations being as high as three times the steadystate value. They have clearly brought out the utility of LES in identi-fying the stagnant zones, mixing inhomogeneities due to the vesselgeometry and optimal location for the feed pipe.

Alcamo et al. (2005) presented the predictive capabilities of LESfor flow field generated in an unbaffled cylindrical vessel. The LESpredictions were better compared with the experimental data ob-tained by both inhouse PIV technique and with the available litera-ture data. The simulated values agreed very well with the measuredtangential velocities and fairly well with the radial velocities. The LEScould capture the existence of pair of trailing vortices in unbaffledtanks.

Recently Derksen et al. (2007) extended their previous lattice-Boltzmann scheme based LES solver for industrial scale crystallizer.It has been focused on the assessing the feasibility of using acomputationally efficient LES to quantify the fine scale turbulentstructures in an industrial size crystallizer. The effects of spatialresolution and the subgrid scale model (Smagorinsky and the mixedscale models) were also investigated. No significant difference wasobserved in the predictions of the two subgrid scale models. Further,the particle tracking technique was employed for the study of mix-ing performance. Practically uniform conditions were obtained after5–10 impeller revolutions with respect to concentration of fluidelements, except for the bottom region which gets characterizedby the presence of fresh feed. However, LES predictions have been

discussed more qualitatively because of the non-availability of theexperimental data from the industrial scale equipment.

The foregoing review brings out the superior potential of LES overthe RANS models. However, the LES predictions have been comparedwith the experimental measurements in a limited region (less than10% of the tank volume) close to the impeller. Further, the effect ofimpeller design has been investigated over a limited range of powernumber. Further, it cannot be taken for granted that, if any turbulencemodel predicts the flow in one region (even most complex), it wouldhave the similar ability over the entire computational domain. If thiswas true, the model parameters of k– �, RSM which were selectedto get better prediction in the near impeller zone, the same parame-ters should have been able to capture the flow quantitatively in thebulk region. In order to overcome such a limitation, attempts havebeen made for the standard k– � model to identify the set of modelparameter values in different zones, i.e. zonal modeling. The impor-tant message is to realize the existence of non-similar featured flowsin different zones. For instance, the literature reports clearly showthat various flow structures with certain characteristics (existence,shape and energy content) appear in different regions in the tank.For example, some features include processional vortex motion nearshaft, trailing vortex behind the blade and jet instabilities (below im-peller region). In addition, these characteristic features strongly de-pend upon the impeller design. Therefore, it was thought desirableto investigate the flow patterns generated by a wide spectrum of

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Fig. 7. Comparison between the simulated and experimental profiles of the dimensionless mean axial velocity for PBTD60 at various axial levels. (A) H=0.01m; (B) H=0.044m;(C) H = 0.082m; (D) H = 0.1m; (E) H = 0.118m; (F) H = 0.154m; (G) H = 0.190m and (H) H = 0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

open impellers. For instance, DT produces strong radial and tangen-tial flow with complex trailing vortices, whereas the hydrofoil (HF)impeller generates flow with practically weak trailing vortices andless complexity. In view of such a wide possible variation, flows gen-erated by five impeller designs namely DT, PBTD60, PBTD45, PBTD30and an HF impeller have been investigated in the present work.

Further, in the present work, for the first time, comprehensiveRSM has been employed to simulate both radial and axial flow im-pellers. In the literature, it is quite evident that the predictive ca-pabilities of full fledged RS model has yet to receive due attention.However, there have been quite a few studies, which have used al-gebraic stress model (simplified RSM) which were excellent startingpoints. Further, one of the significant terms in the Reynolds stresstransport equations is pressure strain term, which has been mod-eled using quadratic pressure strain model, which has been demon-strated to give superior performance in a range of basic shear flows,including plane strain, rotating plane shear and axisymmetric ex-pansion/contraction. On LES front, the standard Smagorinsky modelhas been widely used in the literature for stirred tank simulations.However, it is well known that the Smagorinsky model is essentiallyan algebraic model in which subgrid-scale stresses are parameter-ized using the resolved velocity scales. The underlying assumptionis the local equilibrium between the transferred energy through the

resolved scale and the dissipation of kinetic energy at unresolvedsubgrid scales. In order to overcome this limitation, the dynamic ki-netic energy subgrid-scale model has been used in this study. In thismodel, transport equation for subgrid-scale turbulence kinetic en-ergy is explicitly solved.

It is well known that the precessing vortex is an intermittent,non-stationary, pseudo periodic and large scale structure. This en-tire phenomena is named precessing instability. This is mainly dueto a precessional motion of a vortex around the shaft. These arelow-frequency in nature with high amplitude of oscillatory motionsin the low turbulence regions of a vessel which have the capabil-ity of transporting substances over relatively long distances. Theselow-frequency motions can severely affect the transport phenom-ena (local mean flow pattern heat and mass transfer rates, gas/solidhold-up, etc.). Apart from the transport phenomena, these frequen-cies at higher amplitude can affect the solid components inside thereactors causing permanent damage to baffles, internals, sensors, etc.Therefore, it was thought desirable to quantify the energy content ofthe precessing vortex with respect to all the impeller designs underconsideration. In addition, scanty information is available in the lit-erature on the possible interaction between themean circulation andthe precessing instability which subsequently leads to intermediateinstabilities (between circulation and precessional vortex). Hence, in

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this study an attempt has been made to identify the intermediateinstabilities with respect to various impeller designs.

In this work, all the simulations have been performed using un-steady sliding mesh approach, second-order discretization schemesand finemesh throughout the computational domain. All the compu-tations have been performed using FLUENT 6.2. (2005), whose suc-cessful application has been documented in the literature (Bakkeret al., 2001; Rocchi and Zasso, 2002; Roussinova et al., 2003; Hu andKazimi, 2006; Wang et al., 2006; Murthy et al., 2007). The details ofthe turbulence modeling are discussed in Section 4.

3. LDA measurements

LDA measurements were carried out in a 0.29m i.d. transparentacrylic vessel equipped with various impellers located at a clearance(C) equal to H/3 from the tank bottom. The vessel was fitted withfour baffles having width 1/10 of the vessel diameter (fully baffledconditions). To minimize the effect of curvature on the intersect-ing beams, the vessel was placed inside a square vessel. A standardDT, PBTD60, PBTD45, PBTD30, and HF (D = 0.1m) were used as theimpellers. The impeller was centrally located and driven by a vari-able speed DC motor. The flow measurements were performed inthe midbaffle plane for all the configurations. LDA measurements

were carriedwith a single point, two-component LDA system (Ar-ion,5W laser system) following the work of Ranade and Joshi (1989) andRanade et al. (1992). A large number of samples (300,000–500,000)were recorded for each run in which the data rate varied from 300to 1500Hz.

LDA data collected for all the five impellers contain periodic-ity due to the rotary motion of impeller. The periodic componentcontributes significantly to the flows, especially in the near im-peller region. Hence direct processing of the data may lead to theoverestimates of the turbulent kinetic energy. Thus it is essential toremove the signal corresponding to the impeller blade passage fre-quency and its simple low-frequency harmonics. For the validationof the turbulence models, which essentially model the turbulence,data free of regular (i.e. periodic) component are required. In viewof this, the periodic component arising out of the impeller rotationfrom the velocity–time series has been removed. The following arethe details.

The data obtained due to LDA are random in nature with re-spect to time. For any kind of transformation, it is desirable that thetime series should be expressed in terms of the variable at equaltime intervals. This conversion of a random time series to a timeseries in terms of discrete data at equal time intervals is termedhere as equispacing of the data. For this purpose, linear interpolation

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technique was used. Further, the signal obtained after equispacingwas again processed using fast Fourier transform (FFT). The powerspectrum was obtained and the frequencies corresponding to theimpeller blade passage frequency (6 × 4.5Hz) and its harmonicswere identified. The data corresponding to these frequencies wereremoved and the time series was reconstructed using the inverseFourier transformation. The data free of periodicity was subsequentlysubjected to the noise removal using the multiresolution analysis(Kostelich and Yorker, 1988).

Fig. 2 shows eight axial locations at which LDA data were col-lected. The data were processed for the estimation of all the threemean components of the velocity and the turbulent kinetic energy.It may be emphasized again that, in this work, only random part ofthe turbulent kinetic energy has been presented which was madefree from periodic component.

4. Turbulence modeling and flow governing equations

4.1. Standard k– � model

In a turbulent flow, if we assume that the turbulence respondsrather quickly to changes in the mean flow we would expect the

Reynolds stresses themselves to be related to the mean rate of strainas given by

−〈u′iu

′j〉 = �t

(�〈ui〉�xj

+�〈uj〉�xi

)− 2

3�k (1)

It involves six unknown eddy viscosities. Unfortunately the viscosi-ties are related to each other in a manner that would be difficult todescribe in general. Hence, the most commonly used assumption isto treat the eddy viscosity as a scalar quantity.

The standard k– � model (Jones and Launder, 1972; Sahu et al.,1999) is essentially a high Reynolds number model and assumesthe existence of isotropic turbulence and the spectral equilibrium. Itproposes the relation for the eddy viscosity in terms of k and � withthe help of the turbulence parameter C�. In addition, the triple ve-locity correlations in the transport equation for the energy dissipa-tion rate are modeled with the help of two more constants (C�1 andC�2) whose values have been derived from the measurement of flowcharacteristics of simple two-dimensional equilibrium flows. In thek– � model, the length and the time scales are built up from the tur-bulent kinetic energy and the dissipation rate using the dimensionalarguments.

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Fig. 10. Comparison between the simulated and experimental profiles of the dimensionless mean radial velocity for PBTD60 at various axial levels. (A) H = 0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

The k– � turbulence models suffer from the necessity of model-ing a number of quantities for which reliable experimental data aredesirable under a large number of flow conditions. While this ne-cessity is a fundamental weakness of the k– � approach, a furtheruncertainty lies in the assumption that the turbulent kinetic energyand its dissipation rate are necessary and sufficient turbulence vari-able for the simulation of turbulent flows. Nonetheless, the model iswidely used and has been attributed to some significant simulationsuccesses.

4.2. Reynolds stress model

It is clear that the standard k– � is not equipped to pre-dict anisotropic turbulent flows (Reynolds, 1987; Launder, 1990;Hanjalic, 1994). Further, the modeling of transport equations for kand � clearly brings out the difficulties to account for streamlinecurvature, rotational strains, and the other body-force effects. RSM,in theory, circumvents all the above mentioned deficiencies andalso it has an ability to predict more accurately each individualstress.

The transport equation for the Reynolds stresses is given as

��ij�t

+ 〈uk〉��ij�xk

= −(�ik

�〈uj〉�xk

+ �jk�〈ui〉�xk

)

− ��xk

Cijk + �ij − �ij + �∇2�ij. (2)

The terms Cijk, �ij and �ij in Eq. (2) are given as

�ij =⟨P′�

[�u′j

�xi+ �u′

i�xj

]⟩= Pressure– strain correlation (3)

�ij = 2�

⟨�u′

i�xk

�u′j

�xk

⟩= Dissipation rate correlation (4)

Cijk = 〈u′iu

′ju

′k〉 + 1

�(〈P′u′

i〉�jk + 〈P′u′j〉�ik)

= Third-order diffusion correlation (5)

The third-order diffusion correlation, Cijk, pressure strain, �ij anddissipation rate tensor �ij need to be modeled to close the set of the

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governing equations. The turbulent diffusion is modeled as per Lienand Leschziner (1994) in FLUENT 6.2: �/�xk((�t/k) ��ij/�xk), wherek = 0.82 and the turbulent viscosity, �t , estimated from the k and� with C� = 0.09. It is obvious that the highly anisotropic flow dueto impeller rotation and subsequent interaction between the strongimpeller jet and the bulk flow suggests that the production term andthe pressure strain correlation play a significant role in the predictionof turbulent stresses. In the present work, the quadratic pressurestrain model proposed by Speziale et al. (1991), which is known toimprove the accuracy of flow field with streamline curvature, hasbeen used to model the pressure–strain term of the RSM.

To obtain the boundary conditions for the Reynolds stresses atthe wall, the equation for the turbulent kinetic energy (k) was solved.Further, the equation for the dissipation rate (�) of turbulent kineticenergy was solved to obtain the dissipation rate (�ij) term in theReynolds stress transport equation.

4.3. Large eddy simulations

LES equations of turbulent flows are formally derived by ap-plying a filtering operation to the Navier–Stokes equation and

assuming that filtering and differentiation operators' commute(Leonard, 1974). The filtering process effectively filters out the ed-dies whose scales are smaller than the filter width or grid spacingused in the meshing. The resulting equations have the structureas the original equation plus additional terms, called subgrid scalestresses (SGS). The filtered equations are used to compute the dy-namics of the large-scale structures, while the effect of the smallscale turbulence is modeled using a SGS model. Following are theflow governing equations for LES:

�ui�xi

= 0 (6)

��t

(ui) + uj�ui�xj

= − 1�

�P�xi

+ ��xj

([� + �t]

�ui�xj

)(7)

Where we have used the incompressibility constraint to simplify theequation and the pressure is now modified to include the trace term�kk�ij/3. The SGS resulting from the filtering operation are unknownand require modeling. Most of the past work on LES of stirred ves-sels have employed algebraic models in which subgrid-scale stressesare parameterized using the resolved velocity scales. The underlyingassumption is the local equilibrium between the transferred energy

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through the grid-filter scale and the dissipation of kinetic energyat small subgrid scales. The subgrid-scale turbulence can be bettermodeled by accounting for the transport of the subgrid-scale tur-bulence kinetic energy. The dynamic kinetic energy SGS model inFLUENT replicates the model proposed by Kim and Menon (1997),which has been used in the present work. The subgrid-scale kineticenergy is defined as (Kim and Menon, 1997)

ksgs = 12(u2k − (uk)

2) (8)

The subgrid-scale eddy viscosity, �t , is computed using ksgs as(Kim and Menon, 1997)

�t = Ckk1/2sgs (9)

The unkown, ksgs is obtained by solving its transport equationgiven by

�ksgs�t

+ uj�ksgs�xj

= − �ij�uj�xj

− C�k3/2sgs

+ ��xj

(�tk

�ksgs�xj

)(10)

In the above equations, the model constants, Ck and C�, are deter-mined dynamically.

5. Numerical details

5.1. Geometrical details and grid generation

The geometrical details of the problem investigated are shownin Figs. 1 and 2. The cylindrical vessel of diameter T = 0.30m andfitted with four baffles of width, T/10 was used as the stirred vesselto study the flow characteristics of DT, PBTD and HF. In all the cases,the diameter of the impeller was D = 0.1m. The clearance betweenthe impeller and the vessel bottom was kept at T/3 and the impellerwas centrally located. The height of liquid level (H) was equal to thetank diameter (T). The impeller rotational speed was 4.5 rps and wa-ter was used as the working fluid. Hexahedral elements were usedfor meshing the geometry and a good quality of mesh was ensuredthroughout the computational domain using GAMBIT mesh gener-ation tool. In order to ensure better quality mesh, impeller bladesand baffles were considered with zero thickness. The computationalmesh consisted of 575,000 (≈ 90 × 60 × 106, z × r × �) hexahedralcells for the standard k– � model and the RSM. In the case of LES,

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Fig. 13. Comparison between the simulated and experimental profiles of the dimensionless mean tangential velocity for PBTD60 at various axial levels. (A) H = 0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

1,275,567 (≈ 100×69×184, z×r×�) hexahedral cells were used andwith relatively fine mesh in the impeller region in order to betterresolve the strong velocity flow field. In the present LES, the lengthscales were resolved between 200�m and 4.2mm, which are of theorder of the Taylor microscale (�), i.e. 6.5, 6.7, 7.1, 7.3 and 7.5mmfor DT, PBTD60, PBTD45, PBTD30 and HF impeller, respectively.Therefore, one can expect the realistic LES predictions from this gridresolution. However, the present grid resolution is much higher ascompared with the Kolmogorov's length scale ( ) of 46�m((�3/�)1/4), for the flow generated by DT, i.e. smallest scale in theflow generated among the five impeller designs in the presentstudy. Further, the Kolmogorov length scale is 53�m for PBTD60,57�m for PBTD54, 64�m for PBTD45 and95�m for HF impeller, re-spectively. It may be noted that these estimations have been madeby considering � as the average energy dissipation rate.

5.2. Method of solution

In this work, all the computational work has been carried out us-ing the commercially available software FLUENT 6.2. The discretizedform of the governing equations for each cell was obtained suchthat the conservation principles are obeyed on each cell. The second-order implicit scheme was used for time discretization in all the

turbulence models. Further, the second-order central bounded dif-ference scheme was used for spatial discretization in case of LES andthe second-order upwind scheme for RANS based models. All thediscretized equations were solved in a segregated manner with thePISO (pressure implicit with splitting of operators) algorithm. PISOinvolves one predictor step and two corrector steps and may be seenas an extension of SIMPLE (semi-implicit method for pressure-linkedequations), with a further corrector step to enhance it. PISO is apressure–velocity calculation procedure developed originally for thenon-iterative computation of unsteady compressible flows. There-fore, PISO has better performance in unsteady simulation than SIM-PLE series algorithm. To improve the efficiency of this calculation,the PISO algorithm adopts two additional corrections: neighbor cor-rection and skewness correction. As far as LES run is concerned,to ensure smooth and better convergence initially k– � simulationshave been performed until the complete steady state flow field isobtained and then k– � results have been used as the initial guessvalues for the LES. In the present work, all the solutions were con-sidered to be fully converged when repeated iterations do not de-crease the sum of residuals below 1 × 10−5. Here, the residual Rcalculated as the imbalance in algebraic equation summed over allthe computational cells. FLUENT scales the residual using a scal-ing factor representative of the flow rate of variable through outthe domain. Standard model constants have been used for all the

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turbulencemodels. Grid independent study has been carried out onlyfor the case of DT using two grid resolutions for the standard k– �,Reynolds-stress transport models (RSTMs). The total number of cellsin the three directions for the two cases, 475,000 (≈ 95 × 50 × 100,z× r×�) and 575,000 (≈ 90×60×106, z× r×�), respectively. It hasbeen found that both the grids gave very similar profiles of turbulentkinetic energy and the values of power number were found to bepractically the same. However, all the RANS based simulations havebeen performed for the grid resolution of 575,000 (≈ 90× 60× 106,z × r × �). In the case of LES, the presented turbulent kinetic energyis a sum of resolved and unresolved (obtained fromsolving transportequation for the sgs kinetic energy equation) turbulence kinetic en-ergy. The periodic component of all the variables was removed in allthe simulations. In the present work, simulations have been initiallyperformed with the time step size of 0.0001 s and the correspond-ing CFL number is 0.7. As the solution progressed the time step sizehas gradually been increased to 0.001 s for LES, and 0.01 s for RANSin order to save computational time. For the same reason implicittime stepping scheme has been employed in the present work. Themajor advantage of this kind of methods is that the time step is notlimited by stability reasons, i.e. CFL condition. This means that, con-trary to explicit schemes, stability is ensured for any value of thetime step. In this way a smaller number of iterations are required tocomplete the simulation, leading to an important gain in CPU time.However, the level of accuracy of the solution at high CFL numbers is

low. Therefore, when a high level of accuracy is needed, for instancethe resolution of unsteady and transient phenomena, the time stepmust be kept small. In this situation, explicit codes will perform bet-ter than implicit ones because of their smaller computational costper iteration. The simulation was performed for a time span of 90 sin the case of LES. In the case of RANS the simulations were per-formed for the flow time of 9 s, which corresponds to 44 impellerrevolutions. All the computations have parallely been performed onan AMD64, 32 (16 nodes) processors cluster with a total 32GB RAM,2.4GHz processor speed.

6. Results and discussions

CFD simulations have been performed with three turbulencemodels, namely standard k– �, RSM and LES for the five differentimpellers. The simulation results of the dimensionless mean axialvelocity, mean radial velocity, mean tangential velocity and turbu-lent kinetic energy have been plotted against the LDA experimentaldata. The comparison of radial profiles of these parameters hasbeen made at eight different axial levels, z = 0.01m (A), 0.044m(B), 0.082m (C), 0.1m (D), 0.118m (E), 0.154m (F), 0.190m (G) and0.244m (H). It may be pointed out that the exercise of comparisoncovers practically the entire region in the vessel both near and awayfrom the impeller.

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The following section discusses the performance of various tur-bulence models for all the five impellers. However, in order to makethe explanation clear and simple, the discussion on the predictivecapabilities of all the three models is divided into the DT, all theaxial flow impellers together, i.e. PBTD (60, 45 and 30) and HF. Itis well known that a DT produces strong radial discharge with in-tense swirling. In the case of axial flow impellers, PBTD60 developsa strong downward flow with strong tangential velocity componentin the impeller region. On the other hand, PBTD45, PBTD30 and HFproduce tangential component in the decreasing order of magni-tude. Therefore, it would be easy to assess the inadequacy of themodel formulation and underlying assumptions involved in differentmodels.

6.1. Disc turbine

In case of DT, the high speed impeller stream impinges on thetank wall and changes the direction three times to return to impelleragain. This recirculatory flow exists in the bulk region of the tank.Near the eye of circulation, very small mean velocities exist. Theradial profiles of axial velocities at various axial locations are shownin Fig. 3(A–H). It can be seen that maximum axial velocities existnear the wall and are of the order of 0.20–0.30 times the tip speed.However, the axial velocity changes more sharply in the near wall

region compared to that in the near axis region of the vessel. Asone moves away vertically from the impeller swept region, axialvelocity initially increases, attains a maximum and then decreases.It is evident that the predictions of all the axial velocity profilesby all the three turbulent models are in good agreement with theexperimental data.

With regard to the radial component, impeller rotation generatesradially outward flow through the vertical surface of swept volume.This high speed radial jet entrain surrounding fluid and slows downas they approach the tank wall. It can be noted from Fig. 4(A–H)that both RSM and LES predictions agree well with the experimentalmean radial velocity profiles at all the axial levels. Whereas, standardk– � model exhibits some disparity at all the levels particularly abovethe impeller.

Radial profiles of the mean tangential velocity are depicted inFig. 5(A–H). Again, LES and RSM simulations capture the experimen-tal mean tangential velocity profiles quite well. In case of standardk– � model, predictions mainly deviate in the near impeller region.

Fig. 6(A–H) illustrates the comparison for turbulent kinetic en-ergy throughout the tank. In the impeller central plane (Fig. 6D) theprofile of turbulent kinetic energy is practically similar to those ofradial velocity and tangential velocity. Below the impeller (Fig. 6C),turbulent kinetic energy also exhibits two maxima as in the case oftangential velocity. The first maximum is at r = 0.27 and the secondone occurs closer to the wall at r = 0.92. The former is generated by


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Fig. 16. Comparison between the simulated and experimental profiles of the dimensionless turbulent kinetic energy for PBTD60 at various axial levels. (A) H = 0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

the rotating impeller and latter is because of the shear flow. The ki-netic energies below the impeller are much smaller than those in theimpeller center plane and above the impeller (Fig. 6E), it is mainlybecause of the upward inclination of trailing vortex pair. It can beobserved that at many axial levels, significantly in the impeller re-gion, the standard k– � and RSM models consistently underpredictthe turbulent kinetic energy. The good predictive capabilities of LEScan be clearly seen for the turbulent kinetic energy predictions. Hereone can justify the utility of LES even at the expense of computationalresources. As the LES provides macro and the reliable predictions ofthe turbulent flow, the predictions can further be useful for betterpredictions of macro- and micromixing, heat and mass transfer.

6.2. Axial flow impellers

Pitched blade down flow turbine (PBTD) develops a downwardjet below the impeller. The jet entrains fluid and the velocity gra-dients become less sharp as they approach the tank bottom. Theaxial jet impinges on the base and moves radially along the baseand after approaching the wall, turns upward in a well defined walljet. The following section discusses the mean axial, radial and tan-gential velocity profiles and the turbulent kinetic energy profiles forall the three axial flow impellers. Figs. 7–9 show the comparisonfor the radial profiles of the mean axial velocity, whereas the mean

radial velocity comparison is depicted in Figs. 10–12. Figs. 13–15show the mean tangential velocity and the turbulent kinetic PBTD30,respectively.

Figs. 7C–9C clearly show that the maximum axial velocity in-creases with an increase in the blade angle upto 60◦. The width ofhigh speed jet issuing from the impeller also increases with an in-crease in the blade angle. The magnitude of the maximum velocitydecreases with a decrease in the blade angle (0.25Utip for 30◦ to0.55Utip for 60◦). In the bulk (Figs. 7–9A, B, F–H), the mean axial ve-locity is relatively small near the axis at all the axial levels. Further,the mean axial velocity increases steadily till a maximum is reached.The maximum shifts towards the wall as the flow approaches thebase and the magnitude of the maximum decreases as the flow be-comes predominantly radial. After the peak value, axial velocity de-creases steadily and the flow turns upwards near the wall. RegardingCFD modeling, in all the three cases, axial velocity predictions by allthe three turbulence models are in good agreement with the exper-imentally measured values over the entire computational domain.

The comparison of the radial profiles of the mean radial velocityis shown in Figs. 10–12. In the impeller center plane (Figs. 10D–12D)radial flow is towards the center at all the radial locations and itincreases gradually from the wall and a maximum occurs near theimpeller blade tip. Just below the impeller (Figs. 10C–12C), the meanradial flow near the axis is feeble where the axial flow is dominant.It increases gradually and a maximum occurs at r/R = 0.3 and just


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Fig. 17. Comparison between the simulated and experimental profiles of the dimensionless turbulent kinetic energy for PBTD45 at various axial levels. (A) H = 0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

beyond the tip of the impeller (at r/R=0.4) radial velocity is very smalllike the velocities near the axis. Radial profiles of the mean radialvelocity in the bulk region below the impeller (Figs. 10–12A and B)show that, the radial flow is towards the wall and velocity increasesright from the axis and attains a maximum. The radial location ofthis maximum shifts towards the wall as one moves towards thevessel bottom. The magnitude of the maximum increases near thebase. After attaining the maximum value, radial velocity decreasesand becomes small near the wall where the axial flow is dominant.Above the impeller in the bulk (Figs. 10–12F–H), the flow is towardsthe center at all the radial locations. In all the cases, all the turbulentmodels are able to capture the mean radial velocity profiles at all theaxial levels. However, in the impeller region, standard k– � modelunderpredict the mean radial velocity (though it is weak) generatedby the PBTD60.

Figs. 13–15 show the mean tangential velocity comparison. Itcan be noticed that the tangential flow is in the direction of theimpeller rotation at all the axial levels. Just below the impeller(Figs. 13C–15C), it increases and decreases steeply between r = 0.15and 0.3. Beyond this point, it is almost constant from r = 0.4 on-wards. The maximum velocity is comparable with the maximum ax-ial velocity at this level. Impeller with blade angle of 60◦ generatessignificant tangential velocity components along the vertical surfaceof the impeller region which do not exist for the impellers of bladeangles 45◦ and 30◦. These tangential velocities result into a local

maximum in the resultant velocity along the vertical periphery. Theposition of this maximum appears slightly below the impeller cen-ter plane. This shift may be because of the strong axial downwardflow. The entering and leaving angles of the flow also decrease witha decrease in the blade angles. It can be seen that the LES and RSMpredictions are in good agreement with the strong tangential flowgenerated by the PBTD60. However, due to strong swirling motion,standard k– � model predictions show disparity with the experimen-tal data mainly in the impeller region. In case of PBTD45, standardk– � model is unable to capture the tangential flow field. However, inthe case of PBTD30, standard k– � model is quite successful in simu-lating the relatively weak tangential velocities. Therefore, this studyclearly brings out quantitatively the limitations of the standard k– �model.

The profiles of turbulent kinetic energy are shown in Figs. 16–18.In the impeller center plane (Figs. 16D–18D) turbulent kinetic energyis more or less constant. Just below the impeller (Figs. 16C–18C), thevalue of k increases sharply from the axis upto r=0.3 and decreasesdrastically beyond that point. From r=0.4 onwards turbulent kineticenergy is very small and constant. It can be seen that, the values ofk behave similar to the mean velocity with respect to blade angle.Impeller with blade angle of 60◦ generates intense turbulent flow.The maximum value of resultant intensity is 0.45Utip, whereas it is0.15Utip for the impeller with blade angle of 30◦. There is a goodagreement between the experimentally measured turbulent kinetic


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Fig. 18. Comparison between the simulated and experimental profiles of the dimensionless turbulent kinetic energy for PBD30 at different axial levels. (A) H = 0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

energy and LES predictions for all the axial flow impellers. However,there are discrepancies as shown in Fig. 16F, and it may be attributedfor not long enough statistics. On the other hand, both the RANSbased models fail to simulate the turbulent kinetic energy associatedwith the unsteady large scale motion generated by PBTD60 and 45.However, the RSM, standard k– � models can be seen to be successfulalong with LES in predicting the relatively weak turbulent kineticenergy generated by the PBTD30. It can further be noticed that, theturbulent kinetic energy has been well captured in the bulk regionof the tank (both below and above impeller) by all the three models.

Since, the validation of blade angle from 60◦ to 30◦ shows atrend in the variation of all the flow parameters, it was thoughtdesirable to investigate the flow generated by a HF impeller. It maybe pointed out that the power number of HF, PBTD30, PBTD45 andPBTD60 are 0.27, 1.3, 2.24 and 3.1, respectively. Further, the rate ofmean to turbulent kinetic energy (at all the locations) is the highestfor a HF and decreases with an increase in the blade angle. Further,though all these axial flow impellers generate amixed flow, the radialcomponent generated by HF is the weakest and increases with anincrease in the blade angle. In view of the strong convective motiongenerated by an HF, these impellers are widely used for the flowcontrolled operation.

Fig. 19(A–H) depict the comparison of experimental axial velocitywith the predictions of the three turbulence models. It can be seenthat, at all the axial locations, all the three models give excellent pre-dictions. A comparison of radial profiles of the mean radial velocity

are shown in Fig. 20(A–H). In the impeller center plane (Fig. 20D)radial flow is towards the impeller center at all the radial locationsand it increases gradually from the wall and a maximum occurs nearthe impeller blade tip. Just below the impeller (Fig. 20), the meanradial flow near the axis is feeble where the axial flow is dominant.It increases gradually and a maximum occurs at r/R = 0.2 and justbeyond the tip of the impeller (r/R=0.4), the radial velocity remainsmore or less constant. Radial profiles of the mean radial velocity inthe bulk region below the impeller (Fig. 20A and B) show that theradial flow is towards the wall and the velocity increases right fromthe axis and attains a maximum. The radial location of this maxi-mum shifts towards the wall (at z = 0.046m it was r/R = 0.3 and atz=0.01m it was r/R=0.6) as one moves towards the vessel bottom.The magnitude of the maximum increases near the base. After at-taining the maximum value, radial velocity decreases and becomessmall near the wall where the axial flow is dominant. Above the im-peller in the bulk (Fig. 20F–H), the flow is towards the center at allthe radial locations. In all the cases, all the three turbulence modelsare able to capture the mean radial velocity profiles.

Fig. 21(A–H) show themean tangential velocity comparison. It canbe noticed that the tangential flow is in the direction of the impellerrotation at all the axial levels. Just below the impeller (Fig. 20C), itincreases and decreases steeply between r = 0.15 and 0.3. Beyondthis point, it remains practically constant from r=0.5 onwards. Thesetangential velocities result into a local maximum in the resultantvelocity along the vertical periphery. However, the tangential flow

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Fig. 19. Comparison between the simulated and experimental profiles of the dimensionless mean axial velocity for HF at various axial levels. (A) H=0.01m; (B) H=0.044m;(C) H = 0.082m; (D) H = 0.1m; (E) H = 0.118m; (F) H = 0.154m; (G) H = 0.190m and (H) H = 0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

field generated by HF impeller is relatively weaker than the PBTD30impeller. Further, the measured mean tangential velocities at all theaxial locations except at a few locations (Fig. 21D), have been foundto be in good agreement with the experimental data.

The profiles of turbulent kinetic energy are shown in Fig. 22. Inthe impeller center plane (Fig. 22D) turbulent kinetic energy is moreor less constant. Just below the impeller (Fig 22C), the value of kincreases sharply from the axis up to r=0.2 and decreases drasticallybeyond that point. From r = 0.4 onwards turbulent kinetic energy isvery small and constant. It could be noticed that the turbulence levelsgenerated by HF impeller are relatively low than that generated bythe PBTD30. And the turbulent kinetic energy has been well capturedin the bulk region of the tank (both below and above impeller) byall the three models.

6.3. Discussion

In case of DT, the flow pattern generated by the impeller is amixed radial–tangential flow. The values of radial and tangentialvelocities are maximum at the impeller tip. In the impeller center

plane, with increasing radial distance, the tangential velocitydecreases more rapidly than the radial component. As far as themean radial and tangential velocity predictions are concerned theRSM outperforms the standard k– � model in the impeller region(Figs. 4 and 5). It can be attributed to the estimation of all thecomponents Reynolds stress which takes care of anisotropy, an im-provement captured by RSM models and not captured by the scalareddy viscosity approach employing a Boussinesq constitutive rela-tion. In addition, RSM accounts for the effects of streamline curva-ture, and the rapid changes in the strain rate more rigorously thanthe standard k– � model. In the bulk region where isotropy exists,both the velocity components get satisfactorily predicted well byboth the standard k– � and the RSM model. In case of LES, since itdirectly solves for the instantaneous scales, the entire mean flowhas been captured well.

PBT impellers generate a mixed axial–radial flow in the form ofa jet which spreads radially as it progresses towards the base of thevessel entraining fluid adjacent to the impeller. After hitting the base,part of the axial momentum gets converted to radial componentalong the base towards the wall of the vessel. As it can be seenfrom the results, the axial mean velocity flow field of all the three

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Fig. 20. Comparison between the simulated and experimental profiles of the dimensionless mean radial velocity for HF at various axial levels. (A) H=0.01m; (B) H=0.044m;(C) H = 0.082m; (D) H = 0.1m; (E) H = 0.118m; (F) H = 0.154m; (G) H = 0.190m and (H) H = 0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

impellers has been predicted well by all the three turbulence models.However, the mean radial and tangential velocities in the impellerregion have not been well predicted by the standard k– � due to thestrong anisotropic nature of the flow as discussed above. Both theRSM and LES faithfully captured the flow associated with the swirlcomponent like that generated by a DT.

The total kinetic energy is high in the jet, in the vicinity of theimpeller. Indeed, in the region near the impeller blade, the tangen-tial velocities of the trailing vortices are high and the associated pe-riodic energy is significant. The trailing vortices vanish far from theimpeller region. Just away from the impeller blade, kinetic energyassociated with the periodic motion decreases, however, the totalturbulent kinetic energy remains practically constant. Which indi-cates a continuous conversion of kinetic energy (periodic compo-nent) into turbulent kinetic energy. For this situation of flow, boththe standard k– � and RSM fail to capture such a transfer processdue to unsteady and complex nature of flow structures in the im-peller region. Therefore, in the near impeller regions, these modelsconsistently underpredicted the turbulent kinetic energy.

In the bulk region, all the three models were found to predictclose to the experimental data. These deviations can be attributedto the energy transfer mechanism in each of the model and its abil-ity to handle intermittency in the flow. The best agreement with

the data comes from the LES predictions which are able to cap-ture the dynamic behavior of the coherent structures. Therefore, LESpredictions for all the five impellers, have been in good agreementwith the experimentally measured turbulent kinetic energy. How-ever, the results reaffirm that the RSM outperforms the k– � modelwhen swirling flows and recirculations are present, i.e. flow in thevicinity of the DT and PBTD60 impellers. In the cases of PBTD30 andHF, it has been shown that, due to the relatively much lower inten-sity of swirl motion, the results show that the standard k– � modelcan even predict the flow fields equally well along with the RSMand LES.

6.4. Instantaneous snap shots of flow field

The instantaneous vector field obtained from LES in a verticalmidbaffle plane of the stirred vessel is depicted in Fig. 23A–D forDT, PBTD60, PBTD45 and PBTD30 impellers, respectively. These re-ports instantaneous velocity vector plots at the time instant of t=5 s(corresponding to 23.5 impeller revolutions). As can be seen fromFig. 23 that the instantaneous flow field is highly complex and ran-dom in nature. First of all the flow is never symmetry. Further,it should be noted that the flow field is unsteady, and that these

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Fig. 21. Comparison between the simulated and experimental profiles of the dimensionless mean tangential velocity for HF at various axial levels. (A) H = 0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

images show snapshots of the flow field only. In the case of DT, astrong radial discharge flow can be observed, which is most intensein the plane containing the impeller. Large and irregular secondaryrecirculation structures are present throughout the vessel. For theaxial flow impellers, PBTD60 tend to generate much complex andirregular instantaneous flow than PBTD45 and PBTD30. It is mainlydue to strong interaction between the axial and tangential flow com-ponents. With a decrease in the blade angle, the flow can be seen tobe less and less complex. This particular feature provides a justifica-tion for the good predictive capabilities of the standard k– � modelfor the PBTD30 and HF impellers.

The mean flow and turbulence fields in a fully baffled vesselstirred by various impeller designs (DT, PBTD60, 45 and 30, and HF) atone clearance have been investigated with laser-Doppler anemom-etry (LDA) and LES to characterize the instabilities present in suchflows. Time-resolved velocity measurements were made and the fre-quency content of the velocity recordings was analyzed with FFTtechniques. The study aims to identify the flow instabilities and asso-ciated energy with them as well as frequency with respect to the im-peller design. The frequency of the precessing vortex instability wasfound to be linearly related to the rotational speed (∼ 0.015– 0.02N,Hz) of the impeller and to be essentially independent of the im-peller design. The LDA data and LES predictions indicated clearly that

the precessing vortex instability stems from a precessional motionabout the vessel axis, similar to the precession encountered in mostswirling flows.

In the present study, the Reynolds number of the flow was 45,000for all the impeller designs under consideration. To estimate theenergy, time-resolved velocity measurements were made and thedata obtained due to LDA are random in nature with respect to time.For any kind of transformation, it is desirable that the time seriesshould be expressed in terms of the variable at equal time intervals.This conversion of a random time series to a time series in terms ofdiscrete data at equal time intervals is termed here as equispacing ofthe data. For this purpose, linear interpolation technique was used.Further, the signal obtained after equispacing was again processedusing FFT. The power spectrum was obtained and the frequencies ofinstabilities were identified and corresponding energy on y-axis isenergy associated with that instability.

The energy of precessing vortex instability for both DT andPBTD60 impellers near the vessel surface was found to be highestand about 9m2 s−2

. And for PBTD45 and PBTD30 impellers the en-ergy was found to be 1.8 and 1.35m2 s−2 which is one-fifth of DT andPBTD60. Whereas for HF impeller the energy content is 0.8m2 s−2,which is relatively less among the impeller designs considered inthis study. It can be mainly attributed to the intensity of swirl


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Fig. 22. Comparison between the simulated and experimental profiles of the dimensionless mean turbulent kinetic energy for HF at various axial levels. (A) H = 0.01m; (B)H=0.044m; (C) H=0.082m; (D) H=0.1m; (E) H=0.118m; (F) H=0.154m; (G) H=0.190m and (H) H=0.244m: �—Experimental; (1) LES model; (2) RSM and (3) k– � model.

generated by DT and PBTD60 than the other impeller designs. It canbe realized that the flow generated by HF is mainly a convectiveflow.

Also, the macroinstability is triggered by the complex interac-tion of the impinging jet from the impeller discharge stream witheither the tank wall or bottom. The resulting oscillation in the cir-culation pattern is called as jet instability. The frequency of the jetinstability was found to be linearly related to the rotational speed(∼ 0.13– 0.2N, Hz) of the impeller and to be essentially independentof the impeller design for a given impeller clearance. The energiesassociated with the jet instability for all the five impellers, i.e. DT,PBTD60, PBTD45, PBTD30 and HF, have been found to be 42, 38, 0.64,8 and 12m2 s−2, respectively.

Further, there has been an apparent discrepancy in the literatureon the possible interaction between the jet instability and the pre-cessing instability which subsequently leads to intermediate insta-bilities (between jet/circulation and precessional vortex). Therefore,in this study, an attempt has been made to identify the intermediateinstabilities with respect to various impeller designs. It can be seenin Fig. 24(A–E) that the occurrence of some intermediate instabili-ties in the frequency range of 0.2–0.3 (i.e. f/N=0.04– 0.07) for all theimpeller designs. Energy content of these intermediate instabilityhave been found to be 8m2 s−2 (DT), 8m2 s−2 (PBTD60), 3m2 s−2

(PBTD45), 2m2 s−2 (PBTD30) and 1m2 s−2 (HF). These kind of

instabilities might be part of the cascading of the large scale insta-bilities. Further a separate detailed study of identification, quantifi-cation and relating these instabilities to design objectives is underprogress.

6.5. Energy balance

It was thought desirable to assess the performance of the differ-ent turbulence models by establishing the energy balance. The totalpower dissipation rate was calculated by volume integration of thepredicted turbulent kinetic energy dissipation rate (�) as

P =∫ 2�0

∫ H0∫ R0 �r dr dzd�∫ 2�

0∫ H0∫ R0 r dr dzd�

(11)

In LES model, � has been estimated based on the SGS viscosity ob-tained as

� = �ijSij = −2�SGSSijSij = −(CS)2|Sij|3 (12)

By assuming local equilibrium between production and dissipationat and below subgrid-scale level, the energy dissipation rate in theLES can be coupled to the deformation rate.

From the predicted energy dissipation rate (Eq. (13)), the valuesof power number (NP) for all the five impellers were obtained using


Fig. 23. The instantaneous vector field in a vertical midbaffle plane: (A) DT; (B) PBTD60; (C) PBTD45 and (D) PBTD30.

the following equation:

NP = P

�N3D5(13)

Table 2 shows the comparison of power numbers predicted by thestandard k– �, RSM and LES models for all the five impellers with theexperimentally measured values. The power consumption was mea-sured by measuring torque on table, below which the tank remainsstationary. For this purpose, the torque table was restrained from ro-tating by a string and the force on the string was then measured byconnecting it to a cantilever type load cell. The pre-calibrated loadindicator displays the load on the torque table. Ten readings weretaken for each set and average was used for power measurement.The predicted NP by LES can be seen to be in close agreement withthe experimental values which implies good overall energy balance.Whereas, both the standard k– � and RSM models consistently un-derpredicted the power numbers for all the impellers.

The energy balance for all the impellers was also established usingan alternative procedure. The power consumption P is calculated asthe product of torque on the impeller blades and the angular velocity.

This is then used for the estimation of power number and it can beexpressed as follows:

NP = 2�NM�N3D5

(14)

where torque (M) exerted in all blades was computed by summingthe cross product of the pressure and viscous forces vectors withcorresponding distance vector for each computational cell on theimpeller surface. The predictions of the power numbers for all thefive impellers are shown in Table 3. The power number (NP) hasbeen well predicted by all the three turbulent models. It can benoted that, the power number calculated from the integration of localepsilon value obtained from RANS based models tends to be under-predicted in the range of 20–25% for DT and PBTD60 impellers, a sim-ilar behaviour has already been reported in the literature (Brucatoet al., 1998; Patwardhan, 2001). For the other two pitched blade im-pellers (PBTD45 and PBTD30) and HF impeller a close agreementbetween predictions and the experimental values can be seen. Onthe other hand, LES found to closely predict the power numberwith the maximum of 7% deviation in the case of DT. However,


Fig. 24. Power spectra for the different impeller designs (Z/T = 0.27, r/R = 0.6): (A) DT; (B) PBTD60; (C) PBTD45; (D) PBTD30 and (E) HF.

for PBTD60, PBTD45, PBTD30 and HF impellers close agreement hasbeen obtained between the experimentally measured values and LESpredictions.

7. Conclusions

Three-dimensional mean flow field and the turbulent kinetic en-ergy of the baffled stirred vessel agitated by five impellers (DT, PBT

(60◦, 45◦ and 30◦) and an HF) have been measured using LDA. Forall the above cases three-dimensional CFD simulations have beenperformed by the standard k– �, the Reynolds stress transport andlarge eddy simulation turbulence models. The pressure strain termsin RSM was modeled based on the proposal in the literature that issuitable for the present flow configuration. In case of LES, for thefirst time, one equation dynamic subgrid scale model has been suc-cessfully employed for the stirred vessel geometry. The predicted


Table 2Comparison of the power number predicted (integral � based approach) by standard k– �, RSM and LES turbulence models

Turbulence model Disc turbine (Exp NP = 5.1) PBTD60 (Exp NP = 3.1) PBTD45 (Exp NP = 2.24) PBTD30 (Exp NP = 1.3) HF (Exp NP = 0.27)

Standard k– � 3.9 2.6 1.9 1.1 0.25RSM 4.1 2.75 1.9 1.1 0.25LES 4.7 2.85 2.1 1.2 0.26

Table 3Comparison of the power number predicted (torque based approach) by standard k– �, RSM and LES turbulence models

Turbulence model Disc turbine (Exp NP = 5.1) PBTD60 (Exp NP = 3.1) PBTD45 (Exp NP = 2.24) PBTD30 (Exp NP = 1.3) HF (Exp NP = 0.27)

Standard k– � 4.9 2.95 2.3 1.35 0.28RSM 5.0 3.2 2.2 1.25 0.28LES 5.2 3.0 2.1 1.38 0.28

flow fields by all the three turbulence models were comprehensivelycompared with the experimental data. The following conclusions canbe drawn from the present work.

1. As for as mean flow predictions are concerned, the RSM per-formed better than the standard k– � model when comparisonswere made with DT. The results reaffirm that the RSM can out-perform the k– � model when recirculations are present. This isdue to the overestimation of the eddy viscosity which is a generalcharacteristic of the k– � model. In general, it can be concludedthat the standard k– � model performs well when the flow is uni-directional that is with less swirl and weak recirculation.

2. The standard k– � model tends to satisfactorily represent all theflow parameters generated by a hydrofoil (HF) impeller. Since theHF impeller produces such a weak swirl flow, these results showthat all the three turbulence models can predict the flow fieldequally well.

3. Both the standard k– � model and anisotropy RSM fail to predictthe turbulent kinetic energy profiles in the impeller region whenthe flow is dominated by the unsteady coherent flow structures.

4. Using both the LDA and LES tools, the strength of the precess-ing vortex instability has been quantified for all the five impellerdesigns. It has been observed that the impeller which generatesstrong swirl flow generates equally strong was the main reasonfor such instabilities. Therefore, among the impeller designs con-sidered in the present study, the DT produces strongest instabil-ities and the HF generates the weakest instabilities.

5. The frequency of the jet instability was found to be linearly relatedto the rotational speed (∼ 0.13– 0.2N, Hz) of all the impeller de-signs under consideration. The energies associated with the jet in-stability for all the four impellers, i.e. DT, PBTD60, PBTD45, PBTD30and HF, have been found to be 42, 38, 0.64, 8 and12m2 s−2, re-spectively.

6. Further, consistent occurrence of intermediate instabilities havebeen observed having frequency of 0.04–0.07N which lies inbe-tween the precessional and the jet instability. The origin of thesekind of instabilities could be due to the interaction of precess-ing vortex instability with either the mean flow or jet/circulationinstabilities.

7. Having demonstrated that the LES model provides predictionsthat agree well with the measurements, and these simulationscapture relatively many flow features, these simulations can befurther utilized to explore a variety of issues with a reasonableconfidence.

8. Energy balance has been established using both integral � andtorque based approaches. It can be concluded that the torquebased approach seems to be more promising for the estimationof power number for a new impeller design and using by compu-tationally economical RANS based simulations.

Notation

C impeller clearance from the tank bottom, mC�, C�1, C�2 turbulence model parameters in the k– � modelCk, C� SGS model constantsdw distance of a grid point closest wall, mD impeller diameter, mDT disc turbineDij diffusion of Reynolds stresses, m2 s−3

D� turbulent diffusion of dissipation, m2 s−4

f frequency, s−1

H liquid height, mHF hydrofoil impellerk turbulent kinetic energy, m2 s−2

lo characteristic length scale, mM torque, NmN impeller rotation speed, s−1

NP power number of the impellerNQP primary flow number of the impellerP power dissipation, m2 s−3

PBTD30 pitched blade downflow turbine with a blade angleof 30◦



P filtered pressure term for LES model, Nm−2

P� production of dissipation, m2 s−3

Pij production in Reynolds stress transportequation,m2 s−3

r radial coordinate, mR radius of the vessel, mRij Reynolds stress tensor, Nm−2

Sij strain rate of the resolved scales, s−1

t time, sT tank diameter, m〈ui〉 time average of velocity, ms−1

〈u1〉 time averaged radial velocity, ms−1

〈u2〉 time averaged axial velocity, ms−1

〈u3〉 time averaged tangential velocity, ms−1

ui filtered velocity, ms−1

Utip impeller tip velocity, ms−1

V volume of the computational cell, m3

z axial coordinate, m

Greek letters

� Kronecker delta filter width, m


� turbulent energy dissipation rate, m2 s−3

Kolmogorov length scale, m� tangential coordinate,rad� Taylor microscale, m� kinematics viscosity, m2 s−1

�ij pressure strain term in RSM equations, m2 s−3

� density of the fluid, kgm−3

k turbulent Prandtl number for the turbulent kineticenergy

� turbulent Prandtl number for the dissipation rate�ij shear stress in i-direction, Nm−2

�o characteristic time scale, s�sgs subgrid-scale stress, Nm−2

�sgs subgrid scale eddy viscosity, m2 s−1

�� turbulent destruction of dissipation, m2 s−4

Subscripts and Superscript

i, j, k axis indexes of space coordinatesl molecularsgs subgrid scalet turbulence′ fluctuating quantity

Acknowledgment

One of the authors, Mr B.N. Murthy gratefully acknowledge thefinancial support during this work by the Department of AtomicEnergy (DAE), Government of India.

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Assessment of standard k–ε, RSM and LES turbulence models in a baffled stirred vessel agitated by...

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