simulation of laminar and turbulent impeller stirred tanks using immersed boundary method and large...

Chemical Engineering Science 62 (2007) 1351–1363www.elsevier.com/locate/ces

Simulation of laminar and turbulent impeller stirred tanks using immersedboundary method and large eddy simulation technique in multi-block

curvilinear geometries

Mayank Tyagia, Somnath Royb, Albert D. Harvey IIIc, Sumanta Acharyaa,b,∗aCenter for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA

bMechanical Engineering Department, Louisiana State University, Baton Rouge, LA 70803, USAcShell Exploration and Production Company, Houston, TX 77079, USA

Received 7 February 2006; received in revised form 6 August 2006; accepted 1 November 2006Available online 15 November 2006

Abstract

Impeller stirred tanks are commonly used in the chemical processing industries (CPI) for a variety of mixing and blending technologies.Such processes require accurate modeling of the turbulent flow in the tank over a range of operating conditions (e.g. impeller speed), and inaddition, require a computationally efficient solution strategy that can represent moving rigid geometric parts (impellers) in the tank. In thepresent study, a methodology is proposed that combines the advantages of the immersed boundary method (IBM) to represent moving rigidgeometries with the efficiency of multi-block structured curvilinear meshes (to minimize wasted grid points) for the representation of overallcomplex domains. The IBM implementation on a multi-block curvilinear mesh is advocated for the simulations of impeller stirred tank reactors(STR) and has distinct advantages over other competing methods. In the present work, the curvilinear-IBM methodology is further combinedwith the curvilinear coordinate implementation of large eddy simulation (LES) technique to address the issue of modeling unsteady turbulentflows in the STR. To verify the implementation of IBM in a multi-block curvilinear geometry, a laminar STR with a stack of four pitchedblade impellers on a single shaft is simulated and compared against experimental data. Verification of the combined IBM–LES implementationstrategy in curvilinear coordinates is done through comparisons with the measurements of turbulent flow in a baffled STR with a single pitchedblade impeller. For both laminar and turbulent STR, the predictions are in very good agreement with measurements. It is suggested here thatthis methodology can be reliably used as a predictive tool for the flow fields in STRs with complex geometries.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Simulation; Turbulence; Mixing; Chemical processes; Immersed boundary method; Large eddy simulation

1. Introduction

Mixing, blending, reaction, and extrusion of polymers areessential technologies of the chemical industry to form plas-tics, adhesives, rubbers, insulating materials, and many otherproducts that are essential to everyday life. These technolo-gies are estimated to produce several hundred billion dol-lars of polymer-based products annually. Improvements inexisting technologies can therefore potentially translate toseveral billion dollars in annual cost savings. According to

∗ Corresponding author. Mechanical Engineering Department, LouisianaState University, Baton Rouge, LA 70803, USA. Tel.: +1 225 578 5809;fax: +1 225 578 5924.

E-mail address: [email protected] (S. Acharya).

0009-2509/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2006.11.017

Tatterson et al. (1991), half of the $750 billion per year outputof the U.S. chemical industry is circulated through stirred tankreactors (STR), and nearly $1–20 billion per year is potentiallylost due to inadequate design of the mixers. Most polymeriza-tion reactions are carried out in tanks and extruders of com-plex geometric shapes. Modeling the relative movement of thecomplex surfaces is still considered a major challenge for thecurrent grid generators and flow solvers. Therefore, it is highlydesirable to find a computationally efficient approach that in-corporates the description of such complex and rigid movinggeometries to accurately simulate turbulent flows of industrialrelevance.

Most of the experimental studies for complex industrial pro-cesses are enormously costly and often fail to scale from the

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1352 M. Tyagi et al. / Chemical Engineering Science 62 (2007) 1351–1363

laboratory to the plant designs. Computational fluid dynamics(CFD) has matured over past several decades into a “routinelyused” design tool to provide cost-effective and accurate designsolutions in the chemical industries. With high performancecomputing (HPC) platforms available, it is now possible to han-dle the problems of industrial relevance using super-computingresources. However, the algorithmic challenges posed by theproblems for the efficient representation of complex impellersand their relative movement in the tank are still enormous,and require special techniques and significant computationalresources. For example, a deforming mesh methodology thatregenerates the fluid mesh around moving rigid geometrieswould be extremely time-consuming for large problems. More-over, in some problems, moving parts (impellers) might sweeponly a small fraction of the total computational domain. Asliding mesh approach may be a good alternative for problemswhere the interface can be defined clearly between the blocksthat are moving relative to the fluid mesh at all times (Murthyet al., 1994). However, for complex impeller designs and in-termeshing extruders, the sliding mesh approach might not befeasible or computationally very demanding (several slidingblocks in the computational domain) (Tanguy and Thibault,2002). On the contrary, immersed boundary methods (IBM)add the influence of moving geometries through “forcing”terms in the momentum equations for the fluid, and thereforethe IBM does not face the issues of regenerating the fluid mesharound the moving geometries. Further, there is no constrainton the prescribed motion of the rigid geometries (unlike slid-ing meshes) to get an interface to exchange information acrossmoving and stationary blocks. However, a Cartesian coordinateimplementation of the traditional IBM to represent the outerboundary of a cylindrical tank would potentially waste about20% of the total grid points in the vicinity of the corner edgesin an underlying square cross-sectional Cartesian mesh. Thus,from this perspective, a body-fitted grid remains an accurateand economic discretization for the bulk of the computationaldomain such as STR. An IBM in curvilinear grids is proposedhere to alleviate the issues of sliding meshes and deform-ing grids that are otherwise needed to represent the movingimpeller, while retaining the advantages of the body-fittedgrids that accurately and economically represent the outerboundary of the tank with baffles. The proposed immersedboundary concepts and algorithms in this study are developedand implemented in a body fitted multi-block curvilinear finitevolume solver.

The issue of turbulence modeling in STR is addressed hereby large eddy simulation (LES). Jones et al. (2001) tested sixdifferent two equation turbulence models for STR, and foundthat none of the model predictions compared well with pub-lished data. In STR flows, a range of energetic flow scales thatare inherently unsteady are encountered, and LES is well-suitedfor such flows since they resolve the more energetic scales. Bycombining the curvilinear IBM with the LES methodology, asolution procedure is obtained that can efficiently address theproblems in the laminar-to-turbulent regime and complex ge-ometries, for example, those with multiple independently mov-ing impellers.

There are only very few studies that have exploited IBM incurvilinear grids. Ghias et al. (2004) implemented IBM in non-conforming curvilinear meshes to study the tip flows around arotor in hover. You et al. (2004) used a combined LES–IBMmethodology in curvilinear grids to study the tip-clearanceflows in turbo-machines. In this paper, we present a specific im-plementation strategy for combining IBM on curvilinear gridswith LES, and demonstrate its application for STR flows. Theadvantages of this approach will be further discussed in thepresentation of the results.

2. Computational methodology

The non-dimensional governing equations for the conserva-tion of mass and momentum for an incompressible Newtonianfluid are given as

�uj

�xj

= 0,

�ui

�t+ �uiuj

�xj

= − �p

�xi

+ 1

Re

�2ui

�x2j

+ fi ,

where ui is velocity field, p is pressure, Re is the non-dimensional Reynolds number defined in terms of character-istic velocity and length scales of the problem and fi is thebody force term due to immersed boundary.

In this study, a parallel multi-block chemical reacting com-pressible flow code for generalized curvilinear coordinates isused. For incompressible flows, low Mach number precon-ditioning is used (Weiss and Smith, 1995). A second orderbackward three-point physical time differencing is used forthe temporal derivatives in conjunction with Euler differenc-ing for pseudo-time derivatives. Second order low diffusionflux-splitting algorithm is used for convective terms (Edwards,1997). Second order central differences are used for the viscousterms. An incomplete lower–upper (ILU) matrix decompositionsolver is used. Domain decomposition and load balancing areaccomplished using a family of programs for partitioning un-structured graphs and hypergraphs and computing fill-reducingorderings of sparse matrices, METIS (Karypis and Kumar,1998). The message communication in distributed computingenvironment is achieved using Message Passing Interface, MPI(Gropp et al., 1999). All the multi-block structured curvilineargrids presented in this paper are generated using commercialgrid generation software GridPro�. Surface meshes to modelthe moving rigid geometries are generated using another com-mercial package Gambit�.

3. Immersed boundary method

At this point, a brief overview of the IBM is presented. Inthe IBM, the complex geometrical features are incorporated byadding a forcing function in the governing equations. With theIBM, grid points internal (or in the vicinity, in case of mov-ing geometries) to a solid surface have body force terms addedsuch that the no-slip boundary conditions at the interior surfaceis exactly satisfied (Peskin, 1972; Fadlun et al., 2000; Tyagi,

M. Tyagi et al. / Chemical Engineering Science 62 (2007) 1351–1363 1353

Fig. 1. Schematic to illustrate (a) immersed surface (Lagrangian), (b) fluid mesh (Eulerian) and identification/tagging of different cells in computational domainand (c) details of the interpolation scheme showing the points in the interior of the immersed body (solid), on the solid surface (lagrangian marker), solvedpoints (fluid) and the immersed boundary points (forced). (a) Immersed surface can be located using the surface normals defined at the triangulated facets.(b) Tagging of the Eulerian cells (fluid mesh) defines the region for application of immersed boundary interpolation schemes.

2003). The forcing function is zero everywhere except at thesurface where the influence of the solid boundaries is assigned(Fig. 1). Details of identifying “forced points” around the mov-ing rigid geometry in a fixed mesh are illustrated in Fig. 1.The surfaces of moving rigid geometries are represented as La-grangian markers and the surface normals can be used to definethe interior and exterior of the moving immersed solids withrespect to fixed fluid mesh (Fig. 1a). Appropriate interpolationstencils can be formed using the various “tagged-points” in thecomputational domain. These points include: the “solved” flowfield points, the “forced” points inside the moving immersedobjects without any influence from “solved” fluid points, andthe “forced points at the immersed boundary” using an appro-priate interpolation scheme incorporating the influence from allother tagged points except immersed boundary points (Fig. 1c).There are several other ways to achieve similar forcing (Tyagiand Acharya, 2005). Also, for moving geometries, these forcingterms can be prescribed in a general time-dependent fashion.On a fixed Cartesian mesh with geometry defined on moving

Lagrangian markers (surface points), the need for time consum-ing grid-generation methods is therefore not required. How-ever, for several different forcing strategies at the immersedboundary, the intersection of these immersed boundaries withthe underlying Cartesian mesh need to be computed in an ef-ficient fashion. Several researchers (Iaccarino and Verzicco,2003; Tseng and Ferziger, 2003) have used the inverse-distanceweighted interpolation to evaluate the body forces on the im-mersed grid points. A general interpolation scheme can bewritten as

Vim =n∑

m=1

wmVm/q, wm =(

R − hm

Rhm

)p

,

q =n∑

i=1

(R − hi

Rhi

)p

,

where Vm are the computed solution around the immersed point,wm is the weight, p is the power exponent (usually set to 2),


hm is the distance of corresponding grid points from the forcedpoint, and R is the maximum of hm (Franke, 1982). This inverse-distance forcing strategy is adopted in the present study.

The computed velocity field needs to be consistent with theno-slip requirement at the geometric features of the immersedsolid object. As a first step, the exact location of the geometricfeatures to be rendered is solved or specified at the Lagrangianmarkers (Fig. 1a). Note that in general, these locations will notcoincide with computational grid nodes. The weights can beevaluated by an appropriate interpolation scheme to satisfy theno-slip condition on these solid walls (Fig. 1c). Thus, the influ-ence of the moving complex geometric features is distributedon the computational mesh through these body force terms.

4. Implementation of immersed boundary method incurvilinear geometry

As a first step, a description of complex moving geometriesis needed for the IBM. Description of complex geometric fea-tures can be achieved using a stereo-lithography (STL) format(Ito and Nakahashi, 2002). The STL representation of any sur-face is a collection of unconnected triangles of sizes inverselyproportional to the local curvature of the original surface. STLformat is the standard for Rapid Prototyping and all the CADsystems have the capability to automatically export any givensurface in STL format. Therefore, complex geometries can bedescribed without the need of a surface mesh (with the restric-tion that these surfaces must be closed manifolds). However, inthis study, various surface grids are generated using Gambit�to define the moving rigid geometries. Preprocessor for theimmersed boundary flow solver generates all the interpolationdata by separating the computational domain into fluid, solidand immersed cells. The properties of surface normal vector areused to tag these cells (Figs. 1a and b). Alternatively, a simpleray tracing technique can also be used. By tracking the movingsurfaces using these tags in a multi-block curvilinear mesh, us-age of deforming meshes or sliding meshes can be completelyavoided. Appropriate interpolation schemes for the forcing termin the momentum equations are used around these immersedcells (Fig. 1c). The proposed “optimal” approach can capitalizeon the advantages offered by the block structured curvilinearmeshes, the STL (or any other convenient) format for definingthe surface grid and the interpolation schemes using the IBM. Italso avoids the substantial computational overhead associatedwith deforming meshes.

In the impeller stirred tanks, each impeller (in case of mul-tiple impellers) geometry is stored as a separate input file andhence provides the fidelity to move them independently. Thesearch/tagging is performed only in a fraction of the com-putational domain that contains the moving impeller and ittherefore saves a considerable amount of computation (Fig. 2).It is only in these blocks that the immersed boundary forcingis also applied. In all the current studies, the impeller (im-mersed) zones are disjoint and therefore, the forcing schemesand search/tagging strategies do not face any conflictingscenarios.

5. Large eddy simulation

At this point, a brief overview of the LES governing equa-tions in curvilinear coordinates is presented. LES is a cost-effective approach to turbulence simulation in which the gov-erning equations are spatially filtered to resolve the dynamics ofthe large scales, and modeling is done only for the “universal”small scales (Tyagi, 2003; Sagaut, 2001). However, LES incomplex 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 gov-erning equations are spatially filtered, with the filter width (pro-portional to the grid size) representing the scales in the flowfield that are resolved. The non-dimensional filtered governingequations for the conservation of mass, momentum and energyfor an incompressible Newtonian fluid in curvilinear coordinatesystem are given as (Jordan, 1999; Tafti, 2004):

Continuity equation:

�

��j

(√

gUj ) = 0.

Momentum equations:

�

�t(√

gui) + �

��j

(√

gUj ui)

= − �

��j

(√

g(aj )i p) + �

��j

((1

Re+ 1

Ret

) √ggjk �ui

��k

),

1

Ret

= C2s (

√g)2/3|S|,

where√

gUj =√g(aj )kuk , (aj )k =��j /�xk are the contravari-

ant velocity components and associated metric terms, respec-tively. The term

√g is the Jacobian of the transformation, and

gij are the elements of the contravariant metric tensor. Thestrain rate tensor is given by

Sik = 1

2

((am)k

�ui

��m

+ (am)i�uk

��m

)

and |S| is the magnitude of the strain rate tensor. In the aboveequations the over-bar represents the filtered quantities.

The anisotropic subgrid (—) and subtest scale (∼) stresstensor is formulated in terms of the Smagorinsky eddy viscositymodel (Smagorinsky, 1963), and is given by

�aij = −2C2

s (√

g)2/3˜|S|Sij ,

T aij = −2C2

s �(√

g)2/3| ˜S| ˜Sij ,

Cs is the model coefficient and is assumed to be same at bothsubgrid as well as subtest level. � denotes the ratio of filterwidths at the test level to the grid level. The Germano Identity(Germano et al., 1991) relates the SGS stresses at different filterlevels in terms of the filtered fields only

Tij = uiuj − ˜ui˜uj , �ij = uiuj − ui uj ,

Lij = Tij − �ij = ˜uiuj − ˜ui˜uj .


Fig. 2. Identification of immersed zones to localize the search for the immersed points in the Eulerian curvilinear mesh and illustration of the static curvilinearsolid boundaries with respect to the moving impeller blades (Lagrangian surface mesh). *Complex surfaces can be created using any CAD or surface mesher.STL files, Gambit neutral files, GridPro triangulations, ICEM, etc.

Therefore, using the Smagorinsky’s model for SGS terms, theGermano Identity becomes

Laij = Lij − 1

3�ijLkk

= − 2C2s (

√g)2/3(�| ˜S| ˜Sij − ˜|S|Sij ),

Laij

�= −2C2s (

√g)2/3Mij ,

therefore,

C2s = −1

2

1

(√

g)2/3

Laij · Mij

Mij · Mij

.

For numerical stability, the coefficient Cs is limited to posi-tive values only or smoothed in a general fashion (Tyagi andAcharya, 2005).

6. Parallel implementation

The flow code solves the flow equations in a zonal manner.The solution domain is divided into an arbitrary number of

hexagonal grid zones, Z, and the parallel solution strategy par-titions these domains among an arbitrary number of processes,P, with the requirement that P �Z. Each process obtains asolution on its pre-assigned portion of the domain, subject tothe boundary conditions for that part of the domain. Domainboundaries which are not physical boundaries obtain their datafrom the neighboring domains, either by in-memory data trans-fer when the domains reside on the same process or throughinter-process data transfers when they reside on different pro-cesses. The inter-process communication is accomplished viathe industry-standard MPI protocol (Gropp et al., 1999).

Parallel efficiency is highly dependent on the application andcan be controlled by the user in the grid generation step. It is ex-pected that applications using zones of approximately the samesize will exhibit a higher level of parallel efficiency due to uni-form communication and computation load. Data on the bound-aries of each block are communicated to its adjacent blocks dur-ing every solution iteration. The amount of information neededto update these inter-block values is kept to a minimum. Theparallel setup routines precompute the memory addresses of


both the sources and destinations of all data transfer events,whether in-memory or inter-process. Using these precomputedaddresses, only the data values must be exchanged during eachiteration, minimizing the communications overhead.

The computational effort must be divided among each pro-cess such that each completes its work in about the same lengthof time. By ensuring that the number of zones describing thesolution domain is significantly larger than the number of pro-cesses, and that the zones are fairly uniform in size, a goodcomputational load balance can be achieved. Computationalload balancing alone is not sufficient to ensure good parallelperformance. For a given number of computational zones, theinter-processor communications increase with the number ofprocesses assigned to the problem. Also, if the sizes of thezones are made too small, the amount of data transferred, rel-ative to the amount of CPU work required, will also increase.Assignment of grid zones to a user-defined number of processesis automatic and is accomplished using METIS (Karypis andKumar, 1998).

In the implementation of immersed boundaries techniques,each immersed surface is described by a surface grid file. Ingeneral, these immersed surfaces are free to move around inthe entire computational grid (all zones). To increase overallefficiency and limit the extent of the search algorithm, each zonein the computational grid is assigned a logical switch indicatingif a search is required for the presence of immersed surfaces.

Tagging and searching of immersed boundary points andconstruction of weighting functions need to be done efficientlyto avoid this becoming a large computational task. One prob-lem encountered was that the immersed surfaces often (as is thecase for the present application) reside in only a small portionof the computational domain. The processors on which theseflagged zones (for searching and tagging) resided become over-loaded while all other processors (containing no flagged zonesat all) sat idle, thus, significantly decreasing overall parallelperformance (Harvey et al., 2004). For example, in the laminarSTR problem presented in the next section, out of 1728 totalgrid zones, only 128 where tagged for searching (in the imme-diate vicinity of the impellers). For computations involving 64processors, only five processors contained tagged zones. Obvi-ously, the magnitude of this computational speed-up is depen-dent on the geometry. To address this, an algorithm has beendeveloped which subdivided all zones flagged for searching toall available processors, thus permitting full processor partici-pation in the search and tagging. During preliminary runs forthe laminar tank problem, the redistribution of the cell taggingtasks to all processes resulted in a 10-fold decrease in compu-tational time.

7. Results and discussions

7.1. Laminar multiple impeller stirred tank (IBM verificationstudy)

As a benchmark for the IBM implementation in complex ge-ometries with moving boundaries, a laminar multi-impeller tankis simulated. For the simple configuration of a Rushton impeller

Fig. 3. Schematic diagram of the multi-impeller stirred tank correspondingto the experimental setup of Harvey et al. (1997). Different axial locationsare at z/R2= (a) 3.58, (b) 3.23, (c) 2.89, (d) 2.55, (e) 2.21, (f) 1.87,(g) 1.53 and (h) 1.19.

in a stirred tank, a sliding mesh approach has been commonlyemployed (Luo et al., 1993; Perng and Murthy, 1993; Murthyet al., 1994). An approximate steady state method called themethod of snapshots, applied to laminar pitched-blade simula-tion was presented by Harvey et al. (1995). Harvey and Rogers(1996) compared results from the method of snapshots withthose of unsteady computations and found reasonable agree-ment. Harvey et al. (2000) further applied this technique to mul-tiple impeller blades, and by using a combination of methodof snapshots and multiple reference frames they demonstratedfairly good results.

A schematic of the multi-impeller geometry of interest ispresented in Fig. 3 and corresponds to the experimental setupof Harvey et al. (2000). Rotation speed of the impeller stacksis 92 rpm and Reynolds number based on rotation speed andthe diameter of the largest impeller is 86.

The computational grid for the multi-impeller tank contains1728 blocks and about 3.5 million grid points and was gener-ated using commercial gird generation software Gridpro�. Thesurface meshes for all the impeller blades have total 68,415elements and were generated using Gambit�. The computa-tion was performed on 64 processors of HPC cluster usingMyrinet� interconnect for about 120 CPU hours. A total of7667 time iterations were obtained during this entire compu-tation. Time averaging was performed over 16 revolutions ofthe impellers. Physical timestep for the computation was set to2.173 ms.

The comparison of the computational results with theexperimental data of Harvey et al. (2000) is presented inFig. 4. Data are compared at different axial stations at z/R2=(a) 3.58, (b) 3.23, (c) 2.89, (d) 2.55, (e) 2.21, (f) 1.87, (g) 1.53and (h) 1.19. A good agreement is obtained for all the velocity


Fig. 4. Comparison of the computed (lines) averaged velocity components with the experimental measurements of Harvey et al. (1997) (symbols) along theradial direction at different axial locations z/R2= (a) 3.58, (b) 3.23, (c) 2.89, (d) 2.55, (e) 2.21, (f) 1.87, (g) 1.53 and (h) 1.19. Axial (dash-dot line, ◦), radial(solid line, �) and azimuthal (dashed line, �) velocity components.

components at these axial locations in the tank. In particular,the axial as well as azimuthal components of velocity field arepredicted remarkably well at all the axial locations. To furtherillustrate the complex flow field, streamlines are presented fora baffle-plane in Fig. 5. These streamlines are projected on anaxial cross-section of the stirred tank, and therefore the pitchedblades are represented by their rectangular projections from theside-view. It is because of this projected view that several ofthese streamlines appear to enter the baffles and the impellers.The flow field in the vicinity of the impellers is shown in vari-

ous insets. Downward axial pumping of the fluid along the shaftin the vicinity of impellers is due to the pitch of the impellerblades. Around the tip of the largest impeller, the axial flowcompartmentalizes into lower and upper recirculation zones.

Clearly, the present method provides predictions that are ingood agreement with the measurements. In terms of specificadvantages, the method does not suffer from any overhead as-sociated with the deforming mesh approach. The computationsassociated with the immersed boundary search and interpo-lation are small compared to the calculations that would be


Fig. 5. Details of the flow field in a multi-impeller stirred tank: streamtraces are shown in a baffle plane and the velocity vector details are shown in the insetsin the vicinity of each impeller.

mean axial velocity

y

0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3A B C B C

vrms

y

0.05 0.1 0.15 0.2 0.25 0.3

0.5

1

1.5

2

2.5

3A

a b

Fig. 6. Comparison of the LES computations with the experimental data of Jovic and Driver (1994) at streamwise stations x/h= (A) 4.0, (B) 6.0 and(C) 10.0 for (a) Mean streamwise component of velocity and (b) rms normal component of velocity.

required in the regeneration of fluid mesh and/or interpola-tion of flow fields associated with deforming meshes. Also, thecurvilinear IBM does not have any restriction in defining aninterface between relatively moving and stationary blocks as-sociated with sliding mesh method. In fact, the moving rigidgeometries can be specified by any arbitrary motion as long asit does not violate the time-integration constraints. Further, theproposed method does not need the steady state flow assump-tion associated with method of snapshots.

7.2. Backward facing step (LES verification study)

To verify the implementation of SGS model and filters in thegeneralized LES methodology, a simple problem of turbulentflow over a backward-facing step (BFS) is simulated. Reynoldsnumber (=Uh/�) based on the freestream velocity (U) and thestep height (h) is 5100. The length of the inlet section before

the BFS is about 10 times of the step height (h). A fully devel-oped turbulent boundary layer satisfying the “log-law” is pre-scribed at the inlet for mean velocity components. At the out-flow, a convective boundary condition is used, while the lateralboundaries are prescribed as symmetric boundaries. No-slipconditions are applied on the walls. The Cartesian mesh usedfor this simulation is composed of only three blocks and about15,000 grid points. A total number of 7891 time iterations wereobtained within 84 CPU hours on two processors. Physicaltimestep is taken as 0.01 s. Averaging of the turbulent flowfieldis performed over 15 flow-through times in the computationaldomain. Computed reattachment length (in terms of stepheight, h) is 6.30h as compared to 6.28h reported in the directnumerical simulations (DNS) by Le et al. (1997). Mean stream-wise velocity component as well as the turbulent statistics (rmsnormal component) are presented in Fig. 6 and match satisfac-torily with the experimental data of Jovic and Driver (1994).


Fig. 7. Schematic diagram for the pitched blade impeller stirred baffled tank corresponding to the experiments of Schafer et al. (1998).

7.3. Turbulent impeller stirred tank (combined LES and IBMverification study)

There are a large number of computational studies dealingwith turbulent flows in impeller stirred tanks. Eggels (1996)used lattice-Boltzmann method (LBM) to simulate the turbu-lent flow in a baffled tank stirred with a Rushton turbine andthe agreement between the computed averaged velocity compo-nents as well as turbulent kinetic energy and the experimentaldata was good. Revstedt et al. (1998) used a surface force dis-tribution scheme to simulate impeller stirred tank flows. Thisapproach uses Lagrange polynomials to distribute the surfaceforce onto the fluid mesh. Wechsler et al. (1999) presentedthe computational results for a pitched blade impeller stirredflow using a two-equation turbulence model. Jones et al. (2001)presented results with improved turbulence models for stirredtank flows with no baffles. Under-prediction of the turbulentkinetic energy in the vicinity of impeller is a typical behaviorof these turbulence models. Verzicco et al. (2000) used IBMto simulate an unbaffled tank stirred by a paddle type impellerusing coarse-DNS. However, these predictions were only inqualitative agreement with experimental data. Dersken (2001)presented an assessment of LES for pitched blade stirred tankflows using the LBM and showed that the predictions matchedwith the experimental data very well. More recently, Dersken(2003) has used LES in conjunction with the Lagrangian track-ing of solid particles to model the suspensions in the stirredtanks. Hartmann et al. (2004a,b) revealed a flow macroinstabil-ity by means of LES in a Rushton turbine stirred tank. Stronglynon-homogeneous distribution of turbulent kinetic dissipationrate suggests that for turbulent cases, better turbulence closure

models such as LES would be needed for STRs (Bakker et al.,2000; Jones, 2003). Note that most of the above-stated LESstudies were performed using LBM and it is in contrast to thecurrent study that utilizes the curvilinear LES equations in con-junction with the IBM. In addition, LBM based simulationsneed many more degrees of freedom (approximately five timesmore than conventional LES) to attain reasonably accurate andcomparable predictions. Further, most of the above-mentionedmethods are not easily extendable to the multiple moving ge-ometries of interest, and hence, an implementation of IBM inmulti-block curvilinear meshes is advocated. To address the is-sue of turbulence modeling in the stirred tank situations, thecurvilinear LES approach is adopted here.

The problem selected is that of Schafer et al. (1998) whopresented comprehensive LDA measurements for the pitchedblade turbine stirred tanks. Schematic diagram for the flat bot-tom single pitched blade stirred tank geometry used for thecomputational study is presented in Fig. 7 and corresponds tothe experimental setup of Schafer et al. (1998). Rotation speedfor the impeller is 2673.6 rpm and the Reynolds number basedon the rotation speed and impeller diameter in the stirred tankis 7280.

The computational grid for the single-pitched blade stirredtank is composed of 3024 blocks comprising of about 2.3million grid points. The surface mesh for the impeller bladecontained 7128 triangular elements and is generated usingGambit�. The computation is performed on 64 processors for120 CPU hours. A total of 7923 time iterations were obtainedduring this computation. Phase averaging is performed over85 complete revolutions of the impeller. Physical timestep isset to 0.2 ms for this simulation.


Fig. 8. Comparison of the computed (lines) averaged velocity components (left) and turbulent kinetic energy (right) along the radial direction at differentaxial stations with the experimental measurements (symbols) (Schafer et al.1998) z/T = (a) 0.145, (b) 0.330, (c) 0.460 and (d) 0.670. Axial (dotted line, ◦),azimuthal (dashed line, �) and radial (solid line, �) velocity components.

Comparisons of the computed averaged velocity componentsand turbulent kinetic energy with the experimental measure-ments of Schafer et al. (1998) are presented along the radialdirection at different axial stations z/T = (a) 0.145, (b) 0.330,(c) 0.460 and (d) 0.670 in Fig 8. Agreement between computa-

tions and experimental data is satisfactory with minor discrep-ancies close to the impeller shaft for axial and radial compo-nents of velocity at the axial station below the impeller. Notethat the agreement between the predicted turbulent kinetic en-ergy from LES with the experimental data is excellent. Clearly,


Fig. 9. Phase-averaged velocity vectors at two different phase angles: (a) 30◦; (b) 60◦.

Fig. 10. (a) Instantaneous streamtraces in the baffle-plane for single pitched blade impeller stirred tank. (b) Details of the velocity vectors near the impeller.

the current LES can resolve the energy containing scales betterthan most RANS results presented in the literature so far (e.g.,Jones et al., 2001; Hartmann et al. 2004a,b). It should also benoted that the agreement shown by the predictions in Fig. 8are comparable in accuracy with LBM calculations of Derksen(2001) who utilized nearly five times as many points in theircalculation. Phase-resolved velocity vectors near the impellertip are presented for the phase angles of 30◦ and 60◦ in Fig. 9.Position, size and shape of the trailing tip-vortex obtainedfrom the simulation agree with the experimental observationsof Schafer et al. (1998). Cross-section of the trailing tip-vortexis oval in the shape and the vortex core-size increases with thephase angle.

To show the details of the complex flow field, streamtracesof the instantaneous velocity field in a baffle-plane are pre-sented along with the detailed view of velocity vectors aroundthe impeller tip in the inset (Fig. 10). These streamtraces areprojected on an axial cross-section of the stirred tank and thepitched blade is represented by a simple rectangular projec-tion from the side-view. Patterns of the instantaneous stream-

traces reveal several critical points in the flow field at thebaffle planes. Dynamical description of these critical points canquantitatively explain the mixing in the tank, and will be pre-sented in a later paper. The velocity vectors show the details ofdownward pumping action by the pitched impeller.

Fig. 11 shows the distribution of off-diagonal compo-nents of Reynolds stress tensor in the impeller plane. Theseoff-diagonal components (u′

ru′�, u′

rw′and u′

�w′) have larger

magnitudes in the downward-directed impeller-jet region. Theu′

�w′ component appears to be substantially greater than the

u′rw

′ component implying an anisotropic turbulence field.Asymmetry in the Reynolds stress components is observed,and is primarily attributed to the various macro-instabilities inthe flow.

8. Concluding remarks

A curvilinear coordinate implementation of the IBM com-bined with LES on multi-block meshes is presented in this


Fig. 11. Off-diagonal Reynolds stress components in a mid-baffle impeller plane: (a) u′ru

′�, (b) u′

rw′ and (c) u′

�w′.

paper. The present method does not suffer from any draw-backs of various other alternatives such as deforming mesh,sliding mesh or method of snapshots. Computations usingthe curvilinear IBM are verified against the laminar flowmulti-impeller stirred tank data of Harvey et al. (2000). Thepredictions match the data quite well confirming that the curvi-linear IBM approach is suitable for STR. For turbulent STRflows, the predictions of the combined IBM–LES methodol-ogy in curvilinear coordinates compared satisfactorily againstthe experimental data of Schafer et al. (1998). Present re-sults required only one-fifth of the computational grid sizesas compared to the LBM calculations of Derksen (2001),where 2403 grid points were used for LES of STR flow inan identical configuration. Further, the computed results forturbulent kinetic energy were in better agreement than mostof the earlier RANS calculations for STR flows (Jones et al.2001).

Acknowledgments

All the computations presented in this paper were carriedout on the HPC resources of the Center for Computation andTechnology (CCT) at LSU. Dr. R.M. Jones (DOW) and Mr. M.Rangitsch (DOW) provided help in the multi-block curvilineargrid generation for the stirred tanks and impellers. Discussionswith Dr. A. Gilmanov (CCT) are also acknowledged. One of theauthors (S.R.) was supported financially by Louisiana Boardof Regents (LA-BoR) LEQSF-ITRS grant. Louisiana Boardof Regents (BoR) and DOW’s financial support during thisresearch effort is gratefully acknowledged.

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