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Computational analysis of performance deterioration of a wind turbine blade strip subjected to environmental erosion

  • Alessio CastorriniEmail author
  • Alessandro Corsini
  • Franco Rispoli
  • Paolo Venturini
  • Kenji Takizawa
  • Tayfun E. Tezduyar
Original Paper

Abstract

Wind-turbine blade rain and sand erosion, over long periods of time, can degrade the aerodynamic performance and therefore the power production. Computational analysis of the erosion can help engineers have a better understanding of the maintenance and protection requirements. We present an integrated method for this class of computational analysis. The main components of the method are the streamline-upwind/Petrov–Galerkin (SUPG) and pressure-stabilizing/Petrov–Galerkin (PSPG) stabilizations, a finite element particle-cloud tracking method, an erosion model based on two time scales, and the solid-extension mesh moving technique (SEMMT). The turbulent-flow nature of the analysis is handled with a Reynolds-averaged Navier–Stokes model and SUPG/PSPG stabilization, the particle-cloud trajectories are calculated based on the computed flow field and closure models defined for the turbulent dispersion of particles, and one-way dependence is assumed between the flow and particle dynamics. Because the geometry update due to the erosion has a very long time scale compared to the fluid–particle dynamics, the update takes place in a sequence of “evolution steps” representing the impact of the erosion. A scale-up factor, calculated in different ways depending on the update threshold criterion, relates the erosions and particle counts in the evolution steps to those in the fluid–particle simulation. As the blade geometry evolves, the mesh is updated with the SEMMT. We present computational analysis of rain and sand erosion for a wind-turbine blade strip, including a case with actual rainfall data and experimental aerodynamic data for eroded airfoil geometries.

Keywords

Wind turbine Blades Rain erosion Sand erosion SUPG and PSPG methods Particle-cloud tracking model Erosion scale-up 

Notes

Acknowledgements

This work was supported in part by Sapienza University of Rome “Progetti Grandi 2017” grant “Development of advanced modeling techniques for coupled multi-physics in open and ducted rotor fluid machines” - n. prot. RG11715C81D7D03A. The mathematical model and computational method parts of the work were also supported in part by Grant-in-Aid for Challenging Exploratory Research 16K13779 from JSPS and Grant-in-Aid for Scientific Research (S) 26220002 from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT) (for the 5th author) and ARO Grant W911NF-17-1-0046 and Top Global University Project of Waseda University (for the last author).

Supplementary material

References

  1. 1.
    Wood K (2011) Blade repair: closing the maintenance gap. Composites Technology. https://www.compositesworld.com/articles/blade-repair-closing-the-maintenance-gap
  2. 2.
  3. 3.
    Castorrini A, Corsini A, Rispoli F, Venturini P, Takizawa K, Tezduyar TE (2016) SUPG/PSPG computational analysis of rain erosion in wind-turbine blades. In: Bazilevs Y, Takizawa K (eds) Advances in computational fluid–structure interaction and flow simulation: new methods and challenging computations. Modeling and simulation in science, engineering and technology. Springer, Cham, pp 77–96CrossRefGoogle Scholar
  4. 4.
    Castorrini A, Corsini A, Rispoli F, Venturini P, Takizawa K, Tezduyar TE (2016) Computational analysis of wind-turbine blade rain erosion. Comput Fluids 141:175–183.  https://doi.org/10.1016/j.compfluid.2016.08.013 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Corsini A, Castorrini A, Morei E, Rispoli F, Sciulli F, Venturini P (2015) Modeling of rain drop erosion in a multi-MW wind turbine. In: ASME turbo expo, MontrealGoogle Scholar
  6. 6.
    Hussein MF, Tabakoff W (1974) Computation and plotting of solid particle flow in rotating cascades. Comput Fluids 2:1–15CrossRefzbMATHGoogle Scholar
  7. 7.
    Hamed AA, Tabakoff W, Rivir RB, Das K, Arora P (2005) Turbine blade surface deterioration by erosion. J Turbomach 127:445–452CrossRefGoogle Scholar
  8. 8.
    Hamed A, Tabakoff W, Swar R, Shin D, Woggon N, Miller R (2013) Combined experimental and numerical simulations of thermal barrier coated turbine blades erosion. NASA report NASA/TM-2013-217857Google Scholar
  9. 9.
    Ghenaiet A (2009) Numerical study of sand ingestion through a ventilating system. Proc World Congr Eng 2:1–3Google Scholar
  10. 10.
    Suzuki M, Yamamoto M (2011) Numerical simulation of sand erosion phenomena in a single-stage axial compressor. J Fluid Sci Technol 6:98–113CrossRefGoogle Scholar
  11. 11.
    Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32:199–259MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28:1–44.  https://doi.org/10.1016/S0065-2156(08)70153-4 MathSciNetzbMATHGoogle Scholar
  13. 13.
    Baxter LL, Smith PJ (1993) Turbulent dispersion of particles: the STP model. Energy Fuels 7:852–859CrossRefGoogle Scholar
  14. 14.
    Venturini P (2010) Modelling of particle-wall deposition in two-phase gas-solid flows. Ph.D. thesis, Sapienza University of RomeGoogle Scholar
  15. 15.
    Cardillo L, Corsini A, Delibra G, Rispoli F, Sheard AG, Venturini P (2015) Simulation of particle-laden flows in a large centrifugal fan for erosion prediction. In: 58th American society of mechanical engineers turbine and aeroengine congress, DüsseldorfGoogle Scholar
  16. 16.
    Kaer SK (2001) Numerical investigation of ash deposition in straw-fired furnaces. Aalborg University, Denmark Ph.D. thesisGoogle Scholar
  17. 17.
    Corsini A, Marchegiani A, Rispoli F, Venturini P (1993) Predicting blade leading edge erosion in an axial induced draft fan. ASME J Eng Gas Turbines Power 134:042601CrossRefGoogle Scholar
  18. 18.
    Tezduyar TE, Park YJ (1986) Discontinuity capturing finite element formulations for nonlinear convection-diffusion-reaction equations. Comput Methods Appl Mech Eng 59:307–325.  https://doi.org/10.1016/0045-7825(86)90003-4 CrossRefzbMATHGoogle Scholar
  19. 19.
    Corsini A, Rispoli F, Santoriello A, Tezduyar TE (2006) Improved discontinuity-capturing finite element techniques for reaction effects in turbulence computation. Comput Mech 38:356–364.  https://doi.org/10.1007/s00466-006-0045-x MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Corsini A, Menichini C, Rispoli F, Santoriello A, Tezduyar TE (2009) A multiscale finite element formulation with discontinuity capturing for turbulence models with dominant reactionlike terms. J Appl Mech 76:021211.  https://doi.org/10.1115/1.3062967 CrossRefGoogle Scholar
  21. 21.
    Corsini A, Iossa C, Rispoli F, Tezduyar TE (2010) A DRD finite element formulation for computing turbulent reacting flows in gas turbine combustors. Comput Mech 46:159–167.  https://doi.org/10.1007/s00466-009-0441-0 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Corsini A, Rispoli F, Tezduyar TE (2011) Stabilized finite element computation of NOx emission in aero-engine combustors. Int J Numer Methods Fluids 65:254–270.  https://doi.org/10.1002/fld.2451 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tezduyar TE, Hughes TJR (1983) Finite element formulations for convection dominated flows with particular emphasis on the compressible Euler equations. In: Proceedings of AIAA 21st aerospace sciences meeting, AIAA Paper 83–0125, RenoGoogle Scholar
  24. 24.
    Hughes TJR, Tezduyar TE (1984) Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Comput Methods Appl Mech Eng 45:217–284.  https://doi.org/10.1016/0045-7825(84)90157-9 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tezduyar TE, Ganjoo DK (1986) Petrov-Galerkin formulations with weighting functions dependent upon spatial and temporal discretization: applications to transient convection–diffusion problems. Comput Methods Appl Mech Eng 59:49–71.  https://doi.org/10.1016/0045-7825(86)90023-X CrossRefzbMATHGoogle Scholar
  26. 26.
    Le Beau GJ, Ray SE, Aliabadi SK, Tezduyar TE (1993) SUPG finite element computation of compressible flows with the entropy and conservation variables formulations. Comput Methods Appl Mech Eng 104:397–422.  https://doi.org/10.1016/0045-7825(93)90033-T CrossRefzbMATHGoogle Scholar
  27. 27.
    Tezduyar TE, Osawa Y (2000) Finite element stabilization parameters computed from element matrices and vectors. Comput Methods Appl Mech Eng 190:411–430.  https://doi.org/10.1016/S0045-7825(00)00211-5 CrossRefzbMATHGoogle Scholar
  28. 28.
    Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43:555–575.  https://doi.org/10.1002/fld.505 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tezduyar TE (2004) Finite element methods for fluid dynamics with moving boundaries and interfaces, chapter 17. In: Stein E, Borst RD, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 3: Fluids. Wiley, New YorkGoogle Scholar
  30. 30.
    Tezduyar TE (2007) Finite elements in fluids: stabilized formulations and moving boundaries and interfaces. Comput Fluids 36:191–206.  https://doi.org/10.1016/j.compfluid.2005.02.011 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tezduyar TE, Senga M (2006) Stabilization and shock-capturing parameters in SUPG formulation of compressible flows. Comput Methods Appl Mech Eng 195:1621–1632.  https://doi.org/10.1016/j.cma.2005.05.032 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tezduyar TE, Senga M (2007) SUPG finite element computation of inviscid supersonic flows with YZ\(\beta \) shock-capturing. Comput Fluids 36:147–159.  https://doi.org/10.1016/j.compfluid.2005.07.009 CrossRefzbMATHGoogle Scholar
  33. 33.
    Tezduyar TE, Senga M, Vicker D (2006) Computation of inviscid supersonic flows around cylinders and spheres with the SUPG formulation and YZ\(\beta \) shock-capturing. Comput Mech 38:469–481.  https://doi.org/10.1007/s00466-005-0025-6 CrossRefzbMATHGoogle Scholar
  34. 34.
    Tezduyar TE, Sathe S (2006) Enhanced-discretization selective stabilization procedure (EDSSP). Comput Mech 38:456–468.  https://doi.org/10.1007/s00466-006-0056-7 CrossRefzbMATHGoogle Scholar
  35. 35.
    Tezduyar TE, Sathe S (2007) Modeling of fluid-structure interactions with the space-time finite elements: solution techniques. Int J Numer Methods Fluids 54:855–900.  https://doi.org/10.1002/fld.1430 CrossRefzbMATHGoogle Scholar
  36. 36.
    Rispoli F, Corsini A, Tezduyar TE (2007) Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD). Comput Fluids 36:121–126.  https://doi.org/10.1016/j.compfluid.2005.07.004 CrossRefzbMATHGoogle Scholar
  37. 37.
    Tezduyar TE, Ramakrishnan S, Sathe S (2008) Stabilized formulations for incompressible flows with thermal coupling. Int J Numer Methods Fluids 57:1189–1209.  https://doi.org/10.1002/fld.1743 MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rispoli F, Saavedra R, Corsini A, Tezduyar TE (2007) Computation of inviscid compressible flows with the V-SGS stabilization and YZ\(\beta \) shock-capturing. Int J Numer Methods Fluids 54:695–706.  https://doi.org/10.1002/fld.1447 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2007) YZ\(\beta \) discontinuity-capturing for advection-dominated processes with application to arterial drug delivery. Int J Numer Methods Fluids 54:593–608.  https://doi.org/10.1002/fld.1484 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Rispoli F, Saavedra R, Menichini F, Tezduyar TE (2009) Computation of inviscid supersonic flows around cylinders and spheres with the V-SGS stabilization and YZ\(\beta \) shock-capturing. J Appl Mech 76:021209.  https://doi.org/10.1115/1.3057496 CrossRefGoogle Scholar
  41. 41.
    Hsu M-C, Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2010) Improving stability of stabilized and multiscale formulations in flow simulations at small time steps. Comput Methods Appl Mech Eng 199:828–840.  https://doi.org/10.1016/j.cma.2009.06.019 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Corsini A, Rispoli F, Tezduyar TE (2012) Computer modeling of wave-energy air turbines with the SUPG/PSPG formulation and discontinuity-capturing technique. J Appl Mech 79:010910.  https://doi.org/10.1115/1.4005060 CrossRefGoogle Scholar
  43. 43.
    Corsini A, Rispoli F, Sheard AG, Tezduyar TE (2012) Computational analysis of noise reduction devices in axial fans with stabilized finite element formulations. Comput Mech 50:695–705.  https://doi.org/10.1007/s00466-012-0789-4 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Kler PA, Dalcin LD, Paz RR, Tezduyar TE (2013) SUPG and discontinuity-capturing methods for coupled fluid mechanics and electrochemical transport problems. Comput Mech 51:171–185.  https://doi.org/10.1007/s00466-012-0712-z MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Corsini A, Rispoli F, Sheard AG, Takizawa K, Tezduyar TE, Venturini P (2014) A variational multiscale method for particle-cloud tracking in turbomachinery flows. Comput Mech 54:1191–1202.  https://doi.org/10.1007/s00466-014-1050-0 MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Rispoli F, Delibra G, Venturini P, Corsini A, Saavedra R, Tezduyar TE (2015) Particle tracking and particle-shock interaction in compressible-flow computations with the V-SGS stabilization and YZ\(\beta \) shock-capturing. Comput Mech 55:1201–1209.  https://doi.org/10.1007/s00466-015-1160-3 MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Takizawa K, Tezduyar TE, McIntyre S, Kostov N, Kolesar R, Habluetzel C (2014) Space-time VMS computation of wind-turbine rotor and tower aerodynamics. Comput Mech 53:1–15.  https://doi.org/10.1007/s00466-013-0888-x CrossRefzbMATHGoogle Scholar
  48. 48.
    Takizawa K, Tezduyar TE, Kuraishi T (2015) Multiscale ST methods for thermo-fluid analysis of a ground vehicle and its tires. Math Models Methods Appl Sci 25:2227–2255.  https://doi.org/10.1142/S0218202515400072 MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Takizawa K, Tezduyar TE, Mochizuki H, Hattori H, Mei S, Pan L, Montel K (2015) Space-time VMS method for flow computations with slip interfaces (ST-SI). Math Models Methods Appl Sci 25:2377–2406.  https://doi.org/10.1142/S0218202515400126 MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Takizawa K, Tezduyar TE, Otoguro Y (2018) Stabilization and discontinuity-capturing parameters for space-time flow computations with finite element and isogeometric discretizations. Comput Mech 62:1169–1186.  https://doi.org/10.1007/s00466-018-1557-x MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Otoguro Y, Takizawa K, Tezduyar TE, Nagaoka K, Mei S (2018) Turbocharger turbine and exhaust manifold flow computation with the space-time variational multiscale method and isogeometric analysis. Comput Fluids.  https://doi.org/10.1016/j.compfluid.2018.05.019 Google Scholar
  52. 52.
    Kuraishi T, Takizawa K, Tezduyar TE (2018) Tire aerodynamics with actual tire geometry, road contact and tire deformation. Comput Mech.  https://doi.org/10.1007/s00466-018-1642-1 Google Scholar
  53. 53.
    Hughes TJR (1995) Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles, and the origins of stabilized methods. Comput Methods Appl Mech Eng 127:387–401MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Keegan MH, Nash D, Stack M (2013) On erosion issues associated with the leading edge of wind turbine blades. J Phys D Appl Phys 46:383001CrossRefGoogle Scholar
  55. 55.
    Springer GS, Yang C-I, Larsen PS (1974) Analysis of rain erosion of coated materials. J Compos Mater 8:229–252CrossRefGoogle Scholar
  56. 56.
    Oka YI, Okamura K, Yoshida T (2005) Practical estimation of erosion damage caused by solid particle impact: part 1: effects of impact parameters on a predictive equation. Wear 259:95–101CrossRefGoogle Scholar
  57. 57.
    Castorrini A, Corsini A, Morabito F, Rispoli F, Venturini P (2017) Numerical simulation with adaptive boundary method for predicting time evolution of erosion processes. In: ASME turbo expo 2017: turbomachinery technical conference and exposition. American Society of Mechanical Engineers, V02DT48A019–V02DT48A019Google Scholar
  58. 58.
    Castorrini A, Venturini P, Corsini A, Rispoli F (2019) Numerical simulation of the blade aging process in an induced draft fan due to long time exposition to fly ash particles. J Eng Gas Turbines Power 141:011025CrossRefGoogle Scholar
  59. 59.
    Tezduyar T (2001) Finite element interface-tracking and interface-capturing techniques for flows with moving boundaries and interfaces. In: Proceedings of the ASME symposium on fluid-physics and heat transfer for macro- and micro-scale gas-liquid and phase-change flows (CD-ROM), ASME Paper IMECE2001/HTD-24206. ASME, New YorkGoogle Scholar
  60. 60.
    Tezduyar TE (2003) Stabilized finite element formulations and interface-tracking and interface-capturing techniques for incompressible flows. In: Hafez MM (ed) Numerical simulations of incompressible flows. World Scientific, Hackensack, pp 221–239CrossRefGoogle Scholar
  61. 61.
    Stein K, Tezduyar TE, Benney R (2004) Automatic mesh update with the solid-extension mesh moving technique. Comput Methods Appl Mech Eng 193:2019–2032.  https://doi.org/10.1016/j.cma.2003.12.046 CrossRefzbMATHGoogle Scholar
  62. 62.
    Tezduyar TE, Sathe S, Keedy R, Stein K (2006) Space-time finite element techniques for computation of fluid-structure interactions. Comput Methods Appl Mech Eng 195:2002–2027.  https://doi.org/10.1016/j.cma.2004.09.014 MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Bazilevs Y, Takizawa K, Tezduyar TE (2013) Computational fluid-structure interaction: methods and applications. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  64. 64.
    Tezduyar TE, Behr M, Mittal S, Johnson AA (1992) Computation of unsteady incompressible flows with the finite element methods: space–time formulations, iterative strategies and massively parallel implementations. In: New methods in transient analysis, PVP-Vol.246/AMD-Vol.143. ASME, New York, pp 7–24Google Scholar
  65. 65.
    Tezduyar T, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite-element computation of 3D flows. Computer 26:27–36.  https://doi.org/10.1109/2.237441 CrossRefzbMATHGoogle Scholar
  66. 66.
    Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comput Methods Appl Mech Eng 119:73–94.  https://doi.org/10.1016/0045-7825(94)00077-8 CrossRefzbMATHGoogle Scholar
  67. 67.
    Stein K, Tezduyar T, Benney R (2003) Mesh moving techniques for fluid-structure interactions with large displacements. J Appl Mech 70:58–63.  https://doi.org/10.1115/1.1530635 CrossRefzbMATHGoogle Scholar
  68. 68.
    Corsini A, Rispoli F (2005) Flow analyses in a high-pressure axial ventilation fan with a non-linear eddy viscosity closure. Int J Heat Fluid Flow 17:108–155Google Scholar
  69. 69.
    Craft TJ, Launder BE, Suga K (1996) Development and application of a cubic eddy-viscosity model of turbulence. Int J Heat Fluid Flow 17:108–155CrossRefGoogle Scholar
  70. 70.
    Lain S, Sommerfeld M (2003) Turbulence modulation in dispersed two-phase flow laden with solids from a lagrangian perspective. Int J Heat Fluid Flow 24:616–625CrossRefGoogle Scholar
  71. 71.
    Tezduyar TE, Takizawa K, Moorman C, Wright S, Christopher J (2010) Space-time finite element computation of complex fluid–structure interactions. Int J Numer Methods Fluids 64:1201–1218.  https://doi.org/10.1002/fld.2221 CrossRefzbMATHGoogle Scholar
  72. 72.
    Takizawa K, Tezduyar TE, Hattori H (2017) Computational analysis of flow-driven string dynamics in turbomachinery. Comput Fluids 142:109–117.  https://doi.org/10.1016/j.compfluid.2016.02.019 MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Komiya K, Kanai T, Otoguro Y, Kaneko M, Hirota K, Zhang Y, Takizawa K, Tezduyar TE, Nohmi M, Tsuneda T, Kawai M, Isono M (2018) Computational analysis of flow-driven string dynamics in a pump and residence time calculation. In: Proceedings of the 29th IAHR symposium on hydraulic machinery and systems, KyotoGoogle Scholar
  74. 74.
    Kanai T, Takizawa K, Tezduyar TE, Komiya K, Kaneko M, Hirota K, Nohmi M, Tsuneda T, Kawai M, Isono M (2019) Methods for computation of flow-driven string dynamics in a pump and residence time. Math Models Methods Appl Sci.  https://doi.org/10.1142/S021820251941001X Google Scholar
  75. 75.
    Baxter LL (1989) Turbulent transport of particles. Ph.D. thesis, Brigham Young UniversityGoogle Scholar
  76. 76.
    Wang LP (1990) On the dispersion of heavy particles by turbulent motion. Ph.D. thesis, Washington State UniversityGoogle Scholar
  77. 77.
    Litchford LJ, Jeng SM (1991) Efficient statistical transport model for turbulent particle dispersion in sprays. AIAA J 29:1443–1451CrossRefGoogle Scholar
  78. 78.
    Jain S (1995) Three-dimensional simulation of turbulent particle dispersion. Ph.D. thesis, University of UtahGoogle Scholar
  79. 79.
    Borello D, Venturini P, Rispoli F, Saavedra GZR (2013) Prediction of multiphase combustion and ash deposition within a biomass furnace. Appl Energy 101:413–422CrossRefGoogle Scholar
  80. 80.
    Venturini P, Borello D, Iossa CV, Lentini D, Rispoli F (2010) Modelling of multiphase combustion and deposit formation and deposit formation in a biomass-fed boiler. Energy 35:3008–3021CrossRefGoogle Scholar
  81. 81.
    Armenio V, Fiorotto V (2001) The importance of the forces acting on particles in turbulent flows. Phys Fluids 13:2437–2440CrossRefzbMATHGoogle Scholar
  82. 82.
    Schiller L, Naumann A (1933) Uber die grundlegenden berechnungen bei der schwekraftaubereitung. Zeitschrift des Vereines Deutscher Ingenieure 77:318–320Google Scholar
  83. 83.
    Smith PJ (1991) 3-D turbulent particle dispersion submodel development. Quarterly progress report, Department of Energy, Pittsburgh Energy Technology CenterGoogle Scholar
  84. 84.
    Corsini A, Rispoli F, Santoriello A (2005) A variational multiscale high-order finite element formulation for turbomachinery flow computations. Comput Methods Appl Mech Eng 194:4797–4823CrossRefzbMATHGoogle Scholar
  85. 85.
    Arjula S, Harsha A, Ghosh M (2008) Solid-particle erosion behavior of high-performance thermoplastic polymers. J Mater Sci 43:1757–1768CrossRefGoogle Scholar
  86. 86.
    Tezduyar TE, Takizawa K, Bazilevs Y (2017) Fluid-structure interaction and flows with moving boundaries and interfaces. In: Stein E, Borst RD, Hughes TJR (eds) Encyclopedia of computational mechanics second edition, part 2 fluids. Wiley, New YorkGoogle Scholar
  87. 87.
    Stein K, Tezduyar T (2002) Advanced mesh update techniques for problems involving large displacements. In: Proceedings of the fifth world congress on computational mechanics, Vienna (2002). Paper-ID: 81489. http://www.researchgate.net/publication/303737884/. Accessed 25 Mar 2019
  88. 88.
    Sareen A, Sapre CA, Selig MS (2014) Effects of leading edge erosion on wind turbine blade performance. Wind Energy 17:1531–1542CrossRefGoogle Scholar
  89. 89.
    Drela M, Youngren H (2008) Xfoil subsonic airfoil development system. Software Package. http://web.mit.edu/drela/Public/web/xfoil. Retrieved Feb 2011

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Alessio Castorrini
    • 1
    Email author
  • Alessandro Corsini
    • 1
  • Franco Rispoli
    • 1
  • Paolo Venturini
    • 1
  • Kenji Takizawa
    • 2
  • Tayfun E. Tezduyar
    • 3
    • 4
  1. 1.Dipartimento di Ingegneria Meccanica e AerospazialeSapienza University of RomeRomeItaly
  2. 2.Department of Modern Mechanical EngineeringWaseda UniversityShinjuku-kuJapan
  3. 3.Mechanical EngineeringRice University – MS 321HoustonUSA
  4. 4.Faculty of Science and EngineeringWaseda UniversityShinjuku-kuJapan

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