TIME-ACCURATE TURBOMACHINERY SIMULATIONS WITH OPEN-SOURCE CFD; FLOW ANALYSIS OF A SINGLE-CHANNEL PUMP WITH OpenFOAM

V European Conerence on Computational Fluid Dynamics ECCOMAS CFD 2010 J. C. F. Pereira and A. Sequeira (Eds) Lisbon, Portugal,14-17 June 2010 TIME-ACCURATE TURBOMACHINERY SIMULATIONS WITH OPEN-SOURCE CFD; FLOW ANALYSIS OF A SINGLE-CHANNEL PUMP WITH OpenFOAM Mikko Auvinen, Juhaveikko Ala-Juusela, Nicholas Pedersen, Timo Siikonen, Aalto University P.O. Box 14400 FI-00076 Aalto, Finland e-mail: {mikko.auvinen, juhaveikko.ala-juusela, timo.siikonen}@hut.i Grundos A/S Poul Due Jensens Vej 7, DK-8850 Bjerringbro, Denmark e-mail: nipedersen@grundos.com Key words: CFD, OpenFOAM, Transient, Time-Accurate, Turbomachinery, Rotating Machinery, GGI Abstract. This paper presents a time-accurate analysis o a single-channel pump with a low system characterized by highly oscillatory behavior. The analysis is perormed with an open-source numerical analysis library, OpenFOAM, which eatures a recently implemented Generalized Grid Interace method that provides the means to conduct transient, sliding interace simulations on arbitrarily unstructured meshes. Since the complexity o the low ield demands high standards rom the numerical analysis, two computational models are considered and our separate cases analyzed to investigate the solution s sensitivity to turbulence modeling, grid resolution and boundary condition treatment. The report includes a detailed description o the applied CFD methodology and covers a broad range o issues that are relevant to the OpenFOAM analysis and post-processing o the simulations. The obtained time-accurate results are compared against experimental perormance and LDV velocity proile measurements. The comparisons yield a wealth o inormation on dierent aspects o the analysis providing tangible guidelines and recommendations. All the computational cases depicted the characteristic low behavior o the pump distinctly, but, above all, the high-resolution grid model succeeded in capturing the nature o the low system in striking detail. 1

1 INTRODUCTION Single-channel (and single-blade) pumps constitute a special amily o pumps, eaturing geometry designed to operate in waste water conditions without clogging. This requirement complicates the design or high hydrodynamic eiciency as the non-periodic geometry generates a complex, oscillatory low system that is vastly dissimilar to conventional multi-blade water pumps. The demanding nature o the low system necessitates spatially and temporally accurate computational analysis, 1 which utilizes a sliding interace between the rotating and stationary domains. Moreover, the requirements set by reliable design practise demand that these computationally intensive transient analyses are routinely perormed with high resolution to examine potential pump prototypes in greater detail. The role o such high quality CFD in exposing the mechanisms that drive perormance-deteriorating low behavior is becoming increasingly signiicant. Consequently, the drastic increase in demand or such high-idelity CFD analyses, together with the need or more lexible, customizable tools has heightened interest in open-source and license-ree sotware across the CFD community. In response, this study utilizes OpenFOAM, 2 an object-oriented numerical analysis library written in C++ with a recently implemented Generalized Grid Interace 3 (GGI) coupling algorithm, to conduct a comprehensive, time-accurate low analysis o an experimental single-channel pump design. The implementation o the GGI and the associated testing and validation work 4 have been brought together by the OpenFOAM-extend project, which is a collaborative undertaking that also maintains the repository hosting the OpenFOAM library package used in this study. In accordance with the mission o the OpenFOAM-extend project, as this study builds on the contributions o others, the work is intended to provide a user contribution to the open-source community. This paper provides a thorough description o the perormed transient analysis o an experimental single-channel pump, shown in Figure 1. The report covers the relevant computational methodologies together with the associated algorithmic undamentals and describes the analysis o our dierent computational models in detail. In the end, the obtained time-accurate results are presented together with experimental perormance and Laser Doppler Velocimetry (LDV) velocity proile measurements, and the eects o the dierent modeling choices are discussed. Figure 1: An overview o an experimental single-channel pump geometry. http://sourceorge.net/projects/openoam-extend/ 2

2 CFD METHODOLOGY The solution o the time-dependent, turbulent and incompressible low system o the pump is governed by the Unsteady Reynolds-Averaged Navier-Stokes equations (URANS), which can be written or a moving control volume V, bounded by closed surace S with an outward pointing unit normal vector n, in the ollowing orm: t UdV + dv + (U U g ) nds = 0 (1) U(U U g ) nds + pnds = ν e U nds (2) t where U is the luid velocity vector and U g denotes the velocity o the bounding surace S. In Eqn.(2), the inluence o the Reynolds stresses is embedded in the eective kinematic viscosity ν e, according to Boussinesq s approximation, necessitating the use o an appropriate turbulence model to attain closure. In a more general treatment, where the deormation o control volumes is also considered, an additional requirement must be satisied: t dv + (U g n)ds = 0 (3) This is known as the Geometric Conservation Law. 5 In turbomachinery applications, where the moving grid domain undergoes only solid body rotation, this condition is evidently satisied. Employing the extensive library structure o OpenFOAM, two low solver types utilizing segregated velocity-pressure coupling algorithms and eaturing both automatic mesh motion and deormation unctionality, 6 have been developed by the community or simulating the low systems described by Eqns.(1-3). The irst, entitled pimpledymfoam, is based on the PISO 7 algorithm while the second, transientsimpledymfoam, implements a time-accurate SIMPLE 8 pressure correction method. Although the PISO solver acilitates an accurate transient solution, it suers rom ineicient temporal time marching due to a restricting limitation on the maximum time step length. For this reason it has proven impractical or turbomachinery applications. On the other hand, the solver eaturing SIMPLE does allow more aggressive time marching naturally at the expense o temporal accuracy, which is critical in perorming eicient time-accurate analysis o low systems whose transient behavior evolves over a comparatively long time. Thereore, in this study the simulations are carried out with transientsimpledymfoam, whose principal algorithmic description is provided in the ollowing section. Available in OpenFOAM 1.6 and in OpenFOAM 1.5 as turbdymfoam Developed under OpenFOAM-extend project (http://openoam-extend.sourceorge.net) and, thereore, resides in the OpenFOAM-dev release. 3

2.1 Flow Solver In order to provide an inormative description o the transientsimpledymfoam low solver, which implements the segregated SIMPLE pressure-velocity coupling algorithm, a proper groundwork must be laid by presenting a concise derivation o the pressure equation 9 as it is implemented in OpenFOAM. The grid can be considered stationary in the treament because, within a time step, the pressure-correction step operates in an absolute velocity ield. 2.1.1 Theoretical Background Firstly, equations (1) and (2) are written in a discrete orm or a ixed control volume V (i.e. cell) that is bounded by an arbitrary number N o cell aces. Given the surace area o a cell ace S, its normal vector n and the deinition o a ace lux φ = (U n )S, the equations can be written as: N U t V + (U n )S = 0, or simply: φ = 0 (4) N N φ U (ν e U n) S = p n S (5) N The discrete momentum equation (5) can be transormed into a linear system o equations that, or each computational cell center P surrounded by N nb neighboring cells, obtains a orm: N nb a P U P + a nb U nb = RHS (6) nb where the dimensions o the system have been changed due to a division by cell volume V P. The right-hand-side (RHS) o the equation contains the source contributions arising rom the discretizations o the transient, convection and diusion terms and the pressure gradient. For convenience, the contributions are split into velocity- and pressure-dependent parts RHS = rhs(u) p, recognizing that p = ( 1 ) V p n S. Using short-hand notation N nb H(U) = a nb U nb + rhs(u) Eqn. (6) can be expressed as nb a P U P = H(U) p (7) From this ormulation, a new ace velocity can be deined that is interpolated onto the cell aces using cell center values: N 4

U P = H(U) p (8) a P a ( ) ( P ) H(U) 1 U = ( p) (9) a P The discrete pressure equation is obtained by substituting Eqn.(9) into the continuity requirement o Eqn.(4), yielding N [ ( ) ] N 1 ( ) H(U) ( p) n S = n S (10) a P A simpliied orm resembling the implementation in OpenFOAM can be attained by deining an intermediate velocity ield and evaluating the lux ield, which does not satisy the continuity requirement, accordingly ( ) H(U) U = (11) a P a P a P φ = ( U n ) S (12) Thereby, the discrete pressure equation reaches its inal orm: N [ ( ) ] N 1 ( p) n S = φ (13) a P 2.1.2 Solver Description Reerring to the developments in Section 2.1.1 the solution procedure implemented in transientsimpledymfoam can be illustrated by the ollowing procedure: TIME Loop: while (t n < t end ) 1) Increment time: t n = t n 1 + t. 2) Convert ace luxes to correspond to an absolute velocity ield: φ = (U n )S. 3) Apply mesh movement (and/or deormation) utilizing a chosen dynamic mesh library. 4) Correct the lux ield i the mesh has deormed. (Not necessary in turbomachinery applications.) 5) Convert ace luxes to correspond to a relative velocity ield: φ = (U U g ) n S. 5

6) SIMPLE Loop: or( i = 0; i < niter ; i++) 6.1: Build the momentum equation (6) applying relaxation 0 < α u < 1 to increase the diagonal dominance o the coeicients matrix: a P N nb U i P + a nb U i nb = RHS + (1 α u) a P U i 1 P (14) α u α u nb and solve or U i. (Note that at the matrix level the terms are multiplied by cell volume V P beore the relaxation is applied.) 6.2: Deine an intermediate velocity ield U and compute a corresponding lux ield φ according to Eqns. (11) and (12). 6.3: Store the pressure value o the current iteration: p i 1 = p i. 6.4: Build the pressure equation (13) and solve or p i. 6.5: Correct the( lux ) ield such that it ulills the continuity requirement: φ = φ 1 a P ( p i ) n S 6.6: Apply an explicit relaxation to the pressure ield p i = p i 1 + α p (p i p i 1 ), where α p is the under-relaxation actor or pressure that typically takes on values within range 0.1 α p 0.3. 6.7: Convert ace luxes to correspond to a relative velocity ield: φ = (U U g ) n S. 6.8: Correct the velocity ield utilizing a relaxed pressure ield according to Eqn.(8): U i = U i pi a P 6.9: Solve turbulence model equations. 6.10: Return to 6.1 or continue. 7) Return to 1) or exit time loop and terminate simulation. 2.2 Computational Models This study ocuses on simulating the main low system o the pump, which dictates the hydrodynamic quality o the design, and, thereore, some speciic aspects o the pump arrangement are neglected. For instance, the computational models do not include the water-illed cavities which emerge in the spaces separating the impeller hub and shroud rom the pump housing. Consequently, the relatively small gaps which separate the rotating impeller rom the stationary volute and merge the main low path with the cavities are also omitted rom the analysis. The model simpliications are well justiied rom a computational perspective, but inevitably hinder comparability between the numerical and experimental results. 6

Mikko Auvinen, Juhaveikko Ala-Juusela, Nicholas Pedersen and Timo Siikonen However, in design practice the main objective is to reach comparative improvements. Thus, the principal concern lies in securing the quality o the CFD analysis such that the small changes in the hydrodynamic design are relected in the numerical results. For this reason, strong emphasis has been placed on producing high quality grids consistently through a templatable process. These requirements have been achieved with a grid generation tool called GridPro.10 Its technology has been thoughtully exploited to generate hexahedral meshes or a coniguration where the rotating domain is separated rom the stationary by a cylindrical interace, as shown in Figure 2. The coupling across the nonconormal grid interace is handled by the GGI. Figure 2: The CFD models consist o rotating (middle) and stationary (right) domains, which are coupled across the shown grid interace by GGI. To investigate the eect o grid resolution, two dierent grid densities are considered in this study. Both grids are generated or high Reynolds number turbulence models, which employ wall unctions at the solid boundaries. The grids are depicted in Figure 3 and labeled Coarse ( 0.5M cells) and Fine ( 1.5M cells), respectively. The mesh conversion to OpenFOAM ormat was accomplished with GridPro2Foam converter. Figure 3: General view o the Coarse (let) and the Fine (right) GridPro grids. Available at http://www.rtech-engineering.com/news.html 7

From a computational point o view, the treatment o the inlow boundary conditions turns out to be problematic: Imposing constant velocity and turbulence quantities at a location where a developed (or developing) pipe low truly occurs clearly represents a compromising approximation. This introduces uncertainties whose level o severity should be investigated. For this purpose, a second computational model has been prepared by appending an elongated inlet duct to the Coarse model to allow the pipe low proile to develop beore reaching the original inlow boundary. The computational models are shown in Figure 4. The appended duct is part o the stationary domain and is also connected to the rotating domain via GGI. Figure 4: Outline o the standard model (let) and the elongated inlet duct model (right). 2.3 CFD Analysis and Case Speciications The transient simulations are perormed or the pump s design point conditions, which were also present in the experiment perormed in collaboration with the manuacturer Grundos. The associated boundary conditions applied in the CFD analysis are listed in Table 1. Solution Rotating Stationary Variable Walls Walls Inlet Outlet U U wall = (Ω r) Fixed Value: Fixed Value: U in Zero Gradient Ω = 1470 rpm U wall = 0 (ρu in n)s = 27.8 kg/s p Zero Gradient Zero Gradient Zero Gradient Fixed Value: p out k Wall Function Wall Function Fixed Value: k in Zero Gradient ε Wall Function Wall Function Fixed Value: ε in Zero Gradient ω Wall Function Wall Function Fixed Value: ω in Zero Gradient Table 1: Applied boundary conditions. The computational cases included in this study were constructed to yield inormation about numerous aspects concerning transient analysis o turbomachinery with Open- FOAM. Yet, the main elements o consideration, besides overall solution quality, were 8

limited to the eects o grid resolution, turbulence modeling and treatment o the inlet boundary condition. A listing o the essential computational settings characterizing the our dierent cases is shown in Table 2. The labeling o the numerical schemes adheres to the OpenFOAM syntax in which, or example, the suix V denotes a scheme or a vector variable, while the scalar ollowing the name speciies the level o applied limiting. Due to their low numerical diusion characteristics, limitedlinear schemes were utilized with the high resolution Fine grid, while the more robust Gamma 11 schemes were employed with the Coarse models. Despite having numerical accuracy as a high priority, the eect o the dierent discretization methods on the numerical results were not thoroughly investigated in this study. Experience has shown that on high quality grids most higher order discretization schemes perorm well and the need or associated limiters is reduced. Grid, Fine, Coarse, Coarse Long, Coarse, Turbulence Model k ω SST k ω SST k ω SST k ε Time-Step Size θ = Ω t θ = 0.5 o θ = 0.5 o θ = 0.5 o θ = 0.5 o Time Derivative backward Euler Euler Euler Schemes: 2 nd order 1 st order 1 st order 1 st order Convection U: llv 0.35 U: GammaV 0.5 U: GammaV 0.5 U: GammaV 0.5 Schemes: k, ω: ll 1. k, ω: Gamma 1. k, ω: Gamma 1. k, ε: Gamma 1. Relaxation α u =0.75 α u =0.95 α u =0.95 α u =0.95 Factors: α p =0.3 α p =0.1 α p =0.1 α p =0.1 α k,ω =0.5 α k,ω =0.65 α k,ω =0.65 α k,ε =0.65 SIMPLE Loop niter: niter = 8 niter = 6 niter = 6 niter = 6 Table 2: A case-speciic listing o parameters and schemes used in the CFD analysis. Note the abbreviation: ll(v)=limitedlinear(v). The transient simulations considered in this report were carried out exploiting techniques that aim to minimize the total CPU time required to complete the analysis. At irst, using the Coarse model, an initial low ield was solved using a quasi-steady (or rozen-rotor) method or the purpose o providing a starting point or the time-accurate simulation. Unortunately, due to the non-peridic geometry o the single-channel pump, the quasi-steady results misrepresent the true nature o the low system to such degree that in the transient analysis up to nine complete revolutions were required to convect the nonsense out o the system and reach a recurring periodic behavior. Since this lengthy evolution o the low ield was not o principal interest, it was beneicial to use a larger time step to advance the solution eiciently until higher accuracy analysis became easible. In the build-up phase o the Coarse model, the time steps used corresponded up to θ =3 o and were employed together with modiied solution parameters (relaxation actors, inner iterations, etc.) and discretization schemes. Once the low ield had 9

evolved suiciently, the temporally and spatially accurate settigs were activated and the computation was continued (at least our complete revolutions) to ensure two successive revolutions demonstrated identical perormance behavior. Substantial time was saved by taking advantage o a utility called mapfields, which enabled the low solution to be copied rom the Coarse grid to the Fine grid. This provided an excellent jump-start or the computationally intensive case. The same utility was used to initiate the Coarse Long simulation, although the mapping could not inluence the added inlet duct. Nonetheless, the reduction in computational time was notable. 2.3.1 On Relaxation and Convergence When solving the time-accurate Navier-Stokes equations (2), the matrix equation s diagonal dominance is principally due to the time derivate term (V/ t), which becomes insuicient or numerical stability on larger time steps. For this reason, the diagonal dominance is urther increased by applying relaxation to the matrix equation, as shown in Eqn.(14), requiring that the solution must be iteratively solved until convergence within every time step. However, as is well known about iterative pressure-correction schemes, such as SIMPLE, that are mainly applied to steady-state computations, the role o relaxation is not clearly deined due to application- and numerical scheme-speciic dependencies. Yet, the iterative nature o the solution method gives way to permissive standards which simpliy the practical aspects considerably. The picture is slightly more complex with URANS simulations since the role o the physical time step length and grid density are added into the mix. In this study the physical time step length was determined according to the accuracy requirement set by the hydrodynamic system o the pump, and the relaxation actors or the solution variables were set to ensure smooth numerical behavior within each time step. The variation between relaxation actors or the Fine and the Coarse models lays bare the eect o grid resolution with a given time step: The physical time step provides nearly suicient diagonal dominance or the matrix equation on the Coarse grid, while signiicant relaxation is needed or the Fine case in order to ensure robust behavior. Even though the transient SIMPLE algorithm is not sensitive to any Courant number CF L criteria, it is meaningul to look at the mean Courant numbers o the two cases as they clearly relect the underlying numerical dierence. Fine: CFL mean = 0.0377 Coarse: CFL mean = 0.0266 as the Courant number in OpenFOAM is computed per cell ace: CFL = φ t S x cc where x cc is the distance between the adjacent cell centers. The maximum Courant 10

numbers were practically equal or the two models because o nearly identical grid reinement close to the walls. While smooth numerical behavior was achieved or a wide range o time steps and mean Courant numbers the time-accurate evolution o the pressure ield began to demonstrate irregular behavior as the physical time step was reduced such that the mean Courant number reached the reported Coarse range. As will be seen in the pressure results in Section 3, the Coarse model generates small high-requency oscillations which are visible in the hydrodynamic head graphs. These small-scale oscillations become subdued as the mean Courant number is increased, as is the case with the Fine model or when the time step is increased with the Coarse grid. Through numerical testing it was established that, within a time step, a three orders o magnitude reduction in residuals served as a suicient convergence criterion or the velocity and turbulence variables, while the coincident convergence o one order o magnitude (or even less) in pressure was ound to be adequate. Stricter criteria did not have any detectable eects on the solutions. 2.4 Perormance Analysis and Post-Processing The assessment o the hydrodynamic perormance o the pump is based on control volume analysis o the First Law o Thermodynamics, which, when applied to a pump with adiabatic walls and no heat source, yields the ollowing relation: Ẇ s = ĖT + ṁ (h TOUT h TIN ) (15) where Ẇs denotes the rate o shat work done on the system (i.e. shat power), Ė T is the time rate o change o total energy within the system, ṁ is the mass low rate through the pump and h T stands or the total speciic enthalpy. Utilizing notation to indicate dierences between the values at the outlet and inlet, and the deinition o total pressure or incompressible low (p T = p + 1/2ρU U), the energy balance can be written Ẇ s = ĖT + ṁ [ e + (p T /ρ)] (16) as e represents the change o internal energy across the system. Recognizing that the only mechanism contributing to the change in internal energy across the system is dissipation due to viscous stresses (i.e. Φ = ṁ e) and Ẇs = T Ω, where T is the torque on the impeller, Eqn. (16) can be written in a simple, but inormative orm: T Ω = ĖT + E M + Φ (17) where E M = ṁ (p T /ρ) is the rate o mechanical energy change across the system, or power output. The terms in the equation represent dierent energy budgets that allow the hydrodynamic perormance o the pump to be evaluated. In pump analysis this is typically aided by two additional measures which are derived rom (17), namely total hydrodynamic head, H, and eiciency, η: 11

H = E M ṁg η = E M T Ω Here it should be noted that in time-accurate simulations the role o Ė T on the righthand-side o Eqn.(17) complicates the continuous monitoring o perormance because the balance between the budgets is time dependent; the mechanisms that transer energy rom one budget to another do not operate synchronously. However, this does not aect the time-averaged measures, taken over a complete revolution, because the low physics require that ĖT avg = 0. To accomplish the time-accurate monitoring o system perormance in a lexible and convenient manner, OpenFOAM s built-in machinery or unction objects was harnessed to develop a speciic analysis tool or turbomachinery. With this unction object the transient perormance data could be gathered irrespective o the low solver used or the analysis and across any set o user-deined boundary patches. The same boundary patches were used with all the models to extract the data, including the Coarse Long model where the GGI patch, at the interace between the added duct and the original model, unctioned as a monitoring inlet. To construct a velocity proile comparison between CFD results and LDV measurements, a utility called sample in the OpenFOAM library was used to extract velocity values along speciied lines within the domain or every our degrees. A number o low visualization animations were created or the same saved solution states (90 in total), which were then joined together with the velocity comparison animations to yield a highly inormative depiction o the transient behavior o the system. All the data handling and plotting needed or assembling the perormance results and animations were accomplished with simple Python scripts. 3 RESULTS AND DISCUSSION The transient simulations conducted in this study generated a vast volume o data, which, ater proper post-processing, yielded a tremendous amount o inormation on both the hydrodynamic perormance and the behavior o the pump. While the extraction o the perormance data is straight-orward and can be done on-the-ly as the computation progresses, acquiring knowledge about how the geometric eatures o the design aect the low behavior requires a considerably more arduous process and demands appropriate visualization and post-processing tools. This section is arranged so that the results concerning the perormance analysis are irst presented and then ollowed by an account o the velocity proile comparisons and low visualizations, which bring urther insight to the complex low behavior o the pump. http://www.python.org/ (18) (19) 12

3.1 Perormance Comparison The time-accurate behavior o the perormance measures, shown in Figure 5, exhibit distinctly the oscillatory nature o the low system and demonstrate the dierences between the numerical modeling choices. From the hydrodynamic head and impeller orce plots it becomes apparent that both Fine and Coarse grids, regardless o the chosen turbulence model, generate nearly identical pressure solutions, providing only marginally dierent time-averaged head values, as shown in Table 3. The small-scale pressure luctuations, which are hardly visible in the Fine graphs but notable in the Coarse results, are solely due to numerical issues, as discussed in section 2.3.1. One should note, however, that these luctuations show no eect on the impeller orces and very little eect on the shat power, indicating that the luctuations occur on a global scale and thereby have only a small inluence on the solution o the momentum equation. Figure 5: Comparison o computational perormance behaviors: Head (top let), T Ω (top right), η (bottom let) and F x (bottom right). The dierences in modeling choices are most notable in the shat power and eiciency 13

Total Head Shat Power Eiciency Grid, Turbulence Model H avg (m) TΩ avg (W) η avg (%) Fine, k ω SST 9.60 2985 87.8 Coarse, k ω SST 9.57 3037 85.9 Coarse Long, k ω SST 9.62 3041 86.3 Coarse, k ε 9.55 3137 83.0 Experiment 9.2 3250 78.0 Table 3: Comparison o computational, time-averaged (1 rev.) perormance results. Experimental values are included or reerence. results. The standard k ε model, by virtue o its more diuse nature, predicts 7% higher viscous torque on the impeller than the Coarse k ω SST. On the other hand, all the k ω SST simulations show surprisingly good mutual agreement regardless o the marked dierence in grid resolutions. This evidently maniests the positive inluence o having a high grid quality on a comparatively coarse mesh. A comparison o the Coarse and Coarse Long results leads to a welcomed conclusion concerning the inlet boundary condition treatment: The inluence o having a developed pipe low proile at the inlet, instead o ixed values or U, k, ω and ε, is insigniicant or perormance analysis. Thus, the usage o the conventional model is well justiied or design purposes. The low simulations demonstrated a relatively strict temporal accuracy requirement and a signiicant sensitivity to increasing the time step length. The eect o temporal accuracy can readily outweigh the eect o grid resolution or boundary condition treatment. For instance, increasing the time step to correspond to θ = Ω t = 1 o, the Coarse k ω SST perormance results change as ollow: H avg = 9.45m (1.3%), TΩ avg = 3080W (1.4%), η avg = 83.6% ( η =2.3%), where the percentage dierence is shown in parenthesis. Thus, i comparative studies between slightly changed geometries are conducted, it is important to employ Figure 6: Illustration o the changes in solution behavior as t is increased. The change occurs at t = 0.30. the same computational settings and time steps. Otherwise the changes due to design alterations may be blurred by the dierences in the numerical treatment. Since the numerical behavior o the pressure solution also changes with an increasing time step, Figure 6 is added to exempliy this phenomenon. The juxtaposition o computational and experimental perormance measures in Table 3 does not provide an apparently meaningul comparison since the complexity o the computational models has been reduced by neglecting speciic details (see section 2.2) which complicate the CFD analysis. These neglected wet areas and leakages increase the 14

shat power requirement and have a moderate adverse eect on the hydrodynamic head, which accounts or the act that the CFD analysis ends up consistently over-predicting the perormance. Since the eects o these extra loss mechanisms remain both predictable and uniorm or a particular pump coniguration, the utility o the experimental measurements remains essential. 3.2 Flow Behavior and Velocity Proile Comparison While the perormance data is crucial in evaluating the quality o the design, it does not provide any insight into the low behavior o the system. In order to improve the design rules utilized in generating new pump geometries, the dependencies between characteristic low phenomena and geometric attributes must be properly identiied and understood, which, in turn, requires access to detailed inormation about the low system. Experimental means oer a crucial, yet limited and expensive, contribution in the development, but when combined with high accuracy CFD analysis, the capacity to extract meaningul inormation on low system dependencies increases dramatically. Thereore, as the utility o CFD analysis has been well established or perormance predictions, greater emphasis has now been placed on gaining a more detailed description o the low structures developing within the pump. This naturally requires a higher grid resolution, an appropriate turbulence model and a set o discretization schemes that do not suer rom considerable numerical diusion. Despite these computationally demanding requirements, the scope o this analysis is deliberately limited to such URANS simulations that remain both computationally and practically easible. TheFine case has been prepared with the objec- tive that, while the size o the computational grid remains moderate by current standards, the resolution o the simulated low behavior is considerably increased compared to the Coarse model. In order to gain greater conidence in the transient CFD results produced by OpenFOAM, a set o timeaccurate LDV velocity proile measurements were prepared or this study. Two sets o radial and tangential proiles, labeled A and B, were taken at two locations in the pump s volute, shown in Figure 7, such that the second set o measurements were taken at a slightly dierent radial locations. Figure 7: Time-accurate LDV velocity proile measurements were taken at the shown locations, W1 and W3, or comparison. A tick marker ixed to the trailing edge o the blade was used to set θ = 0 where the trailing edge aligns with W1. The animated velocity proile comparisons bring orth a captivating demonstration o the time-dependent behavior o the low system and reveal the striking agreement on the luctuating nature o the tangential and radial velocity ields, which unortunately cannot be properly conveyed in this report. Although Figures 8 and 9 can only provide a glimpse 15

o the time-accurate comparisons, they warrant support or the ollowing deductions: All the models succeed in predicting the tangential velocity behavior with good accuracy. The measured radial proiles indicate that the pump generates a large swirling structure in the volute such that the low bordering Z max is outward while at Z min it is inward. This undesired low behavior is correctly captured by CFD. In the radial velocity proiles, particularly at W1 shown in Figure 8, there is a consistent discrepancy within the range Z = 0.01 0.03 throughout the revolution o the impeller. It is suspected that this is due to leakage between the impeller and the volute, which is not included in the computational model. The dierences between the Coarse and Coarse Long results are insigniicant. The special treatment o the inlet boundary condition does not pay o in this respect either. Results generated by the Fine model demonstrate greater detail in the secondary low structures, which are evident rom the shape and behavior o the radial proiles. Although the Coarse k ε solution captures the main low characteristics, compared to the Coarse k ω SST cases, the results exhibit notably higher numerical damping. As the animated time-accurate velocity proile comparisons are adjoined with a broader visualization o the low ield, as shown in Figure 10, a remarkably descriptive illustration o the low behavior can be achieved. This is particularly striking with the Fine simulation, which lays out the evolution o a wide range o low structures distinctly. Thereore, with the aid o well-prepared transient visualizations, dierent aspects o the oscillating low system can be closely analyzed. For instance, the complex impeller-volute interaction o the pump emerged with an arresting resolution rom the Fine analysis. Through proper visualizations, detailed inormation about this potentially detrimental transient phenomenon could be extracted. The strong interaction between the impeller and volute is exempliied in Figure 11. In conclusion, the presented work, which applies an open-source CFD tool to a complex low problem, leads to a tangible ramiication: The study demonstrates the means to meet the ever-growing demand to conduct high-accuracy transient turbomachinery analysis routinely as a part o the design practice. As a corollary, even urther cost eiciency can be obtained in design as increasingly detailed numerical testing ultimately reduces the need to construct expensive prototypes. 16

Figure 8: Snapshots rom an animation depicting a time-accurate comparison o the radial (U x ) and tangential (U y ) velocity proiles obtained at measurement point W1. Abbreviations: F=Fine k ω, C-L=Coarse Long k ω, C=Coarse k ω, C-Ke=Coarse k ε Figure 9: Snapshots rom an animation depicting a time-accurate comparison o the radial (U y ) and tangential (U x ) velocity proiles obtained at measurement point W3. 4 CONCLUSIONS - With the newly added utility o the GGI, detailed transient analysis o turbomachinery applications has become easible with OpenFOAM. - A transient solver which implements the SIMPLE pressure correction algorithm is shown to acilitate an eicient and robust transient analysis due to the ability to utilize a wide range o time step lengths. This is essential in the single-channel pump analysis because the low solution goes through a lengthy development period beore settling into 17

Figure 10: A combined illustration o the low ield across the y plane. The location o the line along which W1 measurements were taken is marked by arrows. Figure 11: Visualizations o the pressure (top) and velocity (bottom) ields at dierent impeller angles. a recurring periodic pattern. - With high quality computational meshes, grid resolution does not inluence the perormance predictions notably. However, in order to resolve the characteristic low structures o the pump, grid resolution becomes critical. - The analysis indicates that it is suicient to impose ixed value boundary conditions or velocity and turbulence variables at the inlet o the pump. - The pressure solution remains practically unaltered as dierent turbulence models (k ω SST vs. k ε) are applied. On the other hand, clear dierences maniest in the shat power requirement values (T Ω) and in time-accurate velocity proile comparisons. The standard k ε results clearly suer rom excessive diusion. - As the time step is reduced and the mean Courant number CFL o the computation 18

becomes suiciently small, the pressure ield begins to demonstrate small luctuations. However, this numerical behavior does not have a notable eect on the perormance or the velocity ield behavior. - The time-accurate LDV velocity proile comparisons together with detailed visualizations attest that the high grid resolution model succeeds in resolving the low system with striking detail. Although some details o the pump coniguration have been omitted in the CFD model, the complex luctuating behavior o the low can be successully captured. REFERENCES [1] M. Auvinen, J. Ala-Juusela, L. Ilves, T. Siikonen, Dissecting a Complex System; A Computational Study o Flow Behavior in a Single-Blade Pump, 5th International Conerence on Heat Transer, Fluid Mechanics and Thermodynamics, HEFAT 2007, Sun City, South Arica, July 1-4, (2007) [2] H.G. Weller, G. Tabor, H. Jasak, and C. Fureby, A Tensorial Approach to Computational Continuum Mechanics Using Object Orientated Techniques, Computers in Physics, 12(6), pp. 620-631, (1998) [3] M. Beaudoin, H. Jasak, Development o a Generalized Grid Interace or Turbomachinery simulations with OpenFOAM, Open Source CFD International Conerence 2008, Berlin, Germany, December 4-5, (2008) [4] P. Petit, M. Page, M. Beaudoin, H. Nilsson, The ERCOFTAC centriugal pump OpenFOAM case-study, 3rd IAHR International Meeting o the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, October 14-16, Brno, Czech Republic, (2009) [5] P.D. Thomas, C.K. Lombard, Geometric Conservation Law and Its Application to Flow Computations on Moving Grids, AIAA Journal, 17, pp. 1030-1037, (1979) [6] H. Jasak and Z. Tukovic, Automatic Mesh Motion or the Unstructured Finite Volume Method, Transactions o FAMENA, 30(2), pp. 1-18, (2007) [7] R.I. Issa, Solution o the Implicitly Discretised Fluid Flow Equations by Operator- Splitting, Journal o Computational Physics, 62(11), pp. 40-65, (1986) [8] S.V. Patankar, Numerical Heat Transer and Fluid Flow, Hemisphere Publishing Corporation (1980) [9] H. Jasak, Error Analysis and Estimation or Finite Volume Method with Applications to Fluid Flows, Ph.D. Thesis, Imperial College, University o London, (1996) 19

[10] GridPro, Program Development Company, White Plains, NY 10601, USA (www.gridpro.com) [11] H. Jasak, H. G. Weller, A. D. Gosman, High Resolution NVD Dierencing Scheme or Arbitrarily Unstructured Meshes, Int. J. Numer. Meth. Fluids, 31, pp. 431-449, (1999) 20