Antonio Posa

Computers & Fluids 47 (2011) 33–43

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Computers & Fluids

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

Large-eddy simulations in mixed-flow pumps using an immersed-boundary method

Antonio Posa a, Antonio Lippolis b, Roberto Verzicco c, Elias Balaras d,⇑a Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, Bari 70125, Italyb Dipartimento di ingegneria dell’Ambiente e per lo Sviluppo Sostenibile, Politecnico di Bari, Taranto 74123, Italyc Dipartimento di Ingegneria Meccanica, Università di Roma Tor Vergata, Roma 00133, Italyd Fischell Department of Bioengineering, University of Maryland, College Park, MD 20742, USA

a r t i c l e i n f o

Article history:Received 5 January 2010Received in revised form 5 October 2010Accepted 3 February 2011Available online 18 February 2011

Keywords:Large-eddy simulationFinite-difference methodImmersed-boundary methodTurbomachinery flowsMixed-flow pump

0045-7930/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compfluid.2011.02.004

⇑ Corresponding author. Tel.: +1 301 405 8268; faxE-mail addresses: [email protected] (A.

Lippolis), [email protected] (R. Verzicco), b

a b s t r a c t

Computations of turbulent and transitional flows in rotating machinery applications are very challengingdue to complexity of the geometry, which usually consists of multiple rotating and stationary parts. Theapplication of well-established, body-fitted methods frequently utilizes overset grids and different refer-ence frames, which have an adverse impact on the overall accuracy and cost-efficiency of the method. Inthe present work we explore the feasibility of performing computations of such flows using a single ref-erence frame and an immersed-boundary approach. In particular, we report one of the first large-eddysimulation in this class of flows, where a structured cylindrical coordinate solver with optimal conserva-tion properties is utilized in conjunction with an immersed-boundary method. To evaluate the accuracyof the computations the results are compared to the experimental measurements in [1]. Results using thestandard Smagorinsky model and the Filtered Structured Function model are presented. We demonstratethat the overall approach is well suited for the flow under consideration and the results with the moreadvanced subgrid scale model are in good agreement with the experiment. We also briefly discuss someof the features of the instantaneous flow dynamics, to provide a glimpse of the wealth of information thatcan be extracted from such computations.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The development of computational tools to model fluid flowand heat transfer in rotating machinery applications has been atthe forefront of computational mechanics for the past few decades.The primary challenge that needs to be addressed by all numericaltechniques is the complexity of the geometry, which consists ofmultiple rotating and stationary parts. The application of well-established, body-fitted methods to such problems is not trivialdue to the presence of the moving parts. A frequently adoptedstrategy utilizes overset grids and different reference frames tosimulate the flow in the rotor and in the stator respectively (seefor example [14,11]). This approach usually allows for better gridquality, but the transfer of the solution between reference framesinvolves interpolations, which have an adverse impact on the over-all accuracy and efficiency of the formulation. If a single referenceframe approach is adopted on the other hand, frequent grid defor-mation/regeneration is necessary, which makes grid quality con-trol problematic [24].

ll rights reserved.

: +1 301 405 9953.Posa), [email protected] ([email protected] (E. Balaras).

An alternative class of methods that are well suited for applica-tions involving moving boundaries are the so-called immersed-boundary type methods, which have been gaining popularity inthe past few years. In such case the requirement for the computa-tional grid to be conform to the body is relaxed and structuredCartesian or generalized coordinate solvers can be utilized inhighly complex configurations. The presence of a complex bound-ary is introduced by a discrete forcing function, which is designedto mimic the effect of the body on the flow. Compared to theboundary fitted strategies above, immersed-boundary methodsgreatly simplify the grid generation and the same frame of refer-ence can be used for systems with bodies in relative motion. Inaddition, the conservative non-dissipative structured solvers uti-lized by most immersed-boundary approaches can be ideally cou-pled to eddy-resolving methods such as direct numericalsimulations (DNS) and large-eddy simulations (LES), thus substan-tially enhancing the overall predictive capability of the computa-tional tools. Today, immersed-boundary methods have beenutilized in a variety of fields. Early applications were mostly con-fined to biological flows (see for example [17,18]), but over thepast years a variety of applications in very diverse fields has beenreported. Verzicco and co-workers, for example, analyzed the flowin a piston-cylinder assembly [6], and an impeller-stirred tank [26],using a direct-forcing, immersed-boundary approach. More

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Nomenclature

DEk cyclic unsteadinessD local grid sizeh�i phase average operatorhh�ii passage average operators subgrid scale stress (SGS) tensormt eddy viscosityx rotational speed of the impellerxs specific-speed of the pump/ flow coefficientw head coefficient~p resolved pressureeS resolved strain rate tensor~u resolved velocity vectoru angle defining the position of the impellereF 2 second-order filtered velocity structure functionAin

i area of the impeller at the inflowCk Kolmogorov constantCs Smagorinsky model constantDin

d diameter of the diffuser at the inflowDout

d diameter of the diffuser at the outflowDin

i diameter of the impeller at the inflow

Douti diameter of the impeller at the outflow

H headN number of the blades passagesNU number of phase averagesNd number of the diffuser bladesNi number of the impeller bladesNr number of the impeller revolutionsQ flow rater, h, z coordinates of the cylindrical systemRe = UL/m Reynolds number based on a reference velocity, U,

length, L, and the kinematic viscosity m.T period of the passage of the rotor bladest timet0 initial time instant for the evaluation of the phase-aver-

aged fieldsuin

i tangential velocity of the tip of the impeller blades atthe inflow

uouti tangential velocity of the impeller blades at the outflow

34 A. Posa et al. / Computers & Fluids 47 (2011) 33–43

recently applications ranging from complex fluid-structure inter-actions [29,4] to magnetohydrodynamics (MHD) [8] appeared inthe literature. Details on the available immersed-boundary formu-lations as well as their range of applicability can be found in recentreviews in [10] and [15].

Turbomachinery applications are a great challenge for im-mersed-boundary formulations and a stringent test for theirrobustness and accuracy. As of today very few applications canbe found in the literature, which usually deal with simplified con-figurations. You et al. [30,31], for example, performed LES using animmersed-boundary method and a structured generalized curvi-linear coordinate solver, to study the viscous losses associated tothe tip-clearance flow in a linear cascade. To reduce the cost ofthe computations a single vane was considered, using periodicboundary conditions. Their results revealed some interestingdynamics and the importance of the tip-leakage jet on the vorticityand turbulence generation. They also demonstrate the importanceof utilizing eddy-resolving methods in such applications, whereexperimental measurements are very challenging and classicalReynolds Averaged Navier Stokes (RANS) models cannot capturethe highly three-dimensional and complex flow physics. As of to-day, immersed-boundary methods have not been utilized in morecomplex turbomachinery flows. The aim of the present work is toexplore the feasibility of utilizing such an approach in the case ofturbopumps, where the complete machine is simulated. In addi-tion, we will investigate the accuracy and applicability of standardsubgrid scale (SGS) models available in the literature. We shouldalso note, however, that due to cost considerations, the proposedapproach targets primarily laboratory scale pumps. Extension toindustrial scale pumps would require further developments inapproximate wall boundary conditions and sophisticated adaptivemesh refinement methods (see for example [25]).

Currently the majority of turbopump computations are con-ducted with body-fitted meshes utilizing different frames of refer-ence for the stationary and moving parts. These methods usuallyemploy stable, dissipative, discretizations and are therefore bettersuited to RANS modeling strategies. Within this modeling frame-work, Goto and co-workers [7,23] reported three-dimensional sim-ulations of mixed-flow pumps. They focused on the mechanism ofgeneration of secondary flow patterns and the formation of thejet-wake like flow. They also investigated the sensitivity of these

patterns to the tip-clearance. Kaupert et al. [13] investigated theeffects of pressure discontinuities in high-specific-speed radialpumps, showing reasonable agreement between experiments andcomputations. More recent RANS applications include the studyof the flow in a backswept centrifugal impeller by Zhang and co-workers [32,33], the impeller-diffuser interaction in a diffuserpump stage [20] and various configurations in turbopumps [9,3].LES of such flows are very limited. Byskov et al. [2] performedRANS and LES simulations on a six-bladed centrifugal pump impel-ler. Kato and co-workers [11,12] conducted LES using the Smago-rinsky model with van Driest damping near the walls to simulatea high-specific-speed mixed-flow pump. Pouffary et al. [19] andYamanishi et al. [28] used LES to study cavitation phenomena incentrifugal pumps, with reasonable agreement with experiments.

In the present work we report LES in a mixed-flow pump, wherea structured cylindrical coordinate solver with optimal conserva-tion properties is utilized in conjunction with an immersed-bound-ary method. Both the Smagorinsky (SM) model [21] and theFiltered Structured Function (FSF) [5] model were tested. The re-sults are compared to the experimental measurements in [1]. Inthe following sections the mathematical formulation and the prob-lem setup are given. Next the results are discussed followed bysome concluding remarks.

2. Mathematical model and numerical method

In the LES approach the large and energetic eddies, which areresolved directly as in a DNS, are separated from the small scales,which are modeled, by the application of a filtering operation.The resulting governing equations for the case of incompressibleflows are as follows:

r � ~u ¼ 0 ð1Þ@~u@tþr � ~u~u ¼ �r~p�r � sþ 1

Rer2 ~uþ f; ð2Þ

where � denotes the filtered variables, ~u is the velocity vector, ~p is amodified pressure which also includes the trace of the SGS stresstensor, s is the SGS stress tensor, and f is a forcing term. t is the timeand Re = UL/m is a Reynolds based on a reference velocity scale, U,and a length scale, L, which will be defined in the following sections

Fig. 1. (a) Schematic of the ray-tracing procedure used to classify the nodes on the Eulerian grid into � interior, � fluid, and h interface ones. Symbol � is a control point andthe dashed lines represent control rays. (b) Linear reconstruction of the solution in the vicinity of the immersed body [6]. Cyan: fluid domain; Gray: solid body.

1 For interpretation of color in Figs. 1–3, 5–8, 12 and 13, the reader is referred to theweb version of this article.

A. Posa et al. / Computers & Fluids 47 (2011) 33–43 35

(m is the kinematic viscosity of the fluid). Note that we will usecylindrical coordinates to achieve an efficient distribution of thegrid nodes in the present geometrical configuration.

Two different models are considered for the parametrization ofthe SGS stresses: the standard Smagorinsky model [21] and the fil-tered structure function model [5]. The former, despite its limita-tions, is a popular and inexpensive eddy viscosity model that isfrequently adopted in rotating machinery applications. The latteradjusts locally the eddy viscosity based on the available resolvedenergy near the cutoff wavenumber, resulting in a more accuratereproduction of the energy dissipation especially in transitional/relaminarizing flow configurations such as the one considered inthis study. In the both cases the deviatoric part of the SGS tensoris modeled using an eddy viscosity model of the form:

s� TrðsÞ ¼ �2mteS; ð3Þ

where eS is the resolved strain rate tensor. In the case of the SMmodel the eddy viscosity is defined as:

mSMt ¼ ðCsDÞ2jeSj; and jeSj ¼ ð2eS � eSÞ1=2

; ð4Þ

where Cs is the Smagorinsky constant and D is the local filter size.No effort was made to include wall dumping into the model, whichrequires the computation of the distance from the closest wall. Inthis application, where the solid surface is never aligned to anycoordinate direction, this is a rather challenging task with a sub-stantial computational overhead. In addition, the definition of sucha distance is ambiguous due to the presence of multiple rotatingand stationary surfaces. For the case of the FSF model the eddy vis-cosity is defined as follows:

mFSFt ¼ 0:0014C�3=2

k DðeF 2ðx;D; tÞÞ1=2; ð5Þ

where eF 2 is a filtered, second-order, velocity structure function (see[5] for details).

The governing equations are advanced in time using a fractionalstep method. The viscous terms in the momentum equation aretreated implicitly with a Crank–Nicolson scheme and all otherterms are treated with an explicit Adams–Bashfort scheme. All spa-tial derivatives are approximated with second-order finite differ-ences on a staggered grid. The large-band matrix associated withthe solution of the Poisson equation is first reduced to pentadiag-onal problems using FFTs in the azimuthal direction, and then eachproblem is solved with a generalized cyclic reduction algorithm[22]. Details on the overall formulation can be found in [27].

To simulate the flow in a complex configuration, which is im-mersed in the cylindrical coordinates grid, the direct-forcing ap-proach proposed by Fadlun et al. [6] is utilized. The body isrepresented by a series of triangles in stereo-lithography (STL) for-mat, where their density depends on the local curvature of the sur-face. Initially all points on the Eulerian grid are tagged according to

their relation to the immersed body. A ray-tracing procedure isused for this purpose and all grid points are classified as internal,fluid, and interface points. Internal are the points that are locatedwithin the solid body and interface points are fluid points withat least one neighbor in the solid body. The tracking scheme isillustrated in Fig. 1a: if C is a control point on the Eulerian grid out-side the body, a control ray is a segment which connects C to anygrid point in the neighborhood of the body. For each ray, the inter-sections with the triangles on the body are found. If the number ofintersections is even, the corresponding grid point is external tothe body, otherwise it is internal. The interface points are centralto the reconstruction of the solution near the solid body. FollowingFadlun et al. [6], a linear reconstruction of the velocity is consid-ered. As shown in Fig. 1b, the desired velocity, Vi, at the interfacepoints is computed using a linear interpolation along a grid line(the velocity values Vc and ue, respectively on the body and inthe adjacent external point, are known); then the forcing functionin the Eq. (2) can be found by substituting the interpolated valuefor Vi in the discretized momentum equation:

f lþ12 ¼ Vlþ1

i � ~ul

Dt� RHSlþ1

2 ð6Þ

where l; lþ 12 and l + 1 refer to the time level, and the RHS includes the

discrete convective, viscous, and SGS terms in Eq. (2). More details onthe implementation of the above method can be found in [6].

3. Computational setup

We will consider a mixed-flow pump configuration, which hasbeen studied experimentally by Boccazzi et al. [1]. They reportedParticle Image Velocimetry (PIV) measurements inside the vaneddiffuser at two stations. The geometry of the pump is shown inFig. 2a. To better distinguish the different elements each one is rep-resented by a different color: the rotor blades are drawn in red1,the hub in white and the diffuser blades in blue. Note that the shroud(green), the diffuser throat and the volute (gray) are partially shown,for the internal pump elements to be visible. A meridian cut of thevolute geometry is also shown in Fig. 2b. The number of blades forthe rotor and the stator are Ni = 6 and Nd = 7 respectively. The spe-cific-speed, xs, the flow coefficient, /, and the head coefficient, w,are defined as follows:

xs ¼x

ffiffiffiffiQp

ðgHÞ34/ ¼ Q

Aini uin

i

w ¼ gH

uouti

� �2 ; ð7Þ

where x is the rotational speed of the impeller, Q the flow rate, Hthe head, Ain

i is the area of the impeller inlet and uini and uout

i are

Fig. 2. Geometry of the mixed-flow pump used in the experiments by Boccazzi et al. [1]. (a) Cross cut of the volute; (b) meridian cut of the volute.

Table 1Main parameters of the mixed-flow pump used in the experiments by Boccazzi et al.[1].

x = 55.4[rad/s] xs = 1.08 /opt = 0.314 wopt = 0.443 Dini ¼ 0:154½m�

Douti ¼ 0:224½m� Ni = 6 Din

d ¼ 0:233½m� Doutd ¼ 0:361½m� Nd = 7


the tangential velocities at the impeller inflow and outflow respec-tively. For the inflow the speed is referred to the blades tip. All thegeometric and operating parameters are summarized in Table 1,and are the ones considered in all computations below. Note thatDin

i and Douti are the impeller diameters at the inlet and the outlet,

Fig. 3. Pump geometry immersed in the computational mesh. (a) Computational grid inof the structured grid that falls within the fluid domain. The red color marks the ‘wasted’

while Dind and Dout

d are the corresponding values for the diffuser.The spanwise dimension of the diffuser is Ls = 44[mm]. The flowrate corresponding to /opt is Q = 0.025[m3/s]. The Reynolds number,based on the average inflow velocity and the external radius of therotor, is equal to Re = 1.5 � 105.

The pump is fully immersed in the computational domain,which is discretized with a structured cylindrical coordinates grid.An example is shown in Fig. 3, where two grid slices in the r � zand r � h planes are included. The blue portion of the figure repre-sents the part of the structured grid that falls within the fluiddomain. The red portion marks the grid nodes that are ‘wasted’.For clarity the volute, the shroud and the inflow channel are not

the r–z plane; (b) computational grid in the r–h plane. Blue color represents the partgird nodes. Note that only the impeller and the diffuser blades are shown for clarity.

Fig. 4. Detail of the computational grid in the divergent channel between thediffuser blades (an r � h plane is shown).


included in this figure. The inflow and outflow locations are alsoindicated. In all computations reported in this study we prescribea uniform velocity at the inflow plane, while we use a convectivecondition at the outflow [16]. The convective condition is appliedon the local flow direction, using the averaged velocity on the out-flow surface as convective speed. The immersed-boundary tech-nique described in the previous section is used to enforce theboundary conditions on all solid boundaries.

The computational grid consists of 801 � 350 � 101 points inthe azimuthal, radial and axial directions respectively (a total of28 million). It is uniform in the azimuthal and axial directionsand is stretched in the radial one to cluster points in the areas ofhigh velocity gradients. Since the grid does not conform to thebody, a significant number of nodes falls outside the useful compu-tational domain, which in the present case is approximately 50%.This grid, which has a total of 14 million useful nodes, providesadequate resolution in many critical areas of the computationalbox. In the divergent channel between the diffuser blades, forexample, along the direction normal to the blade’s surface, respec-tively 40, 45 and 52 computational cells are utilized over the linesa, b, and c in Fig. 4. Each computation on the above grid requiresapproximately 56 CPU hours/revolution on a single 2-way quad-core Opteron 2.1 GHz node with 16 GB of RAM. All eight cores onthe node were utilized using OpenMP. Due to cost considerations,we could not perform a detailed grid refinement study on the fullcomputational domain, and the above resolution is the maximumwe could achieve with the available computational resources atthe time of the preparation of the manuscript. We did, however,conduct a grid refinement study on a reduced domain, where onlyone diffuser passage is considered using periodic boundary condi-tions. In particular, we compared the results between a coarse gridwith equivalent resolution to the one utilized above in the full do-main, and a fine grid with double resolution in the azimuthal andradial directions. The mean velocities, although are not directlycomparable to the experiment in [1], due to the simplifications inthe geometry, were in good agreement and the largest discrepan-cies (order of 10%) were observed near the walls.

4. Results

In this section we will report a series of computations con-ducted in the configuration presented above. The primary aim ofthe simulations is to establish the accuracy and robustness of theimmersed-boundary methods in rotating machinery applicationsby direct comparisons to the experimental results by Boccazziet al. [1]. We will also present the results from different SGS mod-els, as well as, an overall view of the instantaneous flow dynamicsin the particular configuration. To facilitate comparisons with theexperiment, for any flow variable f(h,r,z, t), which is a function ofspace and time, the following phase average operator is defined:

hf iðh; r; z;uÞ ¼ 1N

XN

n¼1

f ½h; r; z; t0 þ ðn� 1ÞT�; ð8Þ

where h, r and z are the azimuthal, the radial and the axial coordi-nates respectively; u is the angle which defines the position of theimpeller with respect to the stator, for which the phase average isevaluated; in the present case its value is variable between 0� and60�, since after one blade passage the same impeller-diffuser config-uration is repeated; N = NiNr is the total number of blades passages,i.e., the number of times that the impeller is in the same positionrelative to the diffuser (Nr is the number of the impeller revolu-tions); t0 is the initial time instant and T the period of the passageof the rotor blades. We can also define the passage average operatoras follows:

hhf iiðh; r; zÞ ¼ 1NU

XNU

i¼1

hf iðh; r; z;uÞ; ð9Þ

which is simply the average of the phase-averaged fields, with NU

being the number of phase averages, namely the number of u val-ues for which the phase averages are evaluated. In all computationsreported in this study the phase averages were computed over 5impeller revolutions, and NU = 12. We found that this sample wassufficient to obtain smooth first- and second-order statistics. Weshould also note that approximately seven revolutions were re-quired for the effect of the initial conditions to decay and a quasiperiodic state to be reached prior to sampling.

4.1. Comparison to the experiments

First, the head coefficient, w, and the efficiency, g, are evaluatedfor both computations. For the case of the LES with the SM model(SM-LES) they are w = 0.44, g = 0.74, and for the FSF model (FSF-LES), w = 0.47, g = 0.80. The corresponding experimental valuesare w = 0.44 and g = 0.75, respectively. The results from the SM-LES are in very good agreement with the experiment (within1.5%). The FSF-LES slightly over-predicts both w and g by approxi-mately 6%. A possible reason for this behavior is the less dissipativecharacter of the second model.

In Fig. 5 the phase-averaged velocity magnitude at the diffusermidspan is shown, for window A indicated in Fig. 6a. These phaseaverages correspond to a configuration during the revolution forwhich the position of the impeller relative to the diffuser is suchthat the trailing edge of a rotor blade at the midspan is in the sameazimuthal position as the leading edge of the stator blade number4 (see Fig. 6a for the numeration of the diffuser blades and the po-sition of the experimental windows). Results from the LES with theSM model (SM-LES) and the FSF model (FSF-LES) are compared tothe experiment. It can be seen that all computations capture themain flow features in this window. The SM-LES, however, over-predicts the velocity in the shear layers developing at the trailingedge of the impeller and produces higher velocities in the statorpassage between blades 3 and 4. The results of the FSF-LES are ingood agreement with the experiments. Both models, however,seem to underestimate the wakes of the impeller vanes propagat-ing within the diffuser channels.

A detailed quantitative comparison is shown in Fig. 6, where thephase-averaged velocity profiles at the diffuser midspan in thedirection normal to the pressure side of the blade are shown. Inparticular, Fig. 6b shows the tangential velocity at 10% of the chordof the blade 5 and Fig. 6c and d the tangential velocities at 50% and90% of the chord of the blade 4, respectively. The exact locations ofthe profiles, which match the ones in the experimental dataset, areshown in Fig. 6a. Since these locations are not aligned with the gridlines, the velocities were interpolated from the surrounding gridnodes. The spacing on each line was selected to be similar to the

Fig. 5. Phase-averaged velocity magnitude at the diffuser midspan for window A shown in Fig. 6a. (a) PIV measurements [1]; (b) FSF-LES; (c) SM-LES. The velocity magnitudeshown ranges from 0[m/s] (blue) to 4[m/s] (red).

Fig. 6. (a) Experimental measurement windows, blade numbers, and velocity profile locations. Also the following abbreviations are defined: RBSS: rotor blade suction side;RBPS: rotor blade pressure side; SBPS: stator blade pressure side; SBSS: stator blade suction side. (b) Phase-averaged tangential velocity profile at location a (10% of the chordof the blade 5); (c) Phase-averaged tangential velocity profile at location b (50% of the chord of the blade 4); (d) Phase-averaged tangential velocity profile at location c (90% ofthe chord of the blade 4). � experiment [1], 33 SM-LES, FSF-LES.

Fig. 7. Phase-averaged velocity magnitude at the diffuser midspan for window B shown in Fig. 6a. (a) PIV measurements [1]; (b) FSF-LES; (c) SM-LES. The velocity magnitudeshown ranges from 0[m/s] (blue) to 4[m/s] (red).


Fig. 8. Phase-averaged tangential velocity profiles at the diffuser midspan. (a) location d (10% of the chord of the blade 0); (b) location e (50% of the chord of the blade 0).�experiment [1], 33 SM-LES, FSF-LES.

Fig. 9. Passage-averaged velocity magnitude (top part) and cyclic unsteadiness (bottom part) at the diffuser midspan for window A shown in Fig. 6a. (a) and (c) PIVmeasurements [1]; (b) and (d) FSF-LES.


local grid spacing. In general the agreement with the experimentsat all stations is satisfactory. At station a (Fig. 6b) both computa-tions agree with the experiments and only near the pressure sideof the diffuser blade 4 the velocity is under-predicted, most prob-ably due to the lack of resolution in the thin boundary layer. At sta-tion b (Fig. 6c) the differences between the SM model and theexperiments are more profound. This is due to the dissipative char-acter of the SM model near the walls, which results in excessivedissipation and under-prediction of the wall shear stress. To

conserve mass the velocity increases away from the wall and a lo-cal maximum is generated at the edge of the boundary layer. TheFSF is in better agreement with the experiment. At location c nearthe trailing edge of blade 4 (Fig. 6d) both computations agree wellwith the experiment except for a small area near the wall.

Fig. 7 shows the phase-averaged velocity magnitude at the mid-span of the diffuser throat in window B, near the nose of the volute.The results with the FSF model agree well with the experiment. Forthe SM-LES the flow separates much earlier on blade 0 and


generates a large separated region. This has a direct impact on theangle the flow has as it hits the volute nose, producing a large recir-culation area within the volute and between the same volute andthe blade 0. To better quantify the differences between the differ-ent computations, also in this case the corresponding phase-aver-aged tangential velocity profiles are shown in Fig. 8. The locationd at 10% of the chord of the blade 0 (Fig. 8a) is before the separationpoint, and all numerical results near the pressure side of blade 0agree well with the experiment. On the suction side of the blade6, however, the velocity is underestimated, especially with theSM model. It is worth noting that on the pressure side of the blade6 the SM-LES predicts slightly negative velocities, indicating thatthe flow has already separated. The differences between the SGSmodels are more evident at location e at 50% of the chord of theblade 0 (Fig. 8b). Due to the early separation the SM-LES severelyunder-predicts the velocity near the wall. The FSF-LES, on the otherhand, is in good agreement with the experiment.

Due to cost considerations we only computed phase-averagedstatistics for 12 different relative positions between the impellerand the diffuser (every 5�), which also coincide with the corre-sponding ones from the experiments. Similarly to what has beenobserved above, the phase-averaged velocity field for the case ofFSF-LES agrees well with the experiment for all 12 locations (notshown here). The SM-LES always results in under-prediction ofthe wall shear stress due to its overly dissipative character. InFig. 9a and b passage-averaged velocity fields are shown in win-dow A for both the experiment and the FSF-LES. Note that theimpeller blades are not shown, since the passage-averaged fieldis not related to a particular impeller position. The agreement be-tween the computation and the experiment is very good. Anotherquantity usually adopted in rotating machinery applications tocharacterize the unsteadiness associated to the large scales is thecyclic unsteadiness, defined as:

DEkðh; r; z; uÞ ¼ 12½hvi2ðh; r; z; uÞ � hhvii2ðh; r; zÞ�; ð10Þ

Fig. 10. Instantaneous snapshots of the vorticity magnitude at the midspan of the diffuse

where hvi is the phase-averaged velocity magnitude and hhvii thepassage-averaged velocity magnitude. In Fig. 9c and d the cyclicunsteadiness is shown for the PIV measurements and the numericalresults from the FSF-LES. The structure of the impeller blade wakeinside the diffuser is well predicted and the local minima on thepressure side near the leading edge of the stator blade 5 are alsoreproduced by the simulation.

4.2. Instantaneous flow dynamics

Having established the accuracy of our computations we willnow examine the instantaneous flow dynamics. Note that we willonly present results from the FSF-LES, which was found to betteragree with the experiment. In Fig. 10 the instantaneous vorticitymagnitude is shown for four consecutive positions of the impeller,which are 15� apart. Most of the activity, as indicated by the highvorticity values, is concentrated on the suction side of the impellerand stator blades. It is also evident that the vortical structures gen-erated on the suction side of the impeller blades and consequentlyshed in the wake, interact strongly with the suction side of the sta-tor blades. This is clearly seen in the time sequence shown inFig. 10a–d: in part (a) of the figure, the tip of the impeller bladea is close to leading edge of the stator blade 4, generating a distur-bance, as manifested by the local increase of the vorticity magni-tude in the area; as this impeller blade moves forward (see partsb–d in the figure) its wake impinges on the stator blade and dis-turbs the boundary layers on its surface, indicating that transitionto turbulence probably occurs through a bypass transition mecha-nism. On the pressure side of the impeller blades, on the otherhand, the favorable pressure gradient leads to reduced turbulentactivity as shown in Fig. 11, where the impeller-diffuser configura-tion of Fig. 10a is shown from a different angle. In Fig. 11 it is alsoevident that the vorticity magnitude on the pressure side of the dif-fuser blades is lower when compared to the one on the suctionside. We should note, however, that the effect of the impeller wake

r for the FSF-LES. Consecutive positions of the impeller blades are shown every 15�.

Fig. 11. Instantaneous vorticity magnitude at the midspan of the diffuser for theFSF-LES. Representation of the vorticity values on the pressure side of the rotor andthe stator blades. The same instant in time is shown as in Fig. 10a from a differentangle, in order for the pressure side of the blades to be visible.

Fig. 12. (a) Isosurfaces of Q = 5 � 104[1/s2], colored with the instantaneous vorticity mvolute; Cyan: hub; Blue: diffuser blades; (b) magnification of area b shown in part (a)instantaneous vorticity magnitude in the section A–A shown in part (a) of the figure. DW


is noticeable also on the pressure side of the stator blades, but inthis case the action of the wake on the boundary layer on theblade’s surface is not as profound.

To gain a better insight on the dynamics of the coherent struc-tures which are generated in the rotor and interact with the stator,the second invariant of the velocity gradient tensor, Q, is utilized.Fig. 12a shows an instantaneous picture of the flow by means aQ isosurface, colored by the vorticity magnitude. Four differentgroups of coherent vortices can be identified:

(i) The ‘packet’ of vortices within the area a in the figure, whichis produced near the shroud. In general, the divergent geom-etry of the shroud generates large adverse pressure gradientsand at the same time the flow turns rapidly by 90�, resultingin complex separation phenomena. The vortices shown inarea a are probably related to the above phenomena andare not as coherent as the ones in other areas of the machine.

agnitude, for the FSF-LES. Green: shroud; Gray: impeller blades, diffuser wall andof the figure (note that Q = 1 � 105[1/s2] isosurfaces are shown in this case); (c)SS: diffuser wall on the shroud side; DWHS: diffuser wall on the hub side.

Fig. 13. Instantaneous azimuthal (top) and radial (bottom) vorticity fields on the circumferential section at the impeller mean radius from the FSF-LES: (a) 3D view (white:hub, gray: impeller blades, green: shroud, blue: inflow channel); (b) top view of the ‘unrolled’ plane (RBPS: rotor blade pressure side, RBSS: rotor blade suction side).


(ii) The vortices within the area b in the figure, which is locatedat the suction side of the impeller blades. These are muchmore coherent, compared to the ones above. These vorticesappear approximately at the same location in all impellervanes. In Fig. 12b a closer view of the same area is shownand one can identify strong co-rotating vortices, which arealigned with the flow.

(iii) The smaller eddies within the area c in the figure, which areassociated with the wake of the impeller blades. The wakearea is strongly influenced by the interaction of the vorticesin b and the boundary layer on the impeller blades. The wakevortices quickly lose their coherence as they move awayfrom the trailing edge and appear to be the main source ofturbulent fluctuations for the flow entering the diffuserthroat.

(iv) Vortical structures inside the diffuser and particularly in thewake of the diffuser blades (area d in the figure). These formlarger vortices which are shed from the trailing edge, butthey are weaker than the corresponding ones at the rotoroutlet.

To better characterize the coherent structures in the diffuser,Fig. 12c shows the instantaneous vorticity magnitude at a planeperpendicular to the flow direction (section A–A in Fig. 12a). Theselected position is close to the leading edge on the pressure sideof the blade 2, and to the trailing edge on the suction side of theblade 1. On the pressure side there are still high vorticity values,due to the proximity of the wake of the impeller blades. On thesuction side, non-uniformities in the vorticity distribution alongthe blade span can be observed, which are probably caused bythe flow coming from the impeller shroud. Fig. 13 shows theinstantaneous azimuthal and radial vorticity on a circumferentialsection at the impeller mean radius. Part (a) shows the locationof the section inside the impeller, while in part (b) the plane is ‘un-rolled’ and shown from the top. Referring to part (a) in the figure,the flow enters the impeller axially, from left to right, and exitsradially. Consistent with the observations above, close to theshroud (area a) and at the blade suction side (area b), the footprintof strong vortices can be seen, with the latter being more coherentas well as having the same vorticity sign (co-rotating). We should

also note that, even if the machine geometry is not symmetric, sim-ilar eddy patterns can be identified at all impeller vanes.

5. Summary and conclusions

Turbomachinery applications are a great challenge forimmersed-boundary formulations and a stringent test for theirrobustness and accuracy. As of today very few applications of im-mersed-boundary methods to such flows can be found in the liter-ature and usually deal with simplified configurations (see forexample [30,31]). The aim of the present work is to explore the fea-sibility of utilizing such an approach in the case of turbopumps,where the complete machine is simulated. We considered amixed-flow pump configuration, which has been studied experi-mentally by Boccazzi et al. [1]. The pump has been fully immersedin the computational domain, discretized with a structured grid incylindrical coordinates, using approximately 28 million points. Theimmersed-boundary technique by Fadlun et al. [6] was used toenforce the boundary conditions on all solid boundaries.

We tested two different SGS models: the standard Smagorinshymodel [21] and the filtered structure function model [5]. The for-mer was not expected to capture the complex physics in suchflows, due to the lack of proper limiting behavior near walls andits inability to switch off in areas of laminar flow. Also the use ofa single constant cannot accurately capture the energy dissipationin the wall layers as well as in the free shear layers in the machine.Nevertheless it is a popular and inexpensive eddy viscosity model,which is frequently adopted in rotating machinery applications,and therefore a closer look at its performance would be useful.The FSF model, on the other hand, adjusts locally the eddy viscosityon the basis of the available resolved energy near the cutoff wave-number, resulting in a more accurate reproduction of the energydissipation, especially in transitional/relaminarizing flow configu-rations, such as the one in this study.

In general, the overall agreement with the experimental resultsis good, demonstrating the robustness and feasibility of the ap-proach in rotating machinery applications. Both SGS models cap-ture the main physics of the flow, but as expected the excessivedissipation produced by the SM model, especially near walls,


results in the underestimation of the wall stresses and early sepa-ration. The results with the FSF model are in very good agreementwith the experiments, indicating that it is a much better choice forsuch flows, especially considering that its computational overheadis not much higher than the one of the SM model. We should alsonote, however, that due to cost considerations, a comprehensivegrid refinement study was not feasible. Simulations on a reduced,periodic domain, at comparable and finer grid resolutions, pro-vided mean velocity statistics which differed no more than 10%,giving us some confidence in the results reported here. We are cur-rently working on improving the parallel performance of the sol-ver, which will enable us to do detailed grid refinement studiesin the full domain as well.

Finally an outline of the instantaneous flow dynamics is pre-sented. It is evident that the vortical structures generated on thesuction side of the impeller blades, and consequently shed in thewake, interact strongly with the suction side of the stator blades,indicating that transition to turbulence on the latter occursthrough a bypass transition mechanism. The influence of theimpeller wake is noticeable also on the pressure side of the statorblades, but in this case the effects on the boundary layer on theblade’s surface are not as profound.

From a practical perspective, the LES approach is probablymuch more valuable for off-design conditions, for which it can of-fer unique insights into the complex physics of such pumps. Thepurpose of the present study, however, is to demonstrate the fea-sibility/accuracy of the overall simulation strategy, and for this rea-son we only considered optimal operating conditions, wherecomprehensive laboratory data were available. We will consideroff-design and compare with RANS/DES in a future paper.

Acknowledgments

The authors are grateful to Ing. A. Boccazzi and Ing. R. Miorinifor providing their experimental results. Computational resourceswere provided by CASPUR (Consorzio interuniversitario per leApplicazioni di Supercalcolo Per Università e Ricerca). EB is par-tially supported by the National Science Foundation (Grant CBET-0932613).

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