perturbed turbulent stirred tank flows with amplitude and mode-shape variations

Chemical Engineering Science 66 (2011) 5703–5722

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Perturbed turbulent stirred tank flows with amplitude andmode-shape variations

Somnath Roy, Sumanta Acharya n

Mechanical Engineering Department, Louisiana State University, Baton Rouge, LA 70803, USA

a r t i c l e i n f o

Article history:

Received 14 March 2011

Received in revised form

2 August 2011

Accepted 3 August 2011Available online 10 August 2011

Keywords:

Stirred tank

Perturbation

Impeller-jet

Periodic fluctuations

Turbulence

Power number

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ces.2011.08.005

esponding author.

ail address: [email protected] (S. Acharya).

a b s t r a c t

Stirred tank (STR) flows at low and moderate Reynolds numbers show poor mixing behavior due to

formation of segregated zones inside which both magnitude and fluctuation level of velocity

components show lower values compared to the active fluid regime (i.e., impeller jet stream, circulation

loops). Active perturbation of the STR flow using a time-dependent impeller rotational speed can

potentially enhance mixing by breaking up these segregated unmixed zones and enhancing the

turbulence level throughout the tank volume. In the present study, the effect of different perturbation

cycles on an unbaffled turbulent stirred tank flow at a moderate Reynolds number (rotational speed

N¼3 rps) is studied using a large-eddy simulation (LES) technique coupled with immersed boundary

method (IBM). The perturbation frequency (f) is chosen to correspond to a dominant macro-instability

in the flow (f/N¼0.022). Two different perturbation amplitudes (20% and 66%) and two perturbation

shapes (square-wave and sine-wave) are investigated, and changes in the mean flow field, turbulence

level and impeller jet spreading are examined. Large-scale periodic velocity fluctuations due to

perturbations are noticed to produce large strain rates favoring higher turbulence levels inside the

tank. Production of turbulent kinetic energy due to both the mean and periodic component of the

velocity field is presented. Fluctuations in power consumption due to perturbation are also calculated,

and shown to correlate with the perturbation amplitude.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Stirred Tank Reactors (STRs) represent over $350 billion ofproduct yield in the U. S chemical industry with nearly $ 10billion per year lost due to inefficient operation of the mixingdevices (Paul et al., 2004). Therefore, improvements in existingtechnologies can potentially translate to several billion dollars inannual cost savings. Many of these mixing and blending opera-tions (e.g. polymerization, petro-chemical blending) are carriedout at laminar and transitional regime due to high viscosity of theworking fluids. Hindrance to proper mixing arises due to forma-tion of segregated regions in laminar or moderately low Reynoldsnumber turbulent/transitional stirred tank flows (Dong et al.1994; Harvey et al., 1996), which acts as barriers to the impel-ler-jet stream allowing unmixed fluid to reside in the tank andincreasing the mixing time as well as the amount of byproductsgenerated in industrial operations. So, the dispersion of thesegregated regions via enhanced spreading and mixing out ofthe impeller jet-stream is one of the primary objectives of studiesin mixing enhancement.

ll rights reserved.

Aref (1984) and Aref and Balachandar (1986) showed thatperiodic perturbation sets in chaotic advection in low Reynoldsnumber viscous flows diminishing the sizes of the stagnationzones. Franjione et al. (1989) showed that in lid-driven cavityflows, periodic lid velocity can enhance mixing. For a similar lid-driven cavity flow, Liu et al. (1994) observed that unstablemanifolds wrap themselves around the islands preventing forma-tion of segregated low stretching zones. Lamberto et al. (1996)introduced instabilities in the flow field through a time varyingrpm of the impeller, and for a sufficient low Reynolds number(Re�9–18) flow, they performed a direct visualization of acid–base-indicator mixture to observe the break-up of the segregatedtori structures due to the imposed dynamic perturbations. Yaoet al. (1998) also observed an improvement in mixing rate byperturbing the STR flow with a time-varying impeller rotationalspeed. They also obtained a faster mixing by periodically chan-ging the direction of impeller rotation. Lamberto et al. (2001)demonstrated that with change in rotational speed the stable toriformed during one constant impeller speed breaks up and theunmixed fluid disperses in the outer fluid. Ascanio et al. (2002)studied combined effects of variable rpm and off centered shaftsand obtained a high mixing rate. Using impellers mounted on twodifferent off-centered shafts, they obtained a complete break-upof segregated zones. Bar-Ell et al. (1983) and Schneider (1985)

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S. Roy, S. Acharya / Chemical Engineering Science 66 (2011) 5703–57225704

demonstrated that low frequency periodic perturbations of theconcentration field inside the continuous stirred tank reactorsaffects the chemical kinetics and significantly improve thereaction rate.

Stirred tank flows start showing turbulent behavior around aReynolds number (Re¼ND2=n) of 200 (Lee et al., 1957; Distelhoffet al., 1995) and the flow becomes fully turbulent at a much higherReynolds number (Re�70000 for Rushton impellers). However,even in the higher Reynolds number regime (Re�50000), turbu-lence levels are significant only along the impeller jet stream(Kresta et al., 2001); only for a much higher Reynolds number flow(Re�150000), Roy et al. (2010) observed turbulence to spreadoutside the impeller jet-stream and the main circulation regime.The poor distribution of turbulent fluctuations reduces the globalrate of mixing inside the stirred tank and this problem is moresevere for the lower Reynolds number turbulent flows (Re�1000)where the impeller jet stream itself is much narrower and also thepeak turbulent kinetic energy is of smaller orders (Dong et al.,1994). However, at Re�1000–10000, an almost constant non-dimensional circulation time (Holmes et al., 1964) indicates thatthe mean flow patterns do not change significantly with increasesin stirring rate. Gao et al. (2007) also observed that width of theimpeller jet-stream did not increase significantly when Reynoldsnumber was increased from 1150 to 1559. Hu et al. (2010) reportsthat even for the low Re STR flows (Re�900–2000), the degree ofreaction (which is governed by the global mixing rate) is alsoinfluenced by the turbulence levels inside the tank. But, the LDAexperiments of Dong et al. (1994) did not report any improvementin the peak turbulent kinetic energy when Reynolds number isincreased from 3273 to 4008. Therefore, increases in impeller jetwidth as well as enhancements in the magnitude and spreading ofturbulent fluctuations are the key objectives for mixing enhance-ment in this type of flows at the lower Reynolds number. The ideaof periodic excitation of flow field can be utilized in order toaccomplish both of these objectives for a moderately turbulent STRflow. Exploring this idea and the role of the perturbation para-meters for the lower Re STR is the primary motivation of this paper.It is argued that in this regime, both the mean flow and turbulencecan be substantially effected by the external perturbations.

Periodic excitations of turbulent shear layers in simple flows(jets, back-steps, etc.) have been well studied. It has beenobserved that excitation leads to formation of strong discretevortices (Fiedler et al., 1985). Strong excitations at subharmonicfrequencies lead to successive merging of these vortices resultingin vortex pairing by which the shear layer entrains more fluidfrom surrounding and grows in size (Riley et al., 1980). Perturbedshear layers grow significantly in presence of background turbu-lence indicating that turbulent fluctuations couple with forcedperturbation to enhance this entrainment process. As the turbu-lent entrainment of the surrounding irrotational fluid increasesthe vortex radius, the vortices become unstable and anotherdynamic process, referred to as vortex tearing takes place(Moore et al. 1975; Pierrehumbert et al., 1981) due to whichthe larger vortices break up into smaller ones. Wyganski et al.(1987) indicated that in turbulent free shear flows, coherentstructures show different growth patterns based on the frequencyand magnitude of the perturbations and leads to enhancement inturbulent fluctuations. The reattachment length in a turbulentbackward facing step was observed to decrease by 30% in the PIVstudy of Yoshika et al. (2001) when perturbation at an optimumfrequency was applied and Reynolds stresses increased markedlynear the reattachment zone. Turbulent free jets show a preferredmode for the passage of vortical structures during externalperturbations for which the ratio of the peak amplitude to theinitial amplitude reaches a maximum value (Crow et al., 1971).Perturbations are reported to break the axisymmetric nature of

the circular free jet establishing a helical mode of energy transfer(Drubka, 1981). Due to the background oscillations, jet spreadingas well as turbulent fluctuations along the axial directionincreases (Ivanov, 1972). Using a triple decomposition of turbu-lent flow field Hussain (1970) showed that in presence of periodicperturbations, the energy is drained from the organized large-scales of the periodic motion to the smaller turbulent scales. Thus,the vortices originated due to periodic excitation of the shearlayer break down into smaller vortices following an energycascading until they reach the dissipative scales and thus enhancelevels of turbulence throughout the flow domain. For flow past arib, Acharya and Panigrahi (2004) observed that external largescale perturbations enhance spreading of the shear layer as wellas mixing of the fluid elements. Hwang et al. (2003) showed thatmixing and heat transfer between two fluid-streams are stronglyenhanced when one of the input streams is perturbed withsignificant amplitude.

For an unbaffled Rushton impeller turbulent stirred tank flow,Gao et al. (2007) used this idea of actively perturbing the flowwith the objective of obtaining better mixedness. In their PIVstudy, a step-variation of the impeller speed has been used todynamically perturb the stirred tank flow field. They obtainedsignificant improvements of turbulent fluctuations, which con-tributed to an enhanced spreading of the radial jet-stream comingout of the impeller blades and augmented the mixing of fluidinside the tank. Stirred tank flows at constant rotational speedsare reported to show low frequency high amplitude oscillationscalled macro-instabilities (MI) due to (i) instabilities in theimpeller jet-stream and circulation pattern and (ii) formation ofthe precessional vortices, which move around the shaft with afrequency lower than the blade-passage frequency (Kresta et al.,2004; Galletti et al., 2004). Hasal et al. (2004) observed that forbaffled Rushton impeller stirred tanks, two distinct MI frequen-cies are observed over different Reynolds number regime. At avery high Reynolds number (Re�60000) an MI with f/N¼0.025(N is the impeller rotational speed in rps) has been observedwhere as at a lower Reynolds number (Re�9000), another MIwith f/N¼0.085 is also observed along with the lower frequencyMI. Galletti et al. (2004) identified three different ranges ofReynolds number correlated with the dominant MI behavior forbaffled impeller stirred tanks: (a) 400oReo6300: where thedominant MI is at frequency f/N¼0.106, (b) 6300oReo13600:where two different dominant MI frequencies are observed asf/N¼0.106 and f/N¼0.015, and (c) 13600oReo54400: where thedominant MI is at frequency f/N¼0.015. In their LES study Fanet al. (2007) observed a peak MI of f/N¼0.027 in the impeller jetregion for an STR flow of Reynolds number 9720. Kresta et al.(2004) and Yianneskis (2004) explained that the lower frequencyMI (f/N�0.02) arises due to processional vortex formationwhereas the other MI (f/N�0.1) comes from the flow patterninstabilities in baffled impeller tanks. For pitched-blade STRs at aReynolds number of 44000, Roy et al. (2010) showed that theradial baffles assume a very vital role in originating the flowpattern instabilities. In their DNS simulation of unbaffled RushtonSTR at Reynolds number 1636, Verzicco et al. (2000) observedthat the peak amplitude of velocity fluctuations occurs atf/N�0.025. In our present LES–IBM study on an unbaffled Rushtonimpeller stirred tank at Re¼1140, the dominant MI oscillation isobserved to occur at a frequency f/NE0.023 in the near impellerregion (Fig. 1). While macro-instabilities act as added perturba-tions over the impeller rotation contributing to significantamount of kinetic energy of velocity fluctuations (Roussinovaet al., 2000), our earlier studies showed that rate of scalar mixinginside a pitched-blade turbulent stirred tank can be correlatedwith dominant MI oscillations (Roy, 2010). Ducci and Yianneskis(2007) observed that injecting the passive scalars inside the MI

Fig. 1. FFT of radial velocity signals showing macro-instability period for 3 RPS

fixed impeller rotational speed at r/R¼2, z/T¼�0.033.

S. Roy, S. Acharya / Chemical Engineering Science 66 (2011) 5703–5722 5705

vortex reduce the mixing time by 30%. Therefore, low frequencyMI oscillations contribute significantly to the overall mixingperformance of the STR where MI oscillations contribute to18–30% of the fluctuating kinetic energy (Roy et al., 2010). Hence,perturbing the STR flow at a similar frequency but with higheramplitude are expected to produce further enhancement inmixing. STR flows perturbed at a MI frequency is reported toenhance the impeller jet-width and promote turbulence andgrowth of trailing edge vortices (TEV) (Roy et al., 2008; Roy,2010). However, in order to specify optimal operational condi-tions for perturbed STR flow, different perturbation strategies areto be explored to compare their effects on the mean flow andturbulence parameters.

In the present study, the unbaffled Rushton STR flow-field isactively perturbed by varying the impeller speed at a frequencyequal to the MI frequency (f/NE0.02, N being the mean impellerrotational speed). Three different perturbation cycles are chosen atthis frequency (time period of 15 s): (A) a sinusoidal variation of theimpeller speed with amplitude of 20% of the mean impeller speed(named case sine20) where impeller speed is varied from 2.7 to3.3 rps, (B) a step function with impeller speed variation from 2.7 to3.3 rps, i.e., an amplitude 20% of the mean Impeller speed (casestep20) and (C) step function with 66% amplitude of impeller speedvariation (case step66) where minimum and maximum impellerspeeds are respectively 2 rps and 4 rps. The mean impeller speed ischosen as 3 rps, which corresponds to a Reynolds number of 1145for impeller diameter of 0.1 m. The large eddy simulation data isanalyzed for changes in the mean flow pattern of the impeller jet-stream as well as the turbulent fluctuations during different phasesof the perturbation cycle for these three cases and compared withan unperturbed flow at the mean impeller speed. Power consump-tion during a perturbation cycle for each of the cases is calculated inorder to get an estimate of the penalty on operational cost forobtaining better mixing. The energy transfer from the organizedperiodic scales of motion to the turbulent scales is studied bycalculating the production terms of the kinetic energy budgetequations to understand the mechanism of generation of turbu-lence from the fluctuations in the large-scales.

2. Computational method

The non-dimensional governing equations for the conservationof mass and momentum for an incompressible Newtonian fluidare given as

@uj

@xj¼ 0 ð1Þ

@ui

@tþ@uiuj

@xj¼�

@p

@xiþ

1

Re

@2ui

@x2j

þ fi ð2Þ

where ui is velocity field, p is pressure, Re is the non-dimensionalReynolds number defined in terms of characteristic velocity andlength scales of the problem and fi is the body force term. In thispaper, the equations are solved in a stationary reference framewith the impeller surfaces representing a moving boundarycondition. The moving impeller geometry is handled here usinga special technique called the immersed boundary method (IBM)which is briefly discussed later, and described in detail for STRapplication by Tyagi et al. (2007).

In this study, a parallel multi-block compressible flow code forgeneralized curvilinear coordinates is used. For incompressibleflows, low Mach number preconditioning is used (Weiss et al.,1995). A second order 3-point backward time differencing isemployed for the temporal derivatives. A second order low-diffusion flux-splitting algorithm is used for the convective terms(Edwards, 1997), and second order central differences are used forthe viscous terms. An incomplete Lower-Upper (ILU) matrixdecomposition solver is used. Domain decomposition and loadbalancing are accomplished using a family of programs forpartitioning unstructured graphs and hypergraphs and computingfill-reducing orderings of sparse matrices (METIS, Karypis andKumar, 1998). The message communication in distributed com-puting environment is achieved using Message Passing Interface,MPI (Gropp et al., 1999). The multi-block structured curvilineargrids presented in this paper are generated using commercial gridgeneration software GridProTM. Surface meshes to model themoving rigid geometries are generated using another commercialpackage GambitTM.

IBM is used in which the boundary conditions along themoving rigid impeller geometries are incorporated through theaddition of a force term in Navier–Stokes equations (Gilmanovand Acharya 2008; Tyagi et al., 2007). In a simpler implementa-tion of the approach, nodes adjacent to the moving interface aretagged at each time step, and the solution variables (velocity,pressure) at these tagged points interpolated from the knownboundary conditions at the impeller surface (known velocity, andzero-gradient pressure) and the surrounding nodes where thesolution is obtained from the discretized conservation equations.The interpolation error can be controlled by the number of solvedpoints used in the interpolation. In this work, we have used adistance weighted interpolation involving at least 5 immersedsurface points (coming from an unstructured surface mesh) andone fluid point in each direction of the tagged point. Thisinterpolation scheme provides second order accuracy (Franke,1982). We have demonstrated that with this approach goodagreement is obtained with experimental results for STRs (Tyagiet al., 2007).

For representing turbulence, LES is a cost-effective approach inwhich the governing equations are spatially filtered to resolve thedynamics of the large scales, and modeling is done only for the‘‘universal’’ small scales (Tyagi, 2003; Sagaut, 2001). However, LESin complex geometries introduce additional challenges due to thecomputational effort needed for grid generation and commuta-tion errors introduced due to spatial filtering on non-uniformcurvilinear grids (Tyagi and Acharya, 2005). In LES, the governingequations are spatially filtered, with the filter width (proportionalto the grid size) representing the scales in the flow field that areresolved. The non-dimensional filtered governing equations forthe conservation of mass, momentum and energy for an incom-pressible Newtonian fluid in curvilinear coordinate system aregiven as (Jordan, 1999):

Continuity equation:

@

@xjðffiffiffigp

UjÞ ¼ 0 ð3Þ

Fig. 2. Schematic of the Rushton impeller tank (all dimensions are in mm).


Momentum equations:

@

@tðffiffiffigp

uiÞþ@

@xjðffiffiffigp

UjuiÞ ¼�

@

@xjðffiffiffigpðajÞipÞþ

@

@xj

1

Reþ

1

Ret

� � ffiffiffigp

gjk @ui

@xk

� �

with1

Ret¼ C2

s ðffiffiffigpÞ2=39S9 ð4Þ

where

ðffiffiffigp

Uj¼

ffiffiffigpðajÞkukÞ, ðajÞk ¼

@xj

@xk

� �are the contravariant velocity components and associated metricterms, respectively. The term

ffiffiffigp

is the Jacobian of the transfor-mation, and gij are the elements of the contravariant metrictensor. The strain rate tensor is given by

Sik ¼1

2ðamÞk

@ui

@xmþðamÞi

@uk

@xm

� �and 9S9 is the magnitude of the strain rate tensor. In the aboveequations the over-bar represents the filtered quantities.

The anisotropic subgrid (—) and subtest scale (�) stress tensoris formulated in terms of the Smagorinsky eddy viscosity model(Smagorinsky, 1963), and are respectively given by:

etaij ¼�2C2

s ðffiffiffigpÞ2=39S9Sij

eð5Þ

Taij ¼ 2C2

s aðffiffiffigpÞ2=39eS9eSij ð6Þ

The term Cs is the model coefficient and is assumed to be sameat both subgrid as well as subtest level. The ratio of filter widthsat the test level to the grid level is denoted by a. The GermanoIdentity (Germano et al., 1991) relates the SGS stresses atdifferent filter levels in terms of the filtered fields only

Tij ¼ uiujf�uifuif ,tij ¼ uiuj�ui uj

Lij ¼ Tij�etij ¼ ui ujf��uifuif

ð7Þ

where the Leonard stress,Lij, is the difference between subtest (�)and sub-grid (–) stress terms. Using the Smagorinsky’s model forSGS terms, the Germano identity relates the anisotropic compo-nents of Leonard stress with the strain rate tensor as

Laij ¼ Lij�

1

3dijLkk ¼�2C2

s ðffiffiffigpÞ2=3 a eS�� eSij� SjSij

g�� ð8Þ

Laij ¼

D�2C2

s ðffiffiffigpÞ2=3Mij ð9Þ

C2s ¼�

1

2

1

ðffiffiffigpÞ2=3

LaijUMij

MijUMijð10Þ

For numerical stability, the coefficient Cs is limited to positivevalues only and smoothed locally (Tyagi and Acharya, 2005).

3. Geometry and computational details

The configuration of the stirred tank (shown in Fig. 2) is thesame as that of Gao et al. (2007) for which measurements werereported, and consists of a 305 mm (T) tall cylindrical tank with a16 mm diameter shaft located at the center of the tank. A Rushtonturbine with a 2 mm thick disk and six blades symmetricallyarranged along its circumferential direction is fixed on the shaft.The distance between the blade vertical center and the flat baseplate of the tank, c, is equal to 110 mm. The inner diameter of thecylindrical tank, D1, is approximately equal to 294 mm. The six-blade Rushton turbine impeller has a tip to tip diameter (D¼2 R) of100 mm. The blade width w, is equal to 20 mm, its length L, is equalto 25 mm, and its thickness, t is 2 mm. The experimental

configuration is chosen here to enable comparisons of the simula-tion with data in an identical configuration. A light mineral oil isused for operating fluid with density, r¼900 kg/m3 and dynamicviscosity, m¼0.02358 Pa s.

The Reynolds number is determined by Re¼ND2/n where, N isthe impeller rotational speed (rev/s), D is the impeller diameter(m), and n is the kinematic viscosity of the working fluid (m2/s).In this study the mean Reynolds number (calculated for meanimpeller speed 3 rps) is 1145. For perturbation with 66% ampli-tude, Re varied between 764 and 1527, while with 20% perturba-tion amplitude the Re varies between 916 and 1374. In all cases,the mean Re is maintained at 1145.

Our previous studies (Tyagi et al., 2007; Roy et al., 2010) withpitched-blade baffled turbulent STRs showed that for a Reynoldsnumber of 7280, a grid system with 2.3 million cells satisfactorilyresolved the turbulent scales, and for Reynolds number of the orderof 105, grid-independent solutions are obtained for a grid comprising3.1 million cells (by comparing with a 5.2 million cell simulation).Based on these experiences, the computational grid used for thesimulation of the present stirred tank flow (Re¼764�1527) is madecomposed of 3552 blocks comprising of about 4.2 million grid points(generated using GridproTM). The surface mesh for the impellerblade contained 75166 triangular elements and is generated usingGambitTM. One rotation of the impeller blades took approximately90 min on 320 processors (64 bit 2.33 GHz quadcore Xeon). Atypical grid-independence study cannot be performed for LES thatuses a filter size, which is the same as the grid width because themodeled subgrid stresses change with the grid spacing. Therefore, inassessing the grid independence for LES, one has to consider that thedifferences observed may be related to the accuracy of the sub-gridstress modeling. However, finer grid solutions were obtained with6.3 million cells and compared with coarser mesh solutions alongwith the experimental data (Fig. 3) to verify the accuracy of thecomputed solutions. The coarser mesh solutions are seen to be inexcellent agreement with the finer mesh results and are used for theanalysis presented in this paper. However, the impeller powernumber is calculated using the finer mesh solutions as this calcula-tion involves interpolation of pressure near the impeller walls andcomputing the wall shear stresses.

In order to obtain converged time-averaged velocity andturbulent statistics, the flow field is averaged at the blade passagefrequency for 90 s of flow during which 270 complete rotations ofthe impeller blade have been obtained (17 days of computationon 320 processors). This corresponds to 6 cycles of the perturba-tion (at a perturbation cycle time period of 15 s or f/N¼0.022)and a total of 1620 samples that have been averaged during this

Fig. 3. Comparison with experimental data and validation.

Fig. 4. Impeller-plane velocity vectors showing the profile of the radial jet-stream (in solid red lines): (a) 3rps; (b) 4rps; (c) sine20; (d) step20 and (e) step66.

(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


period. Results are confirmed to be statistically converged withthe number of samples used for the averaging process.

4. Results and discussion

4.1. Validation and verification

The mean velocity (i-th component) averaged over all bladepositions is defined as

/UiS¼XNb

i ¼ 1

/Ui9bS

Nbð11Þ

where /U i9bS is the phase-averaged velocity at different angularpositions and Nb is the total number of these angular positions

used in the averaging. The phase-average velocity is defined as

/Ui9bS¼XMp

j ¼ 1

Uji ðtÞ9bMp

ð12Þ

where Uji ðtÞ is the i-th component of instantaneous velocity at

time t (j-th sample), and Mp is the number of samples acquired atthe b-plane during phase averaging (as noted earlier,Mp is 1620 inthe present study corresponding to six cycles of impeller speedperturbation). Fig. 3(a) shows mean radial velocity profile alongan axial line at r/R¼1.6 for both finer mesh and coarser meshsolutions compared with experimental data of Gao et al. (2007)for constant 3 rps impeller speed. Fig. 3(b) shows phase averagedrms of radial velocity fluctuation along the same axial line. Thepredictions are in excellent agreement with experimental results

Table 1Time-averaged flow number for different cases.

Cases Flow number (NQ)

Fixed 3 rps 0.694

Sine20 0.71

Step20 0.74

Step66 1.08

Table 2Percentage of fluid volume occupied by slowly

moving fluid (o20% impeller speed).

Cases % of volume

Fixed 3 rps 30.1

Sine20 21.8

Step20 15.6

Step66 8.7

Fig. 5. Radial variation of the jet-width averaged over six angular locations.


and do not change significantly as the grid resolution changes. Forthe mean velocity the agreement is within 5% and for the rms it isgenerally less than 10–15%. Experimental uncertainties reported forthis data is also in the range of these observed differences, andtherefore this agreement should be judged to be very good.Figs. 5 and 8 also present comparisons with experimental data thatshow good agreement between predictions and measurements.

4.2. Effects of impeller speed variation on the mean flow field

statistics

4.2.1. Mean velocity vectors and jet spreading

The time-averaged velocity vectors at the impeller plane

(/Ui9b ¼ 0S) obtained for the baseline case 3 rps is shown in

Fig. 4(a). A constant 4 rps case is considered (Fig. 4b) as an upperlimit of impeller speed. Vectors for the different perturbationcycles (sine20, step20, and step66) are shown in Fig. 4(c)–(e),respectively. The zero radial velocity iso-contour is depicted bythe solid red line and represents the boundary of the impeller jet-stream. The impeller-jet is observed to spread in the axialdirection as it moves towards the tank wall. The spread of theimpeller jet-stream is visibly much higher due to perturbationseven comparing to the high-speed 4 rps case. For the highamplitude perturbation (step66), the jet entrains greater amountof fluid from the lower portion of the tank and becomes inclinedtowards the bottom wall of the tank (Fig. 4e). For all threeperturbation cases, strong axial velocity vectors (higher than thebaseline) are seen near the side wall of the tank. This implies thatthe jet stream hits the side wall with a higher momentum for theperturbed cases and circulates along the wall forming a higher-speed boundary layer along it. Recirculation bubbles below andabove the impeller are therefore larger in size for the perturbedflows. For the sine20 case, impeller jet spreading is mostlysignificant after r/R¼2.0, which can be associated with theperiodic flapping of the impeller jet due to perturbation at thislevel (discussed later and shown in Fig. 12).

The impeller stream coming out of a Rushton impeller behaveslike a free jet ejected into a bulk of slowly moving fluid, whichspreads out vertically entraining a considerable amount of surround-ing fluid (Dong et al., 1994). The spreading of impeller jet-streamplays a vital role in the fluid mixing process and can be considered asone of the measures of mixing inside the stirred tank. This spreadingof the jet-stream can be characterized by its width calculated as thedistance between the axial locations above and below the impellerwhere the radial velocity diminishes to zero (shown in Fig. 4 by thesolid red line). Using this definition, the widths of the radial jetcorresponding to different perturbation cycles are compared.

Fig. 5 shows the mean impeller-jet spreading (averaged over sixplanes from 01 to 601, each 101 apart) as a function of radial distancefrom the impeller tip. Experimental data for the fixed 3 rps case isshown by the open symbols (Gao et al., 2007); the simulated resultsfor this case (solid line) show excellent agreement with themeasured data. It can be observed that the step66 case shows asignificant increase in impeller jet width throughout the tank radius.For the sine20 and step20 cases, this growth is more enhanced onlyaway from the impeller (r/R41.9). At r/R� 2.2, the perturbed caseshave a nearly 50% greater jet-spreading than the baseline case. Thisrepresents a substantial enhancement in the jet-spreading viaintroduction of the perturbations. All the perturbed cases produceimpeller jet-streams wider than the fixed 4 rps case.

The discharge flow number is defined as

NQ ¼Q

ND3ð14Þ

where Q is the discharge flow from the impeller estimated asradial outflow over a cylindrical area spanning the width of the

impeller blades at a diameter of 1.02D. Table 1 presents flownumber for different cases. All perturbation cases show enhance-ment with these being more modest at the lower amplitudeperturbation cases (about 7% for step20). For step66 case, almost55% increase in flow number is observed which indicates asignificantly higher discharge of fluid due to the perturbation.

The zones with slowly moving fluid can potentially act as abarrier to fast mixing. The zones with slower velocity are identifiedwhere total velocity is less than 20% of the linear speed of theimpeller. Percentage of the STR volume occupied by these slowlymoving fluid volumes is reported in Table 2. Perturbation results ina decrease of the slower fluid volume with nearly a 50% reductionfor the step20 case and a 70% reduction for the step66 case.

4.2.2. Turbulent fluctuations

In order to determine the turbulence statistics, the periodiccomponents due to impeller blade passage are removed from thecalculation of turbulence statistics by averaging at the phase ofthe blade-passage frequency. This phase-averaging is done over270 revolutions of the impeller, i.e., six perturbation cycles. Thephase averaged root mean square (RMS) of velocity fluctuation atany angular location a is found as

/ui_rms9bS¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXMp

j ¼ 1

ðUji ðtÞÞ9

2

b

Mp�/Ui 9bS

2

vuuut ð14Þ

Fig. 6. Non-dimensionalized in-plane turbulent kinetic energy: (a) 3rps; (b) 4rps; (c) Sine20; (d) Step20 and (e) step66.

Fig. 7. Non-dimensionalized radial velocity fluctuation RMS: (a) 3rps; (b) 4rps; (c) sine20; (d) step20 and (e) step66.


Fig. 6 shows contours of the in-plane kinetic energy offluctuations calculated as ke¼ ð1=2Þðv2

rmsþw2rmsÞ9b ¼ 0 in the impel-

ler plane. It can be seen that peak kinetic energy increases slightly(around 9%) for sine20 case and fixed 4 rps case and substantially(around 25%) for step20 case when compared with the base line3 rps case. However, for step66 case, this increase is significantly

more pronounced and almost 87.5% more than fixed 3 rps case.Furthermore, high turbulence regions are observed in the very-near vicinity and directly above and below the impeller blades,which were absent in the baseline and low-perturbation cases. Inorder to better understand these effects, and the contributionsfrom different components of the velocity fluctuations, i.e., the


behavior of the axial and radial components of velocity fluctua-tions are explored in detail.

Fig. 7 shows the contours of the RMS of the radial velocityfluctuations (non-dimensionalized with respect to mean blade-tipvelocity) in the impeller plane. The non-dimensional radialvelocity RMS improves only by 10% in peak value when for4 rps constant impeller speed compared with the 3 rps case. Thesinusoidal perturbation do not contribute to any noticeableimprovement in radial RMS; however, peak RMS increases by10% during step perturbation with 20% amplitude (step20) and byalmost 50% for step perturbation with 66% amplitude (step66).Fig. 7(d) also shows that during this high amplitude perturbationradial RMS fluctuations are higher at the very vicinity of the

Fig. 8. Non-dimensionalized radial RMS velocity fluctuation along an axial line at

r/R¼1.1 averaged over six angular locations.

Fig. 9. Non-dimensionalized axial velocity fluctuation RMS:

impeller blade (r/R�1) and also spreads close to the tank walls (r/

R�2.6). So, clearly, the enhancements in turbulence (Fig. 6) in theradial direction are partly contributed to by the radial velocityfluctuations. These increases in turbulent fluctuations areexpected to contribute to greater diffusional transport and mix-ing. Fig. 8 shows the radial RMS profiles for different cases alongan axial line at r/R¼1.1 that is averaged over all six bladepositions a (as discussed in the previous section). Experimentaldata of Gao et al. (2007) is available at this axial location and isalso shown for constant 3 rps case. In the present calculations, thepredictions of the vrms show the same trends as the experimentswith reasonably good qualitative agreement as well as quantita-tive agreement to within 10% over most of the vertical extentexcept in regions of low turbulence where uncertainty in themeasurements can be expected to be high. This level of agree-ment is considered to be quite good given that the prediction ofturbulent quantities has always remained a challenge in STR flowcalculations. RANS models were reported to underpredict theturbulent fluctuations (Jones et al., 2001), substantially and onlyLES predictions showed a better level of agreement with somediscrepancies especially along the main discharge stream (Yeohet al., 2004a, b; Tyagi et al., 2007). It is clearly seen in Fig. 8 thatthe perturbations enhance turbulence levels with the step66 caseexhibiting peak levels that are nearly 60% greater than thebaseline case in the impeller jet stream. In the recirculationregions above and below the impeller-jet the lower-perturbationlevels appear to do exhibit greater enhancements in turbulence.

Fig. 9 shows the non-dimensionalized RMS fluctuations in theimpeller plane where all three perturbation cases result in anincrease in levels of axial RMS, with the highest increases for thestep66 case (as in Figs. 6 and 7). The key difference for the step66case is the appearance of high axial-velocity fluctuation levelsobserved just below and above the impeller location. Thisbehavior is consistent with that observed for the kinetic energyplot in Fig. 6, and indicate that the axial velocity fluctuations (andnot vrms) are responsible for the higher turbulence levels above

(a) 3rps; (b) 4rps; (c) sine20; (d) step20 and (e) step66.

Fig. 11. Perturbation cycles with different phases marked into them: (a) sine20;

(b) step20 and (c) step66.

Fig. 10. Non-dimensionalized axial RMS velocity fluctuation on the impeller plane (a) along an axial line at r/R¼1.5, (b) along a radial line at 2z/w¼0.


and below the impeller blades. Also, perturbations increase axialRMS along the impeller jet-stream and therefore across the widthof the jet stream an enhanced spread of fluctuations are observedmost significantly for step20 case. Perturbed cases step20 andsine20 show higher spreading of kinetic energy compared to4 and 3 rps constant rotational speeds. The differences betweenthe step20 and step66 cases are quite remarkable, in that, thestep20 case enhances greater axial broadening of the high-turbulence region while with step66 the highest peak levels areobtained but have a narrower footprint in the impeller jet-streamand high turbulence levels in the very-near vicinity of theimpeller, especially below and above the impeller tip. For theperturbed cases, fluctuations extend closer to the tank wall, andthis is most clearly evident for the step66 case where the impellerjet hits the tank walls with a high radial velocity and circulates inthe axial direction in two opposite circulation loops below andabove the impeller height forming higher-speed boundary layersover the tank wall (Fig. 4(b)–(d)). The higher values of the axialRMS along the wall arise due to perturbation of the impinging jetstream on the tank wall that results into significant fluctuations ofvelocity in the vicinity of the tank wall. A similar behavior ofperturbed impinging jet has been observed in a DNS study on flowand heat transfer due to impinging perturbed jet by Jiang et al.(2006). Due to this turbulence propagates in the axial direction.

Fig. 10(a) shows axial RMS along r/R¼1.5 where a 20% and 10%increase in the peak value is observed for sine20 and step20cases. The highest peaks of axial RMS for the step66 case arebelow and above the impeller blade, so it does not show thehighest value along this line with only a 15% greater peak RMS.Fig. 10(b) shows axial RMS distribution along a radial line at2z/w¼0.0 (impeller plane) and a 20–30% increase in peak value isobserved for the sine20 and step20 cases with a smaller increasefor the step66 case. High values for the perturbed cases are worthnoting near the tank boundary (r/R42.5), due to jet-impingementeffects noted earlier, where the step66 shows 80% higher valuethan fixed 3 rps case.

4.3. Changes in the phase-averaged flow field during a perturbation

cycle

It has been observed that perturbation in the impeller speedenhances turbulence level inside the tank and also augments tothe spreading of the impeller jet-stream indicating a potential forbetter mixing. In order to better understand the mechanismbehind this increase in turbulent fluctuations, we investigatethe behavior of the impeller jet-stream at five different phasesof the perturbation cycle. Fig. 11(a)–(c) shows the perturbation

cycles, respectively, for cases sine20, step20 and step66 andlocates these five different phases during the 15 s perturbationperiod in of the each particular cycle. Time-averaged flow-fieldsfor these five equi-spaced phases are obtained by averaginginstantaneous flow field over 48 samples for each phase toeliminate smaller-scale turbulent fluctuations.

The impeller jet-stream behaves like a free jet in the surround-ing of slowly moving fluid. Free jets were observed to be affectedsignificantly due to perturbations in several previous studies (e.g.,Ivanov, 1972; Farell et al., 2003 and Tatsumi et al.,2006), wherethe break-up of the axisymmetric nature of the jet stream andremarkable enhancements of turbulence were observed. Atassiet al. (1993) observed that with perturbations an increase inmean velocity and turbulence in the core of the jet was obtained,and a decrease in turbulence level outside the jet core occurred

Fig. 12. Phase averaged velocity vectors on the impeller plane for five different phases for sine20 case: (a) phase 1; (b) phase 2; (c) phase 3; (d) phase 4 and (e) phase 5.

Fig. 13. Phase averaged in-plane turbulent kinetic energy on the impeller plane for five different phases during sine20 cycle: (a) phase 1; (b) phase 2; (c) phase 3;

(d) phase 4 and (e) phase 5.



due to local velocity decrease. However, the radial jet generatedfrom the impeller tip has intrinsic differences with a freeaxisymmetric jet. Most importantly, the stirred tank being aconfined geometry, recirculated fluid with a slower time-scaleinteracts with the jet-stream. Further, during a perturbation cycle,inertial effects from the recirculated flow that originated duringthe higher impeller velocity phase, interacts with the jet-streamgenerated by the lower impeller-velocity phase and disrupts itsgrowth significantly. This mechanism is very prominent duringsudden changes of impeller speed due to step perturbations(step20 and step66).

Fig. 12(a)–(e) depicts the behavior of the impeller jet-stream ateach of the phases respectively for sinusoidal perturbation of theimpeller speed from 2.7 to 3.3 rps (sine20). Phase-1 represents atime instant when the impeller speed is 3.285 rps and stillincreasing. A strong radial jet is observed which hits the tankside-wall and forms strong upper and lower circulation bubbles.As the impeller speed reduces from its peak, the impeller jetweakens in strength, but due to inertial effects, the circulationbubbles lag in response. Therefore, in phase-2, where the impellerspeed is 3.18 rps, the jet stream is observed to be distorted by itsinteractions with the upper and lower recirculation zones, which,due to inertial effects, has not lost all the momentum associatedwith the higher rps phase. The bigger circulation bubble from theupper half of the impeller pushes the impeller-jet stream down-wards (Fig. 12(b)). This process continues in phase-3, where theimpeller velocity drops to 2.82 rps and the impeller jet loses itsradial strength and becomes more axially downward. Subsequentdecrease in impeller velocity further weakens the impeller jet andkeeps it oriented axially downward (phase-4). At this stageimpeller velocity starts to increase again, and as the jet is inclineddownward, the recirculation in the bottom half of the impellergrows stronger and it pushes the jet-stream upward. In phase-5,the impeller velocity is 3 rps (i.e., same as the mean velocity), theimpeller jet is observed to become stronger and also inclinedupward as it moves away from the blade tip. So, the sinusoidalperturbation results in a periodic strengthening and weakening ofthe impeller jet with its flapping behavior in the axial plane. Thisflapping is more prominent away from the impeller (beyondr/R¼1.7) and this contributes to the spreading of the meanimpeller jet-stream in this region (Fig. 4b). Fig. 13(a–e) portrays

Fig. 14. Phase averaged velocity vectors on the impeller plane for five different phases

the distribution of in-plane turbulent kinetic energy (normalizedwith respect to square of average impeller speed) at these fivedifferent phases. In phase-1, high turbulent kinetic energy ismostly observed along the radial jet-stream. In phase-2, thecirculated fluid from above and below disrupts the radial jet-stream (Fig. 12(b)) and turbulence level increases due to thisinteraction. Also, the turbulent fluctuations spread throughoutthe tank height due to this interaction and turbulent diffusion. Insubsequent phases, turbulent kinetic energy decreases substan-tially (see phases 4 and 5) as the impeller jet and the circulationbubbles lose their strength.

Fig. 14(a)–(e) portrays the impeller jet-stream at differentphases respectively for step perturbation of impeller speed from2.7 to 3.3 rps (step20). In phases 1–3, the impeller is at thehigher-speed phase while phases 4–5 correspond to the lower-speed phase. While the behavior shows distinct similarities withthe sine20 case, there are intrinsic differences in the velocitymagnitudes and unsteady flow patterns. In phase 1, as theimpeller switches to the higher speed, a strong impeller jet andupper and lower recirculation bubbles are created. There is strongflow dynamics leading to a flapping motion of the impeller jet inthe later phases. In phases 4–5, the impeller speed is dropped to2.7 rps and the jet starts weakening. In phase-4 we see theweakened radial jet, which further loses its strength and becomesaxially downward-inclined due to the circulated fluid from theupper circulation loop in phase-5. Fig. 15(a–e) portrays thedistribution of in-plane turbulent kinetic energy at these fivedifferent phases for this step20 case. In phase-1, turbulentfluctuations are high in the vicinity of the radial jet-stream. Inthe next phase, the interaction of the circulated fluid with theimpeller jet increases the peak value and spreading of theturbulent kinetic energy. This phase contributes maximum tothe overall turbulence of the flow. After the impeller speed isdropped to 2.7 rps, in phase 4 and 5, the magnitudes of turbulentkinetic energy decreases and turbulence is observed only in thenear impeller region and in the core of the impeller jet.

In Fig. 16(a–e) velocity vectors for the step66 cycle havebeen depicted for the same five phases. Phase-1 representsan early time-instant during the 4 rps part of the perturbationcycle shortly (1.5 s after) after it transitions from the 2 rps phasewhere the fluid inside the tank had much lesser momentum.

for step20 case: (a) phase 1; (b) phase 2; (c) phase 3; (d) phase 4 and (e) phase 5.

Fig. 15. Phase averaged in-plane turbulent kinetic energy on the impeller plane for five different phases during step20 cycle: (a) phase 1; (b) phase 2; (c) phase 3;



The velocity vectors at this phase are shown in Fig. 16(a) wherethe impeller jet-stream appears to be strongly radial and thealmost vertical vectors below and above the jet-stream, and theircorresponding velocity magnitudes, indicating strong and largerecirculation bubbles in the upper and lower portion of the tank.As the impeller continues to rotate with the higher speed (4 rps),and time advances to phase-2, the trailing edge vortex generatedfrom the previous impeller tip increases in size and strength, andinteracts with the impeller jet-stream, and due to this, the jetloses its radial strength (Fig. 16(b)) is oriented axially downward.The interaction with the impeller jet also weakens the trailingedge vortex, which shrinks in size (phase-3). The jet-stream againbecomes radial as shown in this phase. When the impeller speedis dropped to 2 rps at phase-4, the impeller jet-stream is observedto be completely suppressed (Fig. 16(d)). The Trailing-edgevortex-jet stream interaction (phase-2) and the mechanism ofbreaking-down and vanishing of the impeller jet-stream (as seenin phase-4) are linked to the sudden deceleration of the impeller

jet combined with the slower inertial response of the recirculat-ing bubbles that interact with the jet and break it down (Roy,2010). The flow-field at phase-4 show only a few smaller vorticesformed due to break-up of the radial jet (Fig. 16(d)). In the nextphase (Fig. 16(e)), those vortices break down into further smallervortices and spans toward the tank wall. Fig. 17(a–e) depicts thecontours of phase-averaged in-plane turbulent kinetic energy forphases 1–5, respectively during the perturbation cycle. Duringphase 1 (Fig. 17(a)), the turbulent kinetic energy is high in theimpeller jet-stream. The trailing-edge vortex interacts with theimpeller-jet in phase 2 (Fig. 16(b)) and at the interaction location(r/R� 1.5) a high kinetic energy is produced (Fig. 17(b)), which isalso convected in axial direction across the tank volume. In phase-3 (Fig. 16(c)), as the jet regains part of its radial momentum, thepeak turbulent kinetic energy is convected with the jet-stream.However, during the 4 rps part of the perturbation cycle (repre-sented by phase-1 to phase-3), turbulent kinetic energy far awayfrom the impeller is low (of normalized value around 0.01).

Fig. 16. Phase averaged velocity vectors on the impeller plane for five different phases for step66 case: (a) phase 1; (b) phase 2; (c) phase 3; (d) phase 4 and (e) phase 5.


In phase-4 (Fig. 17(d)) the jet-stream vanishes and small packetsof highly turbulent fluid gets distributed in the tank volume, andalthough the peak turbulence decreases, turbulence level awayfrom the impeller near the bottom and top wall of the tankincreases to non-dimensional values of 0.03–0.035. Therefore, inthis phase turbulence is convected throughout the tank volume.In phase-5 (Fig. 17(e)), these peaks of the turbulent kinetic energyare convected further away from the impeller and they decreasein magnitude indicating that large portion of turbulent kineticenergy is already dissipated. Therefore, during this perturbationcycle, convection and dissipation of turbulence increases resultinginto a potentially better mixing inside the stirred tank.

4.3.1. Effect of perturbation on turbulence

Turbulence level inside the stirred tank is observed to beenhanced significantly by the active perturbation of the flowusing different time dependent impeller speed cycles. Also, thelarge scale structures in the entire tank volume are observed tochange significantly during perturbation cycles resulting periodi-city in the turbulent kinetic energy distribution. Large strain ratesgenerated by the periodic growth, coalescence and dissipation ofthe large-scale structures (as seen in Fig. 16) are responsible forgeneration of small-scale eddies leading to production of turbu-lence (Ho et al., 1984). In several earlier studies, eddy-viscositytypes of models have been proposed to correlate turbulentfluctuations with strain rates due to periodic structures(Reynolds and Hussain, 1972; Davis, 1974). The periodic compo-nent of velocity can be separated out from the mean velocity andturbulent fluctuation using a triple decomposition (Hussain,1970):

Ui ¼Uiþeuiþu0i ð15Þ

where Ui is the mean time-averaged velocity, euii is the periodiccomponent arising due to perturbation and u0iis the turbulentpart. Now, the strain rate generated by the periodic velocitycomponent for different perturbation cycles can be compared inorder to understand the differences in turbulence enhancement.Figs. 18–20 respectively show in-plane components of the strain-rate tensor produced by the periodic velocity component (eui) forsine20, step20 and step66 cases respectively. The in-plane mean

normal strain-rate corresponding to the mean velocity field (Ui)has a maximum value of 20 (in s�1) for constant 3 rps case in theimpeller jet region. It can be observed that for sine20 and step20cases, the periodic component of velocity produces high positiveand negative strains (peak magnitudes showing 40% and 70% ofmaximum mean normal strain-rate for 3 rps case respectively)near the impeller as well as near the tank wall in phases 1, 2 and3 resulting in stretching and shearing of the fluid elements which,in turn, enhances the production of turbulence. However, forthese two cases (Figs. 18 and 19), during the low rps half of thecycle, periodic contribution to strain rates are very low andconcentrated mostly near the impeller tip. Fig. 19 (step20) showsthat during this part of the perturbation cycle, step perturbationproduces higher shear strain eSrz in the upper and lower circula-tion regions, But Fig. 18 (sine20) does not show any periodicstrain in these regions. This explains why the flipping of theimpeller jet during these two phases of the sine20 cycle(Fig. 12(d) and (e)) do not contribute to turbulence enhancementkeeping turbulence level almost similar to the base 3 rps case(Fig. 6b). A very significant enhancement in periodic shear strainrate (30–40% of maximum 3 rps normal strain-rate) is seen forphases 4 and 5 in step66 case (Fig. 20) which is due to the break-up of the impeller jet stream and advection of circulated fluidacross the impeller plane. The large strain rates producedthroughout the perturbation cycle for this case enhance theoverall turbulence to a higher extent.

Hussain (1970) obtained the budget equation for turbulentkinetic energy due to periodic perturbation as

Dðð1=2Þu0iu0i Þ

Dt¼�

@

@xju0j p0 þ

1

2u0iu0i

� �" #þð�u0iu

0j Þ@ui

@xj|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}P1

þð�/u0iu0jSÞ@

eui

@xj|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}P2

�euj@

@xj

1

2u0iu0i

� �þ

1

Re

@

@xju0i

@u0i@xjþ@u0j@xi

� �" #

�1

2Re

@

@xj

@u0i@xjþ@u0j@xi

� �@u0i@xjþ@u0j@xi

� �" #ð16Þ

where (—) signifies time averaging and (/S) signifies phaseaveraging. Now, in the above equation the terms P1 and P2

represent production of turbulent kinetic energy due to mean

Fig. 17. Phase averaged in-plane turbulent kinetic energy on the impeller plane for five different phases during step66 cycle: (a) phase 1; (b) phase 2; (c) phase 3;



and periodic velocity field (i.e., strain rates). Fig. 21 shows theproduction of turbulent kinetic energy from mean as well asperiodic velocity components. Fig. 21(a) shows that the contribu-tion to production due to periodic component has a peak value 8%of the peak mean production for sine20 perturbation. For thestep20 perturbation, the mean production is of same level of thesine20 case, but the production due to periodic componentenhances to 16% of the mean production resulting into a greaterenhancement in turbulent kinetic energy (comparing Fig. 6(b) and(c)). The mean production increases by around four times for thestep66 case compared with the previous two cases as the overalllevels of turbulence is much higher in this case (Fig. 21c). Also, alarger contribution in production by the perturbed field isobserved in this case, peak value of periodic production being80% of the mean production term. We also observe high produc-tion of turbulent kinetic energy by the periodic component aboveand below the impeller where production due to mean velocity is

negligible. This production component significantly enhancesaxial velocity fluctuation here and peaks of axial RMS as seen inFig. 9(d) appear due to this high production.

4.4. Variation in power number

Power number for a stirred tank flow is defines as

Np ¼P

rN3D5ð17Þ

where N is the mean rps of the impeller, D is the diameter and P isthe power required to move the impeller. The power P iscalculated as P¼�

RRpujdsj�

RRGjdsj, where G is the stress due to

fluid motion on the impeller blade. For a STR flow with Rushtonimpellers around Reynolds number 1000–1500 with c/T¼0.33and t/D¼0.02, the value of power number is not explicitly givenin the literatures. The case with c/T¼0.33 and t/D¼0.03 shows a

Fig. 18. Components of strain rate tensor due to velocity fluctuations (sine20 case).


Fig. 19. Components of strain rate tensor due to velocity fluctuations (step20 case).


Fig. 20. Components of strain rate tensor due to velocity fluctuations (step66 case).


Fig. 21. Turbulent kinetic energy production by the mean (left) and the periodic components (right): (a) sine20; (b) step20 and (c) step66.


power number around 3.7–3.8 in this Reynolds number regime(Distelhoff et al., 1995; Nouri and Hockey, 1998). Yapici et al.(2008) show that power number increases by 30% when t/D ratiois decreased to 0.02 from 0.03 for turbulent Rushton STRs.Rutherford et al. (1996) observed a power number around4.7 for t/D¼0.02. As power number increases with Reynoldsnumber in this regime, we can assume extrapolated powernumbers in the range of 4.45 to 4.6 for Re¼1150 (i.e., 3 rps) toRe¼1530 (i.e., 4 rps), respectively. Min et al. (2006) reports thatLES calculations can predict within 6% variation of the experi-mental power number. Our computation gives an average powernumber of 4.43 for a constant 3 rps impeller rotational speed and4.57 for 4 rps case. The variation of power numbers duringperturbation cycles (calculated based on average impeller speed,i.e., 3 rps) are shown in Fig. 22. This variation appears to correlatewell with the impeller speed variation; the average powernumber for sine20 case is 4.63, for step20 case is 4.69 and forstep66 case is 5.31. As the power number ðNp ¼ ðP=rN3D5ÞÞ

scales with the cube of the impeller rotational speed, powerrequirement (P) for 4 rps case is more than twice the powerrequirement for 3 rps case but the perturbed cases exhibit anaverage power input of maximum 19.8% (for the step66 case) ofthe baseline 3 rps case. Grenville (1992) showed that the runningcosts of an impeller mixer is around 70–80% of the total cost, andtherefore perturbed STR-s that can improve mixing without amajor penalty in power input, can reduce the total operating costof industrial mixers.

5. Conclusions

Active perturbation via impeller speed variation can poten-tially lead to mixing enhancement for both laminar (e.g.,Lamberto et al., 1996, 2001) and turbulent (e.g., Gao et al., 2007,Roy et al., 2008) STR flows. However, the effects of perturbationamplitudes and mode-shapes on STR flows were not addressed in

Fig. 22. Power number (calculated based on the average impeller speed) varia-

tions during a perturbation cycle for sine20, step20 and step66 case along with

average power number for constant rps cases shown in red lines. (For interpreta-

tion of the references to color in this figure legend, the reader is referred to the

web version of this article.).


the previous studies. In the present study, large-eddy simulationshave been undertaken to understand the potential of differentdynamic perturbation cycle on enhancement of mixing inside aturbulent STR and their effects on the jet-spreading, turbulenceenhancement and power input are quantified.

Three different perturbation cycles are studied and all of themare observed to produce higher turbulence levels and betterspreading of the impeller jet stream. For the same amplitude ofimpeller speed variation, a step perturbation produced moreturbulence than a sinusoidal wave perturbation. With increasein amplitude of the impeller speed variation, turbulence levelsinside the tank enhanced. Large strain rates are produced duringperturbation cycles due to periodic changes in the mean flowfield. The periodic strain in the fluid motion produced significantamount of turbulent kinetic energy. Production due to periodiccomponent of velocity is observed to be around 8–16% of themean production for low amplitude perturbations and 80% forhigh frequency oscillations. The increase in time-averaged powerinput rate for the perturbation cycles are not very significant(around 20% for the larger amplitude step perturbation). Pertur-bations result in potentially better mixing by increasing jet-spreading and turbulent fluctuations (even compared with ahigher 4 rps constant rotational speed case) and hence can reducethe mixing time. Reduction in mixing time is also helpful from areaction engineering point of view as it reduces the number ofundesired by-products.

Nomenclature

(aj)k metric termsc impeller height (mm)Cs

2 Smagorinsky model coefficientD impeller diameter (m)D1 tank diameter (mm)f perturbation frequencyfi body forcegjk second derivatives of metric quantities

ffiffiffigp

Jacobian of transformationke kinetic energy of fluctuationL length of the blade (mm)Lij Leonard stress termsMp total number of samples for averagingN average impeller rotational speed (rps)Nb number of angular positions used for averagingNp power numberP powerp pressureR blade radiusr radial distance from the shaft axisRe Reynolds numberSij strain rate in Cartesian frame of referenceTij test filtered stressT tank heightt thickness of the blade tipui_rms rms of velocity fluctuationvrms rms of radial velocity fluctuationw blade width (mm)wrms rms of axial velocity fluctuationuj Cartesian velocity component in j directionui filtered velocityu0 turbulent vecocity componenteu periodic velocity componentU time-averaged mean velocityxj Cartesian co-ordinate axisz axial height from the blade axis

Greek

a ratio between test filter width and grid filter widthb blade angle (degree)dij cronecker deltan kinematic viscosity (m2/s)m dynamic viscosity (Pa s)r density of fluid (kg/m3)G fluid stresses on the impeller bladetij subgrid stressxi curvilinear co-ordinate axis

Acknowledgements

This work was supported by DOW Chemical Company and theLouisiana Board of Regents. Their support is appreciated. Compu-tational work has been done using supercomputing facilities ofLSU and LONI.

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