World Academy of Scence, Engneerng and Technology SPH Method used for Flow Predctons at a Turgo Impulse Turbne: Comparson wth Fluent Phoevos K. Koukouvns, John S. Anagnostopoulos, Dmtrs E. Papantons Abstract Ths work s an attempt to use the standard Smoothed Partcle Hydrodynamcs methodology for the smulaton of the complex unsteady, free-surface flow n a rotatng Turgo mpulse water turbne. A comparson of two dfferent geometres was conducted. The SPH method due to ts mesh-less nature s capable of capturng the flow features appearng n the turbne, wthout dffuson at the water/ar nterface. Furthermore results are compared wth a commercal CFD package (Fluent ) and the SPH algorthm proves to be capable of provdng smlar results, n much less tme than the mesh based CFD program. A parametrc study was also performed regardng the turbne nlet angle. Keywords Smoothed Partcle Hydrodynamcs, Mesh-less methods, Impulse turbnes, Turgo turbne. I. INTRODUCTION MPULSE water turbnes operaton s based on the I nteracton of a hgh velocty water et wth the rotatng turbne runner. The turbne runner changes the drecton of the flow and torque develops on ts blades due to the change of water et s angular momentum. The statc pressure does not change before and after the turbne runner. Ths enables the operaton of the turbne n atmospherc envronment, wthout the need of a sealed casng. The water et s formed at the turbne nozzle, where the water hydraulc head s transformed nto knetc energy. An mpulse turbne may have more than one nozzles, at the turbne casng around the turbne, drectng the et tangentally at the runner blades (Turgo turbne), or buckets (Pelton turbne). The nozzles are equpped wth a needle, whch s used to adust the water flow rate through the nozzle to match the requested power at the generator coupled to the water turbne. Apart from the needle the nozzles often have a deflector, used to deflect the water et from the runner, n case of an emergency to quckly reect load from the turbne. The most well known mpulse turbnes are the Pelton and Turgo turbnes. Turgo turbnes are desgned for medum head applcatons. They have a flat effcency curve and provde excellent part load effcency, thus they can be used as an alternatve of other turbne types, especally f there are large flow rate varatons. The smplest Turgo turbne runner looks lke a Pelton runner splt n half at the plane of symmetry. The water et enters from the one sde of the runner and exts from the other (Fg. ). Because of that, the escape of the water does not nterfere wth the ncomng et, or the other turbne blades, enablng the turbne to handle larger flow rates and et dameters than a Pelton runner of the same runner dameter. As a result the Turgo turbne has hgher specfc P. K. Koukouvns, J. S. Anagnostopoulos and D.E. Papantons are wth the Natonal Techncal Unversty of Athens, 5780, Heroon Polytechneou 9 (e-mals: fvoskouk@gmal.com,.anagno@flud.mech.ntua.gr, papan@flud.mech.ntua.gr, respectvely). speed and smaller sze than a Pelton turbne of the same power. The smaller dameter allows the operaton at hgher angular veloctes, whch n turn, makes the couplng between the turbne and the generator easer, avodng the use of a mechancal transmsson system decreasng costs and ncreasng the mechancal relablty of the system []. Fg. Sketch of the flow n a Turgo turbne (up) and Turgo runner (down) In mpulse water turbnes the developng flow of the mpngng et on the runner s unsteady, free-surface wth movng boundares, due to the runner rotaton. Smulaton wth Euleran methods s dffcult [2], snce specal treatments are requred for capturng the underlyng phenomena. To be more specfc, the treatments requred are the Volume Of Flud (VOF) method, combned wth mesh refnement, for trackng the free-surface, and sldng meshes, for the connecton between the movng and statonary meshes. The above treatments ncrease the computatonal cost and requrements of the smulaton consderably. An alternatve way of smulatng the flow would be by adoptng a Lagrangan pont of vew, usng and trackng partcles whch represent the water et and nteract wth the turbne runner. Several attempts of usng Lagrangan framework exst n lterature, wth Lagrangan partcle trackng [3] or wth a Movng Partcle Sem-Implct method []. A relatvely new and promsng method s the Smoothed Partcle Hydrodynamcs (SPH) method whch wll be used n the present work for the smulatons. The SPH method was ntally developed by Lucy, Gngold and Monaghan (977) and has been used for modelng 3
World Academy of Scence, Engneerng and Technology astrophyscal problems n three dmensonal open space [5]. Today SPH s extended beyond ts ntal purpose of astrophyscal phenomena, for modellng the behavour of solds and fluds. The applcaton of SPH to a wde range of scentfc areas has led to sgnfcant extensons and mprovements of the orgnal method [6]. SPH s a Lagrangan, partcle, mesh-less method and has the advantages of tracng and resolvng the free-surface wthout any specal treatment [7] and descrbng movng/deformng boundares easly. II. STANDARD SPH FORMALISM The SPH formalsm reles on the use of kernel approxmaton of feld functons for the calculaton of the operators appearng n the dscretzaton of the flow equatons, nstead of usng a computatonal grd. In ths way t s able to approxmate dervatves or functons from unconnected and randomly scattered computaton ponts. The bass of the SPH approxmatons orgnates from the followng dentty: f ( x) = f(x')δ ( x-x' ) dx' () where (x) Ω f s a functon of a three dmensonal poston δ x-x' s the Drac delta functon and Ω s the vector x, ( ) volume of the ntegral that contans x. The above relaton can be approxmated usng a smoothng kernel functon W x-x',h : ( ) Ω f ( x) = f(x')w( x x', dx' (2) A smlar equaton can be derved for the gradent of a functon: f ( x) = f(x') W ( x x', dx' (3) Ω In order the above approxmatons to be vald, the kernel functon W ( x x', has to fulfl certan requrements, such as: Unty or normalzaton condton : W ( x x', dx = Drac functon property : lmw ( x x', = δ ( x x' ) h 0 Compact condton : W ( x x', = 0, for x x' > k h, where k.h s the kernel s support doman Also the kernel functon has to be even, postve and monotoncally decreasng functon. There are many types of kernel functons. In the present work the fourth order kernel s used [8] for reasons whch wll be explaned later. In the kernel formulaton (eq. ) q= r /h, wth r the dstance between two computatonal ponts and h a characterstc smoothng length. Subscrpt d denotes the dmenson of the problem and the constant term α s for 20π 3-D cases, n order to fulfl the normalzaton condton. Ω W ( q) 5 3 q 5 q + 0 q 0 q 0.5 2 2 2 a 5 3 d = q 5 q 0.5 q.5 d 2 2 () h 5 q.5 q 2.5 2 0 q 2.5 In the SPH method the entre system s represented wth a fnte number of partcles that carry ndvdual mass, occupy ndvdual space and the characterstc quanttes of the flow (e.g. velocty, densty, pressure etc.). Thus the contnuous ntegral relatons can be wrtten n the followng form of dscretzed partcle approxmaton: f(x ) f(x ) = = N m f(x)w ρ m f(x) W ρ W ( x x, h = N = In the above equatons W = ), m s the partcle s mass and ρ s the partcle s densty. Apart from the above relaton for the dervatve, there are the followng alternatve formulatons [6], whch tend to gve better approxmatons than equaton 6 [9]: N f(x ) = m [ f(x ) f(x )] W, (7) ρ = N f(x ) f(x ) f(x ) = ρ m + W (8) 2 2 = ρ ρ Consderng the above procedure for the dervaton of the SPH flow equatons, one can notce two approxmatons. The frst approxmaton has to do wth the approxmaton of the Drac delta wth the kernel functon W (equatons, 2). As t was proved by Monaghan ths nterpolaton s of, at least, second order of accuracy [0], due to the requrement of the kernel functon beng even. The second approxmaton has to do wth the summaton representaton of the ntegral (equatons 5, 6, 7, 8). It s proved [0] that, provded that the kernel functon s smooth enough and partcles are equspaced, the error of the summaton approxmaton s nearly neglgble. Partcle dsorder and kernel nconsstency tends to degrade the accuracy of the method. On the other hand, smooth kernel functons gve better approxmatons and are less senstve to partcle dsorder [6, 0]. Ths s the man reason for usng the specfc kernel (equaton ), snce hgher order functons are smoother than lower order ones. Choce of the smoothng length h s also crucal for the SPH method, snce t drectly affects the support doman of the kernel approxmaton for each partcle and the accuracy of the approxmatons. A small smoothng length would result n low accuracy due to very few partcles n the support doman. On the other hand a large smoothng length would smooth out local propertes. We have found that a smoothng length of (5) (6)
World Academy of Scence, Engneerng and Technology 3 dx s able to gve satsfactory results for 3D smulatons (dx s the nter-partcle dstance, or the partcle dscretzaton). Usng the above approxmatons for a functon and the dervatve of a functon, one can derve the SPH flow equatons [8]: Contnuty equaton: d ρ = m u W (9) dt Momentum equaton: du p p = m + W Π + g (0) dt 2 2 ρ ρ where Π, s a vscosty term suggested by Morrs []. Ths term wll be omtted n the rest smulatons snce they wll be consdered nvscd. An obvous consequence of the above formulatons s that SPH approxmatons are symmetrc. Ths means that the contrbuton of partcle to n the momentum equaton s equal and opposte to the contrbuton of partcle to. Ths enables SPH to conserve both lnear and angular momentum of the system of partcles under consderaton. III. IMPLEMENTATION DETAILS Pressure s calculated from an equaton of state, thus the method s weakly compressble. The Tat equaton of state s commonly used for modellng ncompressble flows. 2 γ ρ 0c0 ρ a = () p a γ ρ 0 In the above equaton ρ 0 s the reference densty and c 0 s an artfcal speed of sound, snce the real speed of sound would requre a very small tmestep. In order to keep densty varatons less than %, the value of c 0 s chosen ~0 Vmax, accordng to Monaghan [0]. SPH s known to produce unphyscal pressure oscllatons due to the stffness of the equaton of state [2]. In order to reduce the large pressure oscllatons at the pressure feld of partcles, a densty rentalzaton technque [2] s adopted, and for movng the partcles, the XSPH varant [6] s used, n whch the veloctes of the nearby partcles are taken nto account: dra mb = u a ε u abw (2) ab dt b ρb where ε s a parameter set at 0.3 for ncompressble flows. Snce n the SPH method there s no connectvty between ndvdual calculaton ponts, a statc matrx neghbor lst [3] s used to fnd neghborng partcles. For the tme ntegraton of the equatons the fourth order Runge Kutta method s used. Also to speed up calculatons and to utlze properly mult-core hardware, OpenMP s used for parallelzaton. Due to the nature of SPH, calculaton of contnuty and momentum terms can be done ndependently for each partcle. Thus SPH algorthm s easly parallelzed, achevng great speed up. IV. BOUNDARY CONDITIONS There are varous ways to defne sold boundares n SPH, such as fcttous partcles, ghost partcles or boundary partcles ([6], [8], [0]). In the present work, the method of boundary forces wll be used [], [5]. The boundary s descrbed wth one layer of partcles whch exerts Lennard- Jones forces when flud partcles come close enough to the boundary. 2 6 r r 0 r r 0 2 f D (3) = [ m / s ] 2 r r r for r0 r In the above formula r 0 represents the range of nfluence of the Lennard-Jones forces and s consdered to be equal to the flud partcle spacng dx. Also r represents the nter-partcle dstance and D s a problem dependent parameter, usually set n the same scale as the maxmum velocty n the smulaton [6]. Apart from the boundary force, the boundary partcles do not nteract wth the flud partcles n any other way. The bucket geometry s represented usng layers of force partcles that exert the boundary forces prescrbed by the above formula. The boundary partcles on the bucket are postoned at a closer nter-partcle dstance than the flud partcles n order to ensure no penetraton of the boundary and also to make the boundary force as smooth as possble. Ths s due to the overlappng nfluence areas of the boundary partcles whch, for fne boundary resoluton, approxmate the actual boundary surface. The force that the flud exerts on the sold boundary wll be calculated from the reacton of the boundary forces on the wall. Snce the forces durng partcle nteractons are symmetrc, the total force on a wall can be calculated by summng the reacton force on all wall partcles. Furthermore, torque on each boundary partcle can be calculated from the r r r defnton: T = F. V. GEOMETRY DESCRIPTION The Turgo runner geometry was created usng specalzed software (Tools for Turbomachnery TT [6]), whch uses as nput the blade angles at each edge of the blade and the dstance between the edges. The geometry generator forms a quadrlateral surface grd of ponts whch represents the surface of the blade, usng NURBS (fg. 2). Each quadrlateral element s flled up wth boundary partcles wth nter-partcle dstance equal to dx/3 (where dx s the nter-partcle dstance for flud partcles). Apart from fllng the grd wth partcles, the algorthm extrudes the surface grd n the normal drecton, at a prescrbed depth, n order to form a full 3D body, f requred. 5
World Academy of Scence, Engneerng and Technology computatonal mesh used n the Fluent program had a dscretzaton sze of 7.5mm, but refned to 5mm n areas of nterest, such as the wall where pressure gradents appear, or the nterface, consstng of ~500000 elements totally. Also the mesh conssted of two dfferent areas (one rotatng and one statc), connected wth an nterface regon. The VOF method was used to track the nterfaces and second order dscretzaton was used, to lmt numercal vscosty. Fg. 2 Two ndcatve Turgo blade geometres created wth the software TT Fg. 3 Turgo runner geometres for the above blade geometres (vewed from the outlet) VI. CASE SET- UP At frst two dfferent geometres were tested and compared wth the results of Fluent n order to determne the performance and the accuracy of the SPH method. The two geometres modeled are those presented n fg. 2. The two geometres dffer both at the nlet and the edges and, consequently, at the respectve angles. In order to keep smulaton cost at a mnmum, only the space between two turbne blades was modeled, assumng perodc flow condtons. Also, snce the geometry used n Fluent had no wall thckness, the blade was modeled wth only a wall layer wth the SPH method. The runner hub was at 260mm radus and the tp at 500mm radus from the center of rotaton. The nomnal dameter of the runner was at 770mm. Also the runner conssted of 22 blades. The water et had a dameter of 5mm and velocty 32m/s (dscharge 596m 3 /s) and entered the runner at 25 degrees from the runner rotaton plane. The rotatonal speed of the runner was set to.8rad/s. Intally the two blades were postoned at -32 from the Y-axs (fg. 5), ust before the mpact of the water et on the runner. The smulaton duraton was 2ms, n whch the runner performs a rotaton of 07, enough to cover the whole nteracton between the et and the runner. The partcle sze used n SPH was 5mm, whch proved fne enough after a partcle dependence study [5]. The numercal speed of sound was set to 350m/s, larger than 0.V et n order to ensure the ncompressblty of the smulated water. The Fg. The ntal set up for the smulated cases usng SPH and Fluent In the followng fgures ndcatve results of the smulatons are shown. Fgures 5 and 6 show a general vew of the free surface flow developng between the blades at a tme nstance at the mddle of the water et and turbne blade nteracton. By comparng the respectve fgures one can notce some dfferences, especally at the area of et mpngement, but generally both the velocty dstrbuton and the free surface shapes are smlar. As t s expected the velocty at the turbne runner outlet s much lower comparng to the water et velocty, due to the energy transfer of knetc energy from the et to the turbne runner. Another mportant remark, regardng the behavor of the flow, s that a porton of the water et leaks out from the nlet edge of the turbne blade (fg. 5, 6). Ths detal could not be effectvely captured wth Fluent, snce the mesh dd not extend beyond the blade (fg. ). 6
World Academy of Scence, Engneerng and Technology Fg. 5 Smulaton results for geometry (a) usng SPH (left) and Fluent (rght), at tme 2ms. SPH: Flud partcles are coloured accordng to velocty magntude, wall partcles are translucent to allow vsualzaton of the flow between blades. Fluent: Isosurface s colored accordng to velocty magntude. Fg. 6 Smulaton results for geometry (b) usng SPH (left) and Fluent (rght), at tme 2ms. SPH: Flud partcles are coloured accordng to velocty magntude, wall partcles are translucent to allow vsualzaton of the flow between blades. Fluent: Isosurface s colored accordng to velocty magntude. In fgures 7 and 8 the area of the blade, for geometry (b), covered by water s shown at two dfferent tme nstances. Agan results by both programs are very close. Smlar results are obtaned for geometry (a) too. The sharpness of the SPH results s attrbuted to the coarseness of the nterpolaton grd used to obtan the respectve data, note though that ths mesh was not nvolved n the flow feld calculatons. Fg. 7 Free surface evoluton on the wall usng SPH and Fluent, at tme 2.6ms. Water s represented wth blue, ar wth red 7
World Academy of Scence, Engneerng and Technology Fnally n fg. 0 the average torque dstrbuton on the turbne blade s shown for the two geometres tested. In both cases maxmum torque develops n curved areas where the drecton of the flow changes. Geometry (b) has a smoother torque dstrbuton at a larger area due to ts desgn. Fg. 8 Free surface evoluton on the wall usng SPH and Fluent, at tme 2ms. Water s represented wth blue, ar wth red In fgure 9 the developng torque (n respect to the angle of the blade from the Y-axs) on the turbne blade s shown for the two geometres tested wth both programs. The torque calculated usng SPH exhbts some oscllatons, but the general trends are smlar. For each geometry, the work of the torque, calculated by SPH and Fluent, on the blade s approxmately the same (~0.5% dfference). Also by comparng the work of the torque for the two geometres, t was found that blade (b) performs better, approxmately by ~%. Torque on geometry (a) develops earler due to the shape of the nlet edge; the nlet edge n ths case s more curved and cuts the water et earler. Fg. 9 Torque developed on the blade surface for the two geometres Fg. 0 Average torque dstrbuton through the water et bucket nteracton Consderng the tme needed for the executon of the two programs, the SPH method s much faster than fluent, due to the embarrassngly parallel nature of the SPH algorthm. The SPH algorthm only needed 0hrs on a 2xQuad Core Xeon 2.Ghz computer (80 CPUhrs), for each smulaton. On the other hand Fluent needed ~0days usng parallel processes on a 7 2.97Ghz computer (960 CPUhrs). VII. INLET ANGLE DEPENDENCE From the prevous results t has been shown that the geometry (b) performs better than geometry (a). For that reason geometry (b) was used to perform a further desgn test regardng the water et nlet angle. Smulaton condtons wll be the same wth those mentoned at part 6 of the current paper, apart from the et nlet angle. The nlet angles whch were tested are 20, 30, 35 and 0. In fgure a comparson between the torque graphs for the dfferent water et mpngement angles s made; from ths fgure t s shown that, by ncreasng the water et mpact angle, the developng torque curve becomes narrower and exhbts a hgher peak value. Ths s further llustrated n 8
World Academy of Scence, Engneerng and Technology fgure 2, where the average torque dstrbuton s shown; agan for larger mpact angle the peak value s hgher and the peak torque value s moved towards the center of the blade. By calculatng the ntegral of the torque curve t was found that the maxmum effcency s acheved for the ntermedate nlet angle of 30, as shown n fgure n the respectve graph of normalzed effcency (effcency s normalzed by the maxmum value,.e. the effcency of the turbne for 30 nlet angle). A smlar trend has been observed wth our nhouse Fast Lagrangan Smulaton algorthm [], used to estmate the effcency for the same condtons. Fg. Comparatve torque (left) and normalzed effcency (rght) graphs dependng on the water et nlet angle Fg. 2 Comparatve torque dstrbuton on the turbne blade. Red corresponds to hgher torque values, blue to lower VIII. CONCLUSION In ths work the SPH method was used to assess the performance of two dfferent Turgo turbne runners, n comparson to a commercal CFD solver. The SPH method proves to be able to produce smlar results n much less tme than the mesh based program. For ths reason t s an attractve alternatve for parametrc studes or even desgn optmzaton n the complex flows appearng n Turgo and other mpulse turbnes. 9
World Academy of Scence, Engneerng and Technology An mportant advantage of SPH algorthm s that ts nherent structure permts the easy parallelsaton and adapton to run n GPU technology, whch can dramatcally enhance executon speed. Ths work s currently underway n our laboratory. REFERENCES [] J.S. Anagnostopoulos, D.E. Papantons, Flow modelng and runner desgn optmzaton n Turgo water turbnes, Interantonal Journal of Appled Scence, Engneerng and Technology, Vol. Number 3 2007 ISSN 307-38 [2] J.C. Marongu, F. Leboef, E. Parknson (2007), Numercal smulaton of the flow n a Pelton turbne usng the meshless method smoothed partcle hydrodynamcs: a new smple boundary treatment, Journal of Power and Energy, Vol. 22, Part A, p. 89-856. [3] J. S. Anagnostopoulos, D. E. Papantons (2006), A numercal methodology for desgn optmzaton of Pelton turbne runners, Proceedngs of the Hydro2006 conference, Porto Carras, Cyprus, 25-27 September 2006. [] Y. Nakansh, T. Fu, M. Mornaka, K. Wach (2006), Numercal smulaton of the flow n a Pelton bucket by a partcle method, Proceedngs of the 23rd IAHR Symposum, Yokohama, October 2006. [5] Prce D.J., (200), Magnetc Felds n Astrophyscs, PhD thess, Unversty of Cambrdge, UK. [6] G.R. Lu, M. B. Lu (2003), Smoothed partcle hydrodynamcs: a meshfree partcle method, World Scentfc Publshng Company, December 2003. [7] P. Koukouvns, J. Anagnostopoulos, D. Papantons (2009) Flow modellng n the nector of a Pelton turbne, Proceedngs of the th Spherc workshop, Nantes, France. [8] D. Voleau, R. Issa (2006), Numercal modellng of complex turbulent free-surface flows wth the SPH method: an overvew, Internatonal Journal of Numercal Methods for Fluds. [9] F. Coln, R. Egl, F.Y. Ln (2005), Computng a null dvergence velocty feld usng smoothed partcle hydrodynamcs, Journal of Computatonal Physcs. [0] J. Monaghan (2005), Smoothed Partcle Hydrodynamcs, Reports on Progress n Physcs, Volume 68 No.8, p. 703-759. [] J. Morrs, P. Fox, Y. Zhu (997), Modellng low Reynolds number ncompressble flows usng SPH, Journal of Computatonal Physcs, Volume 36, p. 2-226. [2] M. Gomez-Gestera, B. Rogers, R. Dalrymple, A. Crespo, M. Narayanaswamy (200), Users gude for the Sphyscs code v2.0, http://www.sphyscs.org. [3] J. M. Domnguez, A. Crespo (2009), Improvements on SPH neghbour lst, Proceedngs of the th Spherc workshop, Nantes, France. [] J. Monaghan, J. Katar (2009), SPH boundary forces, Proceedngs of the th Spherc workshop, Nantes, France. [5] P. Koukouvns, J. Anagnostopoulos, D. Papantons (200) Flow modellng n a Turgo turbne usng SPH, Proceedngs of the 5th Spherc workshop, Manchester, Unted Kngdom. [6] G. Kon, S. Saraknos, I. Nkolos, (2009) A software tool for parametrc desgn of turbomachnery blades, Advances n Engneerng Software, p.-5, DOI : 0.06/.advengsoft.2008.03.008 20