ON THE USE OF A FLUX-SPLITTING SCHEME IN THE NUMERICAL FLUTTER ANALYSIS OF A LOW-PRESSURE TURBINE STAGE
Keywords:Turbine flutter, CFD, time-marching simulation, gradient reconstruction, limiters
The endeavour to increase the power output of steam turbines results in the design of low-pressure stages with large diameters. Such designs, featuring long and thin blades, are increasingly susceptible to unfavourable aeroelastic effects. The interaction of structure and flow may induce blade vibrations, known as flutter, which act detrimentally on the operational life of the machine. The present work employs a time-marching numerical simulation to investigate the flutter behaviour of a low-pressure transonic turbine cascade. Its blades are subject to a harmonic motion based on the results of a structural analysis and its susceptibility to flutter is evaluated via the energy method. The computations are performed with an in-house Finite Volume Method code. The flow model is based on 2D Euler equations in Arbitrary Lagrangian-Eulerian formulation with the AUSM+-up scheme for inviscid flux discretization. A higher-order spatial accuracy is achieved by using a MUSCL approach, for which both the gradient reconstruction and the slope limiting are given a careful examination-by comparing the convergence and accuracy of multiple methods. The computational model is validated by experimental data on the Fourth Standard Configuration turbine cascade.
P. Petrie-Repar, V. Makhnov, N. Shabrov, et al. Advanced Flutter Analysis of a Long Shrouded Steam Turbine Blade. vol. Volume 7B: Structures and Dynamics of Turbo Expo: Power for Land, Sea, and Air. 2014. V07BT35A022, doi:10.1115/GT2014-26874.
W. Höhn. Numerical Investigation of Blade Flutter at or Near Stall in Axial Turbomachines. Ph.D. thesis, Royal Institute of Technology, Stockholm, Sweden, 2012.
I. McBean, K. Hourigan, M. Thompson, F. Liu. Prediction of Flutter of Turbine Blades in a Transonic Annular Cascade. Journal of Fluids Engineering 127(6):1053–1058, 2005. doi:10.1115/1.2060731.
P.-A. Masserey, I. McBean, H. Lorini. Analysis and improvement of vibrational behaviour on the nd37 a last stage blade. VGB Powertech Journal 92:42–48, 2012.
P. Sváček, M. Feistauer, J. Horáček. Numerical simulation of flow induced airfoil vibrations with large amplitudes. Journal of Fluids and Structures 23(3):391 – 411, 2007. doi:10.1016/j.jfluidstructs.2006.10.005.
B. Perry. Report no. 496: General theory of aerodynamic instability and the mechanism of flutter. Technical Report, NASA. Langley Research Center, Hampton, Virginia, United States, 2015.
H. Doi, J. J. Alonso. Fluid/Structure Coupled Aeroelastic Computations for Transonic Flows in Turbomachinery. vol. 4: Turbo Expo 2002, Parts A and B of Turbo Expo: Power for Land, Sea, and Air, pp. 787–794. 2002. doi:10.1115/GT2002-30313.
R. Kamakoti, W. Shyy. Fluid-structure interaction for aeroelastic applications. Progress in Aerospace Sciences 40:535–558, 2005. doi:10.1016/j.paerosci.2005.01.001.
W. Ning, L. He. Computation of Unsteady Flows Around Oscillating Blades Using Linear and Nonlinear Harmonic Euler Methods. Journal of Turbomachinery 120(3):508–514, 1998. doi:10.1115/1.2841747.
J. Donea, S. Giuliani, J. Halleux. An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions. Computer Methods in Applied Mechanics and Engineering 33(1):689 – 723, 1982. doi:10.1016/0045-7825(82)90128-1.
D. Boffi, L. Gastaldi. Stability and geometric conservation laws for ale formulations. Computer Methods in Applied Mechanics and Engineering 193(42):4717 – 4739, 2004. doi:10.1016/j.cma.2004.02.020.
P. H. Saksono, W. G. Dettmer, D. Peric. An adaptive remeshing strategy for flows with moving boundaries and fluid-structure interaction. International Journal for Numerical Methods in Engineering 71(9):1009–1050, 2007. doi:10.1002/nme.1971.
A. Slone, C. Bailey, M. Cross. Dynamic solid mechanics using finite volume methods. Applied Mathematical Modelling 27(2):69 – 87, 2003. doi:10.1016/S0307-904X(02)00060-4.
N. Donini. Aeroelasticity of turbomachines linearized flutter analysis. Ph.D. thesis, Politecnico di Milano, Facoltá di Ingegneria Industriale, Milano, Italy, 2012.
M. May, Y. Mauffrey, F. Sicot. Numerical flutter analysis of turbomachinery bladings based on time-linearized, time-spectral and time-accurate simulations. In IFASD 2011 - 15th International Forum on Aeroelasticity and Structural Dynamics. 2011.
A. Slone, K. Pericleous, C. Bailey, M. Cross. Dynamic fluid-structure interaction using finite volume unstructured mesh procedures. Computers & Structures 80(5):371 – 390, 2002. doi:10.1016/S0045-7949(01)00177-8.
R. Rzadkowski, V. Gnesin. 3-d inviscid self-excited vibrations of a blade row in the last stage turbine. Journal of Fluids and Structures 23(6):858 – 873, 2007. doi:10.1016/j.jfluidstructs.2006.12.003.
V. Gnesin, L. Kolodyazhnaya, R. Rzadkowski. A numerical modelling of stator-rotor interaction in a turbine stage with oscillating blades. Journal of Fluids and Structures 19(8):1141 – 1153, 2004. doi:10.1016/j.jfluidstructs.2004.07.001.
M.-S. Liou, C. J. Steffen. A new flux splitting scheme. Journal of Computational Physics 107(1):23 – 39, 1993. doi:10.1006/jcph.1993.1122.
B. van Leer. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. Journal of Computational Physics 32(1):101 – 136, 1979. doi:10.1016/0021-9991(79)90145-1.
J. Blazek. Computational Fluid Dynamics: Principles and Applications. Elsevier, 2001.
E. Sozer, C. Brehm, C. C. Kiris. Gradient calculation methods on arbitrary polyhedral unstructured meshes for cell-centered CFD solvers. In 52nd Aerospace Sciences Meeting. 2014. doi:10.2514/6.2014-1440.
M. Berger, M. Aftosmis, S. Muman. Analysis of slope limiters on irregular grids. In 43rd AIAA Aerospace Sciences Meeting and Exhibit. 2012. doi:10.2514/6.2005-490.
D. Mavriplis. Revisiting the least-squares procedure for gradient reconstruction on unstructured meshes. In 16th AIAA Computational Fluid Dynamics Conference. 2003. doi:10.2514/6.2003-3986.
F. Mishriky, P. Walsh. Towards understanding the influence of gradient reconstruction methods on unstructured flowsimulations. Transactions of the Canadian Society for Mechanical Engineering 41(2):169–179, 2017.
A. Syrakos, S. Varchanis, Y. Dimakopoulos, et al. A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods. Physics of Fluids 29(12):127103, 2017. doi:10.1063/1.4997682.
M. Hubbard. Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. Journal of Computational Physics 155(1):54 – 74, 1999. doi:10.1006/jcph.1999.6329.
T. Barth, D. Jespersen. The design and application of upwind schemes on unstructured meshes. In 27th Aerospace Sciences Meeting. 1989. doi:10.2514/6.1989-366.
V. Venkatakrishnan. On the accuracy of limiters and convergence to steady state solutions. In 31st Aerospace Sciences Meeting. 1993. doi:10.2514/6.1993-880.
K. H. Kim, C. Kim. Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows: Part ii: Multi-dimensional limiting process. Journal of Computational Physics 208(2):570 – 615, 2005. doi:10.1016/j.jcp.2005.02.022.
J. S. Park, S.-H. Yoon, C. Kim. Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. Journal of Computational Physics 229(3):788 – 812, 2010. doi:10.1016/j.jcp.2009.10.011.
A. I. Delis, I. K. Nikolos. A novel multidimensional solution reconstruction and edge-based limiting procedure for unstructured cell-centered finite volumes with application to shallow water dynamics. International Journal for Numerical Methods in Fluids 71(5):584–633, 2013. doi:10.1002/fld.3674.
A. Delis, I. K. Nikolos. On a solution reconstruction and limiting procedure for unstructured finite volumes. American Institute of Mathematical Sciences (AIMS) Journal 8:491–499, 2014.
M.-S. Liou. A sequel to AUSM, part II: AUSM+-up for all speeds. Journal of Computational Physics 214(1):137 – 170, 2006. doi:10.1016/j.jcp.2005.09.020.
A. Bölcs, T. Fransson. Aeroelasticity in Turbomachines: Comparison of Theoretical and Experimental Cascade Results + Appendix A5. EPFL Lausanne, 1986.
R. W. Smith. AUSM(ALE): A geometrically conservative arbitrary Lagrangian-Eulerian flux splitting scheme. Journal of Computational Physics 150(1):268 – 286, 1999. doi:10.1006/jcph.1998.6180.
D. Darracq, S. Champagneux, A. Corjon. Computation of unsteady turbulent airfoil flows with an aeroelastic AUSM+ implicit solver. In 16th AIAA Applied Aerodynamics Conference. 1998. doi:10.2514/6.1998-2411.
A. C. Haselbacher. A grid-transparent numerical method for compressible viscous flows on mixed unstructured grids, 1999.
W. Anderson, D. L. Bonhaus. An implicit upwind algorithm for computing turbulent flows on unstructured grids. Computers & Fluids 23(1):1 – 21, 1994. doi:10.1016/0045-7930(94)90023-X.
P. Batten, C. Lambert, D. M. Causon. Positively conservative high-resolution convection schemes for unstructured elements. International Journal for Numerical Methods in Engineering 39(11):1821–1838, 1996. doi:10.1002/(SICI)1097- 0207(19960615)39:11<1821::AID-NME929>3.0.CO;2- E.
Fourth standard configuration experimental data. https://www.rpmturbo.com/testcases/STCF/STCF4_ update/stcf4_update.htm. Accessed: 2020-05-27.
J. J. Waite. Physical Insights, Steady Aerodynamic Effects, and a Design Tool for Low-Pressure Turbine Flutter. Ph.D. thesis, Duke University, Durham, North Carolina, 2016.
T. H. Fransson, J. M. Verdon. Updated report on standard configurations for the determination of unsteady flow through vibrating axial-flow turbomachinecascades.
Proceedings of the 6th International Conference on Aeroelasticity in Turbomachines 1991.
D. Schlüß, C. Frey. Time domain flutter simulations of a steam turbine stage using spectral 2D non-reflecting boundary conditions. Proceedings of the 15th International Symposium on Unsteady Aerodynamics, Aeroacoustics & Aeroelasticity of Turbomachines, ISUAAAT15. 2018.
F. Lane, M. Friedman. Technical note 4136: Theoretical investigation of subsonic oscillatory blade-row aerodynamics. Technical Report, New York University, New York, United States, 1958.
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