Journal of Computational and Applied Mathematics. Numerical modeling of unsteady flow in steam turbine stage

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1 Journal of Comutational and Alied Mathematics 234 (2010) Contents lists available at ScienceDirect Journal of Comutational and Alied Mathematics journal homeage: Numerical modeling of unsteady flow in steam turbine stage J. Halama ab J. Dobeš a J. Fořt a J. Fürst a K. Kozel b a Deartment of Technical Mathematics Czech Technical University FME Prague Karlovo nám. 13 CZ Praha 2 Czech Reublic b IT AS CR Dolejškova 1402/5 CZ Praha 8 Czech Reublic a r t i c l e i n f o a b s t r a c t Article history: Received 5 Setember 2008 Received in revised form 9 January 2009 Keywords: Stator-rotor interaction Unsteady flow Two-hase flow This work deals with numerical solution of unsteady flow in turbine stage. We use models of comressible single-hase flow of air and two-hase flow of wet steam. Presented numerical methods are based on different stator-rotor matching algorithms as well as different numerical schemes. Numerical results achieved by both methods and flow models are discussed Elsevier B.V. All rights reserved. 1. Introduction It is generally difficult to secify flow field arameters (boundary conditions) at the inlet and the outlet of a single turbine cascade due to ustream and downstream located structures see e.g. [1]. Therefore it is interesting to coule several cascades together and the flow field at the interfaces is then the result of numerical simulation. Of course this couling can rovide much more information about e.g. blade forces clocking etc. Different couling techniques from mixing lane to fully unsteady interaction are being used. This aer is aimed at fully unsteady interaction between stator and rotor. We have develoed two finite volume methods. The first method has a relatively simle stator-rotor matching algorithm imlemented into already existing in-house 2D finite volume code based on the Roe aroximate Riemann solver for general unstructured grids which has been reviously verified and used to comute one-hase transonic flow in various single turbine cascades see e.g. [2]. The second method has been created from the in-house Lax-Wendroff finite volume code for two-hase flow of condensing steam by addition of Giles s matching algorithm [3] which has been already successfully used for one-hase stator-rotor interaction in high ressure turbine stage see [1]. 2. Governing equations and roblem formulation Both flow models are given by the set of PDE s of the form t W = x F y G + Q where the 2D flow of erfect gas described by the Euler s equations further referred as one-hase flow model is given by vectors W = [ρ ρu ρv e] T Q = [ ] T F = [ ρu ρu 2 + ρuv (e + )u ] T G = [ ρu ρvu ρv 2 + (e + )v ] T (1) (2) Corresonding author at: Deartment of Technical Mathematics Czech Technical University FME Prague Karlovo nám. 13 CZ Praha 2 Czech Reublic. addresses: halama@marian.fsik.cvut.cz (J. Halama) dobes@marian.fsik.cvut.cz (J. Dobeš) fort@marian.fsik.cvut.cz (J. Fořt) furst@marian.fsik.cvut.cz (J. Fürst) kozelk@fsik.cvut.cz (K. Kozel) /$ see front matter 2009 Elsevier B.V. All rights reserved. doi: /j.cam

2 J. Halama et al. / Journal of Comutational and Alied Mathematics 234 (2010) and the closure is = (γ 1)[e ρ(u 2 + v 2 )/2]. The symbol ρ denotes density u and v velocity vector comonents ressure e total energy er unit volume and γ secific heat ratio. The next model describes flow of mixture containing vaor and condensed drolets and it is referred as two-hase flow model. We consider a convection of drolets by vaor. The governing equations consists of Euler s equation for the mixture and transort equations for arameters of drolet sectra (Hill s moments Q 0 Q 1 a Q 2 see [4]) so the vectors are W = [ρ ρu ρv e ρχ ρq 2 ρq 1 ρq 0 ] T F = [ ρu ρu 2 + ρuv (e + )u ρχu ρq 2 u ρq 1 u ρq 0 u ] T G = [ ρu ρvu ρv 2 + (e + )v ρχv ρq 2 v ρq 1 v ρq 0 v ] T [ Q = T 3 πr 3 ρ c lj + 4πρQ 2 ṙρ l r 2 c J + 2ρQ 1ṙ r c J + Q 0 ρṙ J] (3) where ρ now denotes the mixture density u and v velocity comonents common for both vaor and liquid the common ressure e the total energy er unit volume χ wetness (i.e. the mass fraction of liquid). The system of equations is closed by the equation for ressure according to Šejna [5] [ (1 χ) = (γ 1) e 1 ] 1 + χ(γ 1) 2 ρ(u2 + v 2 ) + ρχl (4) with L denoting the latent heat of condensation and the secific heat ratio γ is here taken as a function of temerature. The Hill s moments are Q 0 = N Q 1 = N r i Q 2 = i=1 N i=1 0 χ r 2 i r = 10 6 Q 2 /Q 0 χ > 10 6 with N denoting the total number of drolets er unit mass of mixture r i the radius of i-th drolet and r is the average radius. The limit value 10 6 is chosen to stabilize numerical algorithm. The number J of new condensed drolets er unit volume and er second is based on Becker s work [6] J = 2σ πm 3 v ρ2 v ρ l ex ( β 4πr 2σ ) c r c = 3k B T v 2σ ρ l R v T v ln(/ s ). (6) The new drolet has radius equal to r c. The vaor density is ρ v = (1 χ)ρ. The vaor temerature is calculated using a erfect gas law T v =. The value of surface tension ρ v σ is corrected by the coefficient β for further details see [7]. The Rv drolet growth is given by λ v (T s T v ) ṙ = Lρ l ( Kn) r r c Kn = ν v 2πRv T v. (7) r 2 4r The remaining symbols are: water molecule mass m v Boltzman constant k B saturation ressure s vaor gas constant R v vaor thermal conductivity λ v saturation temerature T s water density ρ l and vaor kin. viscosity ν v. Let s consider comutational domain unbounded in vertical direction see the Fig. 1 on the left. The stator subdomain Ω s has inlet boundary Γ i and interface boundary Γ c and the rotor subdomain Ω r has interface boundary Γ c and outlet boundary Γ o. The equations are solved in the coordinate system attached to the resective blade cascade. Fixed values of total ressure total temerature and the flow direction are secified at Γ i (inlet axial velocity for all resented cases is subsonic). A constant value of ressure is given at Γ o (outlet axial velocity is again in all resented cases subsonic) because steady non-reflecting boundary conditions already imlemented in both codes fail for unsteady stator-rotor interaction. The continuity of solution (in common frame of reference) is considered along the curve Γ c. To resect the different itch of both cascades and to avoid comlicated algorithms like for examle time inclination lane of Giles [3] we aly a commonly used multile blade channel domain with mp S = np R where P S and P R are stator and rotor itch and m and n are small integers see the examle in the Fig. 1 on the right then the eriodicity conditions are simly W(A) = W(B) and W(C) = W(D). (5) 3. Numerical method I The method I is a cell-centered finite volume method for unstructured triangular mesh with non equal number of oints along both sides of interface Γ c see the Fig. 2 left. The line integral from finite volume formulation is aroximated on each cell face in one Gauss oint. A numerical flux is comuted by the modified Roe aroximate Riemann solver [8]. [F(W)n x + G(W)n y ]ds l F(W L W R n x n y ) (8) c

3 2338 J. Halama et al. / Journal of Comutational and Alied Mathematics 234 (2010) Fig. 1. Comutational domain for stage (left) and eriodicity treatment (right). Fig. 2. Method I: Interface between stator and rotor subdomains (stator is on the left). Fig. 3. Matching algorithm for method II grid reconnection (left) and oint corresondence (right). where F is the numerical flux in the face center and l is the length of cell face and W L W R are aroximations of the solution on the left and right side of face. Higher satial accuracy is achieved using linear reconstruction with Barth limiter [9]. Gradient of solution in each element is comuted by the least square method. Temoral discretization is based uon exlicit TVD Runge Kutta method [10]. Details about the method I can be found e.g. in [2]. Rotor subdomain moves with the constant velocity (0 v rotor ) with resect to stator subdomain. The matching algorithm is based on a simle linear interolation of missing right states. The right state W R in the center of face of i-th stator cell coinciding with the interface Γ c is denoted in the Fig. 2 as W R ij and is comuted using the value W j and the gradient of W in the j-th cell. Of course the values have to be roerly recomuted from relative frame into the absolute frame of reference. The right state W R ji for the summation of numerical fluxes in the j-th rotor cell is comuted analogously. Although this technique is slightly non-conservative ractical tests have not shown any roblems. 4. Numerical method II The method II uses structured quadrilateral H-tye grid. It is based on the Ni s cell-vertex scheme [11] stabilized by a conservative artificial viscosity terms of Stringer [12]. The comutational grid has equally distributed oints along both sides of Γ c and therefore the stator and rotor grids are connected directly. The relative movement of stator and rotor is

4 J. Halama et al. / Journal of Comutational and Alied Mathematics 234 (2010) Fig. 4. NASA HP turbine stage ressure isolines at three successive moments during rotor shift of one rotor itch uer row method I and lower row method II. rovided by a deformation of one cell column. The grids are eriodically reconnected in a roer time to revent excessive cell deformation see the Fig. 3(left). This technique was originally roosed by Giles [3]. Thanks to direct grid connectivity the finite volumes along the interface Γ c do not differ from those inside the domain. The stator and the rotor grids are overlaed over three cells due to method stencil see the Fig. 3(right). The oint B s is the last oint in the stator for which we construct a finite volume (denoted by grey color) and similarly the first one for the rotor is the oint C r. The solution from oints C r and D r is transferred after each time ste into ghost oints C s and D s and similarly the information from A s and B s is transferred into ghost oints A r and B r. The only modification of scheme is due to the cell movement i.e. the integral of time derivative of W in the finite volume formulation has the form t WdV = t WdV W u V n V ds (9) V where u V is the velocity of finite volume boundary. V 5. One-hase stator-rotor interaction in HP gas turbine stage V The first resented case is 2D one-hase flow at the midsan (cylindrical cut at diameter d = m) of NASA HP gas turbine stage [13]. We consider a comutational domain with 3 stator and 5 rotor blade channels i.e. the original number of 36 stator and 64 rotor blades has been slightly modified. The comuted case is given by the inlet total ressure 0 = Pa the inlet total temerature T 0 = K and the axial flow direction at the inlet. The outlet ressure is / 0 = The rotor runs at 8081 RPM. The gas constant for air is R = 287 J kg 1 K 1 and the secific heat ratio γ = 1.4. Numerical results in the form of ressure isolines are lotted in the Fig. 4. The flow field ustream the stator throat is steady since the stator cascade is choked. Both numerical algorithms rovide a continuous solution across the stator-rotor interface.

5 2340 J. Halama et al. / Journal of Comutational and Alied Mathematics 234 (2010) (a) 1-hase / 0 t = t 1. (b) 2-hase / 0 t = t 1. (c) 2-hase χ t = t 1. (d) 1-hase / 0 t = t 2. (e) 2-hase / 0 t = t 2. (f) 2-hase χ t = t 2. Fig. 5. Steam turbine stage isolines of denoted variables at denoted time method II. The results of both methods are in overall good agreement. Besides exected interaction of trailing edge shocks with rotor blades we also observe an ustream traveling reflection of right running trailing edge shock. Shock waves obtained by the method II are not so shar as the ones catured by the method I since method II is more dissiative. 6. One- and two-hase flow in HP steam turbine stage The comutational domain of HP steam turbine stage consists of 4 stator and 7 rotor blade assages it corresonds to the stage with 32 stator and 56 rotor blades which is an aroximation of real existing stage with 32 stator and 54 rotor blades. Each blade assage was discretized by a H-tye grid with oints er one stator and oints er one rotor blade assage i.e with the total number arox oints in the whole domain. For the first comutations were chosen following boundary conditions: inlet total ressure 01 = Pa inlet total temerature T 01 = K and axial flow direction at the inlet. The outlet ressure is 2 = Pa. The rotor runs at 3000 RPM. We consider the constant secific heat ratio γ = The inlet total ressure to outlet ressure ratio corresonds to real stage conditions. Values for boundary conditions were chosen with resect to reviously comuted cases of two hase flow in nozzles and in order to kee the start of condensation inside the comutational domain. We recognize two cases: the first where the condensation is artificially switched off i.e. only vaor flow with no influence through the latent heat extraction or addition. This case is referred as one-hase flow. The second case referred as two-hase flow takes into account the condensation terms and models the effect of extraction or addition of latent heat to the flow. Numerical results of one-hase and two hase flow cases are lotted in form of ressure and wetness isolines for two time moments where at time t 2 is rotor shifted about P R /2 with resect to t 1 see the Fig. 5. The flow in turbine stage is subsonic therefore the rotor cascade influences even the flow at the stator inlet. This means a slower convergence to eriodical solution comared to tyical transonic cases with choked stator. The two consequent ressure distributions along stator and rotor blade are lotted in the Fig. 6. It is well visible that ressure fluctuations travel ustream u to the stator inlet and that ressure fluctuations are stronger in the rotor cascade as one would exect. The heat released by condensation in the two-hase flow case results in slightly different ressure

6 J. Halama et al. / Journal of Comutational and Alied Mathematics 234 (2010) (a) stator at time t and t + T S /2. (b) rotor at time t and t + T R /2. Fig. 6. Pressure along the blade at given time (T S and T R are the stator and rotor basic eriods resectively). One-hase flow denoted by dashed line and 2-hase by the full line. distribution comared to the one-hase flow case see the Figs. 5 and 6. The two-hase flow case is more sensitive on the imlementation of outlet boundary condition. We had to modify the former realization of the outlet boundary condition develoed for one-hase flow which was based on the extraolation of conservative variables from domain to boundary and correction of energy to match given ressure. A new version extraolates the rimitive variables to suress temerature oscillations at the outlet since the unsteady two-hase flow is very sensitive to temerature changes. 7. Conclusions Both imlemented stator-rotor matching algorithms work well for subsonic as well as transonic flow cases. The results for two hase flow show imortance of condensation modeling also for subsonic flow of steam. We exect much ronounced effect of latent heat addition/extraction in the case of transonic flow which takes lace e.g. in LP steam turbines. The unsteady results of two hase flow also showed a sensitivity of results to the the temerature oscillations. We had to modify the realization of outlet boundary condition to revent unhysical oscillations of liquid mass fraction at the outlet. Acknowledgments The suort of the Research lan MSM CR no and the grant no. 201/08/0012 of GACR is greatly acknowledged. Some articular work was carried out with the suort of the Research Plan MSM No. Z References [1] E. Valenti J. Halama R. Dénos T. Arts Investigation of the 3d unsteady rotor ressure field in a h turbine stage ASME Paer GT [2] J. Dobeš J. Fořt J. Halama Numerical solution of single and multile-hase internal transonic flow roblems Internat. J. Numer. Methods Fluids 48 (1) (2005) [3] M.B. Giles A numerical method for the calculation of unsteady flow in turbomachinery Tech. Re. GLT Reort No. 205 MIT May [4] P.G. Hill Condensation of water vaor during suersonic exansion in nozzles art 3 J. Fluid Mech. 3 (1966) [5] M. Šejna Numerical modelling of two-hase flow of steam with homogenous condensation Ph.D. Thesis Prague 1995 (in Czech). [6] R. Becker W. Döring Kinetische behandlung der keimbildung in übersättingten dämfen Ann. Phys. 24 (8) (1935) [7] V. Petr M. Kolovratník Heterogenous effects in the drolet nucleation rocess in l steam turbines in: 4th ECT Firenze Italy [8] P.L. Roe Aroximate Riemann solvers arameter vectors and difference schemes J. Comut. Phys. 43 (1981) [9] T.J. Barth D.C. Jeserson The design and alication of uwind schemes on unstructured meshes AIAA Paer [10] S. Gottlieb C.-W. Shu E. Tadmor Strong stability-reserving high-order time discretization methods SIAM Rev. 43 (1) (2001) [11] R.H. Ni A multile grid scheme for solving Euler equations AIAA J. 20 (1) (1981). [12] S.M. Stringer K.W. Morton Artificial viscosity for the cell vertex method Tech. Re. No. 96/08 Oxford University Com. Lab [13] T.P. Moffit E.M. Szanca W.J. Whitney F.P. Behning Design and cold-air test of single uncooled core turbine with high work outut Tech. Re. NASA Technical Paer