NUMERICAL ANALYSIS OF CAVITATION INCEPTION IN FRANCIS TURBINE

Transcription

1 Proceedings of the ydraulic Machinery and Systems 1st AR Symposium September 9-1, 00, Lausanne NUMERCAL ANALYSS OF CAVAON NCEPON N FRANCS URBNE Romeo SUSAN-RESGA, Politehnica University of imişoara / Department of ydraulic Machinery, imişoara, Romania Sebastian MUNEAN, Romanian Academy-imişoara Branch / Center of Advanced Research in Engineering Sciences, imişoara, Romania oan ANON, Politehnica University of imişoara / Department of ydraulic Machinery, imişoara, Romania ABSRAC he paper presents a numerical investigation of the cavitation inception for the GAMM Francis turbine. We introduce the turbine cavitation coefficient, σ, as being a parameter specific to the nachine, and we present two methods for computing it. n doing so, we are separating the turbine cavitation analysis from the ant cavitation coefficient σ (homa number). At cavitation inception, we have σ = σ. We investigate both pressure and velocity formulations for σ, and we validate our numerical results by comparing them with available experimental data. We conclude that the velocity formulation for σ is useful for evaluating the turbine cavitational characteristics at (or in the neighborhood of) design operating point. On the other hand, the pressure formulation for σ is in good agreement with experimental data for all operating points under investigation. RÉSUMÉ Cet article présente une recherche numérique sur le commencement de la cavitation dans les turbines Francis GAMM. Nous introduisons le coefficient de cavitation de la turbine, σ Τ, en tant que paramètre spécifique à la machine, et nous présentons deux méthodes pour le calculer. Ainsi nous séparons l analyse de la cavitation dans la turbine, du coefficient de cavitation liée à l imantation : σ PL (nombre de homa). A l apparition de la cavitation, nous avons : σ Τ = σ PL. Nous travaillons sur les formulations de la pression et de la vitesse à σ Τ et nous validons nos résultats numériques en les comparant aux données expérimentales disponibles. Nous concluons que la formulation de la vitesse à σ Τ, est utile pour évaluer les caractéristiques liées à la cavitation de la turbine, et ceci au point (ou au voisinage du point) de fonctionnement. D autre part, la formulation de la pression à σ Τ est en bon accord avec les données expérimentales pour tous les points de fonctionnement étudiés au cours de ce travail. NOMENCLAURE erm Symbol Definition erm Symbol Definition Specific ydraulic E Level z Energy Discharge Q Cross Section Area S Static Pressure p Radius R, r

2 Proceedings of the ydraulic Machinery and Systems 1 st AR Symposium September 9-1, 00, Lausanne erm Symbol Definition erm Symbol Definition Vapour Pressure p va Water Density ρ Ambient Pressure p amb Gravity Acceleration g Suction ead s Minimum value min Net ead Draft ube nlet Section Ref Plant Cavitation pamb pva ± ρgs σ Draft ube Outlet Coefficient Section Reserve Cavitation pmin pva σ rez Coefficient ail Race Level A urbine Cavitation Coefficient σ σ rez σ NRODUCON Among all the aspects of machine operation, cavitation inception and development ays a fundamental role with respect to the erosion risk. Obviously, it is economically perable to have a cavitation free operation as long as the efficiency is unaffected and the erosion is limited. his exains why the cavitation inception problem receives greater attention in the case of hydraulic machines. he standard cavitation tests are important for the evaluation of the setting level of the machine to the tail-water level. Usually, these tests are designed to determine the influence of the homa number on the efficiency, for different operating points. owever, the homa number (also called ant cavitation coefficient ) σ, defined by the EC standards (Ref. 5) as, pamb pva ± ρgs σ, (1) ρg is not a quantity specific to the machine. n order to locate and to evaluate the presence of cavitation inside the turbine, one computes the reserve cavitation coefficient, σ rez, as pmin pva σrez = σ σ () where σ is the turbine cavitation coefficient. Cavitation development occurs in all the zones where the local static pressure is equal or less than the vapour pressure. heore, the rezerve cavitation coefficient allows one to detect the incipient cavitation points as well as the cavitation and supercavitation regimes, as follows p min > p va σ rez > 0 σ < σ without cavitation p min = p va σ rez = 0 σ = σ cavitation inception p min < p va σ rez < 0 σ > σ cavitation p min << p va σ rez << 0 σ >> σ super-cavitation Equation () introduces σ which is a coefficient specific to the turbine. n doing so, we separate the turbine cavitation analysis from the ant cavitation evaluation. One can see that σ = only at cavitation inception, otherwise there are two cometely different quantities. σ he main goal of this paper is to evaluate σ using D numerical simulations for the GAMM Francis turbine. We investigate the σ behavior for variable operating points, and we compare the computed σ values with measured σ at cavitation inception regimes

3 E URBNE CAVAON COEFFCEN Proceedings of the ydraulic Machinery and Systems 1 st AR Symposium September 9-1, 00, Lausanne From (1) and () one obtains the turbine cavitation coefficient, g ( ) ( zr z ) σ = cp min + ηd c. () E he above equation is customized for our numerical simulation of the turbine flow. he computational domain we have considered does not include the draft tube, Muntean et al. (Ref. 6), theore the erence pressure p is considered at the draft tube inlet. Moreover, the whole turbine specific hydraulic energy (including the draft tube), denoted here by E, while the specific energy considered up to the draft tube is. he difference between E and E corresponds to the draft tube hydraulic losses. As a result, the minimum pressure coefficient in () is pmin p cp min =, but for our numerical simulations we evaluate * pmin p E c p min = = cp min E at each operating point. he p value corresponds to the static pressure at the wall, for the draft tube outlet. Consequently, following the approach developed by Kubota et al. (Ref. 4) we have Q S ρ ρ Q p = p + ρg( z z ) ζ D = S S S. (4) = p + ρg ρ Q ( z z ) η D S Kubota et al. (Ref. 4) obtain the draft tube losses coefficient ζ D as function of the discharge coefficient φ using experimental data for all operating points considered in the hill chart, Fig. 1. E Fig. 1 n equation (4), ydraulic loss coefficient ζ D for the GAMM Francis turbine draft tube, Kubota et al. (Ref. 4). z z is the draft tube inlet/outlet level difference, and S. S = for the GAMM Francis turbine. he draft tube efficiency, defined as in equation (4) is:

4 η Proceedings of the ydraulic Machinery and Systems 1 st AR Symposium September 9-1, 00, Lausanne V S D = 1 ζ D = 1 ζ D V S, and the erence velocity coefficient is 1 Q 1 Q c = =. E S E πr With the above considerations we are able to evaluate the turbine cavitation coefficient σ, equation (), once the pressure field in the runner is obtained from a D flow simulation. he results presented in this paper are based on our D inviscid flow computations for the GAMM Francis turbine, Muntean et al. (Ref. 6). Alternatively, the pressure formulation for σ can be reaced by a corresponding velocity formulation. his is useful especially in evaluating the cavitational characteristics when a quasi-d design approach is emoyed, and the velocity field is computed first. Anton defines the turbine cavitation coefficient Anton (Ref. 1) (Ref. ) using velocity coefficients, by using the Bernoulli equation. Fig. Notations for the Francis turbine sections, Anton (Ref. 1). According to Fig., he first uses the Bernoulli equation for relative flow (M ) pm WM UM p W U + + zm = + + z + hpm γ g γ g then he uses the Bernoulli equation for absolute flow ( A) p V pa VA + + z = + + za + hpa γ g γ g he indices correspond to: the runner blade outlet, M point belongs to the threedimensional inter-blade channel of the runner and A to the tailrace section. According to Anton (Ref. 1), the turbine cavitation coefficient σ will have the expression, Wmax W U M U V VA hpm hpa a MD σ = + +. (5) g g g

5 Proceedings of the ydraulic Machinery and Systems 1 st AR Symposium September 9-1, 00, Lausanne where W correspond to the relative velocity, V absolute velocity, U transport velocity, head of the turbine, p static pressure, hp hydraulic losses between the specified points and a M D is the distance indicated in Fig.. he above formula is then written using the dimensionless velocity coefficients, V = E, U ωr u = =, E E c to obtain σ W cw =, kp E ( cw ) ku ( u ) + ( c ) ( c ) W max = 1 W max, hp hp M A = kpmax M A + ku M U M = 1 U a MD. (6) ere kp max is the dimensionless maximum velocity coefficient and ku M is the dimensionless transport velocity coefficient at M. Next, by assuming that the hydraulic losses hp M are negligible, Anton (Ref. 1) and the draft tube losses can be written by using the draft tube efficiency η D, the following formula is obtained: a D kp ( cw ) ku ( u ) ( c ) M σ = max M + ηd +. (7) One can easily recognize that this velocity coefficient formulation is equivalent to the following pressure formulation ( ) ( zr z ) c c σ = p min + ηd, thus in order to compare numerical results obtained with both formulations for the turbine cavitation coefficient we have used in this work the point to correspond to the erence section. he velocity coefficients formulation of the turbine cavitation coefficient is correct as long as one can identify a relative flow streamline along which the relative flow Bernoulli equation is valid. Note that when introducing (7) and (1) in the right-hand side of (), one can clearly see the separation between ant characteristics and turbine specific quantities. NUMERCAL RESULS Numerical results are presented in this paper for the GAMM Francis model turbine, operating at constant guide vane opening (corresponding to the best efficiency point), and variable discharge. he turbine cavitation coefficient σ is evaluated using both pressure formulation () and velocity formulation (7). he results obtained with equation () are marked with triangles, and a least squares solid curve, on Fig.. On the other hand, equation (7) gives the results presented with squares, together with the corresponding dashed line in Fig.. One can see that the two numerical approaches produce practically the same σ values for a discharge smaller or equal to the best efficiency point value Q Q BEP. owever, for larger discharge, the velocity formulation (7) significantly departs from the pressure formulation ()

6 Proceedings of the ydraulic Machinery and Systems 1 st AR Symposium September 9-1, 00, Lausanne (LM/EPFL, experimental) pressure formulation eq. () velocity formulation eq. (7) 0.6 [ ] σ 0.4 BEP φ [ ] Fig. urbine cavitation coefficient σ versus discharge coefficient φ at constant guide vane opening for GAMM Francis runner. Comparison between the experimental data from (LM/EPFL ) and numerical results ( with pressure coefficient formulation, with velocities coefficients formulation). Fig. 4 Constant pressure coefficient lines (thin labeled lines) and relative flow streamlines (tick lines) on the suction side runner blade, near the band at Q =1.17Q BEP n order to elucidate this discrepancy, we investigated in detail the velocity and pressure fields on the runner blade suction side, in the band neighborhood. Fig. 4 shows the constant pressure coefficient lines (thin solid lines, labeled with the corresponding c values), as well as two streamlines (thick lines) originating from the minimum pressure region. n order for equation (7) to be correct, one has to identify a relative flow streamline which starts at a minimum pressure point and goes downstream up to the runner blade trailing edge. As one can see from Fig. 4, such an open streamline can be found only if the starting point is p

7 Proceedings of the ydraulic Machinery and Systems 1 st AR Symposium September 9-1, 00, Lausanne considered away from the minimum pressure location. n our case such a starting point corresponds to c p = 0.4, while the actual minimum pressure coefficient is = c p min Fig. 5 Photography of the inlet edge cavitation development for GAMM Francis turbine at the best efficiency operating point and = 0., Avellan et al. (Ref. ). σ Fig. 6 Computed cavitational zone ( p < pva ) development for GAMM Francis turbine at the best efficiency operating point and σ = 0.. When attempting to start a streamline from the exact location of minimum pressure, a closed streamline is obtained. his streamline ends at blade half-chord, indicating a possible recirculating flow (not shown in the picture due to the discretization emoyed). As a result, the relative flow Bernoulli equation cannot be written between the minimum pressure point and a point downstream the blade, and the σ values obtained with (7) do not match the values computed with () for Q >Q BEP. n order to validate our computations, we have otted in Fig. 4 the available Finally, Fig. 5 and Fig. 6 present the observed and computed cavitation region for σ = 0. σ values, measured for the GAMM Francis turbine at cavitation inception. As shown in the ntroduction, in this case σ should be equal to σ. t can be seen that a good agreement between our numerical σ and the measured investigated. One should keep in mind that the observing the cavitation inception, theore the relationship within a reasonable approximation range. σ is obtained for all operating points σ values were determined by visually σ σ should be considered at best efficiency operating point. One can observe a very good qualitative agreement for both the location and extent, respectively, of the cavitating zone. CONCLUSONS he paper introduces a methodology for evaluating the turbine cavitation coefficient, σ. his coefficient is practically equal to the homa number,, at cavitation inception. wo approaches are presented for computing σ. he first one uses the computed runner pressure field, as well as the measured draft tube efficiency (pressure formulation). he second one, emoys the Bernoulli equation for absolute and relative flow to obtain the velocity σ

8 Proceedings of the ydraulic Machinery and Systems 1 st AR Symposium September 9-1, 00, Lausanne formulation of σ. t is shown that the results obtained with pressure formulation agree well with available experimental data, in the sense that σ = σ at cavitation inception. On the other hand, both pressure and velocity formulations for σ give the same values for discharge smaller or equal than the best efficiency point. For larger discharge, the velocity formulation cannot predict the correct σ values since no streamline connecting the minimum pressure point on the blade and a point downstream the blade can be found. As far as the location and extent of the cavitation region is concerned, a good agreement is found between our numerical simulation and experimental visualization. n conclusion, the paper advocates the use of the turbine cavitation coefficient σ for evaluating the cavitational performances of the machine. We examine two approaches for computing σ and we validate the numerical results by comparing them with available experimental data. Once σ known from D turbine flow simulation, σ can be set taking into account that σ = σ at cavitation inception. ACKNOWLEDGEMEN he present work has been supported from the Romanian Academy Grant 81/001. Numerical computations have been performed at the Numerical Simulation and Parallel Computing Laboratory from the Politehnica University of imisoara, National Center for Engineering of Systems with Comex Fluids. Experimental data, as well as the Francis turbine geometry have been kindly provided by Prof. François Avellan and Dr. Gabriel Ciocan from the École Polytechnique Fédérale de Lausanne, Laboratory for ydraulic Machines. REFERENCES Ref. 1 Ref. Ref. Ref. 4 Ref. 5 Ref. 6 Ref. 7 Anton., 1985, "Cavitation, Editura Academiei R.S.R, Bucureşti, Romania. Anton., 1964, "Curbe caracteristice de cavitatie la masinile hidraulice (turbine si pompe)", Comunicarile Conferintei de Masini idraulice, imisoara, vol. 1. Avellan F., Dupont P., Farhat M., Gindroz B., enry P., ussain M., 199, "Experimental flow study of the GAMM turbine model" in Proceedings of the GAMM Workshop D-computation of incompressible internal flow Ed. Sottas G. and Ryhming.L., NNFM 9, Vieweg Verlag, Braunschweig, pp. -5. Kubota,., an, F., Avellan, F., 1996, Performance Analysis of Draft ube for GAMM Francis urbine, Proceedings of the18th AR Symposiumon ydraulic Machinery and Cavitation, September , Valencia, Spain, Vol. 1, pp EC 6019, 1999, nternational standard: hydraulic turbines, storage pumps and pump-turbines. Model acceptance tests, Geneva, Muntean S. Susan-Resiga R., Anton., 00, "D Flow Analysis of the GAMM Francis urbine for Variable Discharge", Proceedings of 1th.A..R. Symposium on ydraulic Machinery and Systems, 9-1 September 00, Lausanne, Switzerland. (submitted) Muntean S., 00, Numerical methods for the analysis of the D flow in Francis turbine runners, PhD hesis, imisoara, Romania (in romanian)