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Unstructured Shock-Fitting Calculations of the Transonic Flow in a Gas Turbine Cascade

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Abstract

Even though shock-capturing techniques are the de-facto standard in the CFD simulation of turbo-machinery flows, the accurate estimation of shock-induced losses in transonic flows can be severely hindered by the numerical errors that are generated along a captured shock and convected downstream. Indeed, and despite their widespread use, shock-capturing techniques are known to be plagued by a number of drawbacks that are inherent to the numerical details of the shock-capturing process.

In recent works, the authors have developed a novel shock fitting technique for unstructured grids that has been applied to the computation of transonic, supersonic and hypersonic flows in both two and three space dimensions. In this paper, the proposed technique is applied to two-dimensional, transonic flows around an isolated profile and in a gas turbine cascade.

It is shown that the use of unstructured meshes allows to relieve most of the algorithmic difficulties that have contributed to the dismissal of the shock-fitting technique in the framework of structured meshes. Moreover, it is confirmed that, in contrast to shock-capturing, shock-fitting allows to obtain very accurate solution on coarse meshes.

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References

  1. D. Zaide and P. Roe, “Shock capturing anomalies and the jump conditions in one dimension,” in Fluid Dynamics and Co-located Conferences, American Institute of Aeronautics and Astronautics, June 2011. Paper 2011–3686.

  2. H. W. Emmons, “The numerical solution of compressible fluid flow problems,” NACA-TN 932, NASA, 1944.

  3. G. Moretti and M. Abbett, “A time-dependent computational method for blunt body flows,” AIAA Journal, vol. 4, no. 12, pp. 2136–2141, 1966.

    Article  Google Scholar 

  4. G. Moretti, “Shock-fitting analysis,” in Shock Fitting: Classical Techniques, Recent Developments, and Memoirs of Gino Moretti (M. Onofri and R. Paciorri, eds.), pp. 3–31, Cham: Springer International Publishing, 2017.

    Chapter  Google Scholar 

  5. F. Nasuti and M. Onofri, “Steady and unsteady shock interactions by shock-fitting approach,” in Shock Fitting: Classical Techniques, Recent Developments, and Memoirs of Gino Moretti (M. Onofri and R. Paciorri, eds.), pp. 33–55, Cham: Springer International Publishing, 2017.

    Chapter  Google Scholar 

  6. Y. Ma and X. Zhong, “Receptivity of a supersonic boundary layer over a flat plate. part 1. wave structures and interactions,” Journal of Fluid Mechanics, vol. 488, no. -1, pp. 31–78, 2003.

    Article  MathSciNet  Google Scholar 

  7. K. C. Hall and E. F. Crawley, “Calculation of unsteady flows in turbomachinery using the linearized euler equations,” AIAA Journal, vol. 27, pp. 777–787, June 1989.

    Article  Google Scholar 

  8. J. Xu and W. Ni, “Transonic cascade flow solved by the combined shock-capturing and shock-fitting method,” Journal of Propulsion and Power, vol. 5, pp. 476–481, July 1989.

    Article  Google Scholar 

  9. R. Paciorri and A. Bonfiglioli, “A shock-fitting technique for 2D unstructured grids,” Computers and Fluids, vol. 38, no. 3, pp. 715–726, 2009.

    Article  MathSciNet  Google Scholar 

  10. M. Ivanov, A. Bonfiglioli, R. Paciorri, and F. Sabetta, “Computation of weak steady shock reflections by means of an unstructured shock-fitting solver,” SHOCK WAVES, vol. 20(4), pp. 271–284, 2010.

    Article  Google Scholar 

  11. R. Paciorri and A. Bonfiglioli, “Shock interaction computations on unstructured, two-dimensional grids using a shock-fitting technique,” Journal of Computational Physics, vol. 230, no. 8, pp. 3155–3177, 2011.

    Article  MathSciNet  Google Scholar 

  12. A. Bonfiglioli, M. Grottadaurea, R. Paciorri, and F. Sabetta, “An unstructured, three-dimensional, shock-fitting solver for hypersonic flows,” Computers & Fluids, vol. 73, no. 0, pp. 162–174, 2013.

    Article  Google Scholar 

  13. A. Bonfiglioli, R. Paciorri, and L. Campoli, “Unsteady shock-fitting for unstructured grids,” International Journal for Numerical Methods in Fluids, vol. 81, no. 4, pp. 245–261, 2016.

    Article  MathSciNet  Google Scholar 

  14. A. Bonfiglioli, “Fluctuation splitting schemes for the compressible and incompressible euler and navier-stokes equations,” International Journal of Computational Fluid Dynamics, vol. 14, no. 1, pp. 21–39, 2000.

    Article  MathSciNet  Google Scholar 

  15. A. Bonfiglioli and R. Paciorri, “A mass-matrix formulation of unsteady fluctuation splitting schemes consistent with Roe’s parameter vector,” International Journal of Computational Fluid Dynamics, vol. 27, no. 4–5, pp. 210–227, 2013.

    Article  MathSciNet  Google Scholar 

  16. H. Deconinck, H. Paillère, R. Struijs, and P. Roe, “Multidimensional upwind schemes based on fluctuation-splitting for systems of conservation laws,” Computational Mechanics, vol. 11, no. 5/6, pp. 323–340, 1993.

    Article  Google Scholar 

  17. E. van der Weide, H. Deconinck, E. Issman, and G. Degrez, “A parallel, implicit, multi-dimensional upwind, residual distribution method for the Navier-Stokes equations on unstructured grids,” Computational Mechanics, vol. 23, pp. 199–208, 1999.

    Article  Google Scholar 

  18. R. Abgrall, “Residual distribution schemes: Current status and future trends,” Computers and Fluids, vol. 35, no. 7, pp. 641–669, 2006. Special Issue Dedicated to Professor Stanley G. Rubin on the Occasion of his 65th Birthday.

    Article  MathSciNet  Google Scholar 

  19. H. Deconinck, P. Roe, and R. Struijs, “A Multidimensional Generalization of Roe’s Flux Difference Splitter for the Euler Equations,” Computers and Fluids, vol. 22, no. 2/3, pp. 215–222, 1993.

    Article  MathSciNet  Google Scholar 

  20. P. L. Roe, “Approximate riemann solvers, parameter vectors and difference schemes.,” Journal of Computational Physics, vol. 43, pp. 357–372, 1981.

    Article  MathSciNet  Google Scholar 

  21. E. van der Weide, H. Deconinck, E. Issman, and G. Degrez, “A parallel, implicit, multi-dimensional upwind, residual distribution method for the navier-stokes equations on unstructured grids,” Computational Mechanics, vol. 23, pp. 199–208, 1999. https://doi.org/10.1007/s004660050401.

    Article  Google Scholar 

  22. R. Abgrall, “Towards the Ultimate Conservative Scheme: Following the Quest,” Journal of Computational Physics, vol. 167, pp. 277–315, March 2001.

    Article  MathSciNet  Google Scholar 

  23. D. W. Zaide and C. F. Ollivier-Gooch, “Inserting a curve into an existing two dimensional unstructured mesh,” in Proceedings of the 22nd International Meshing Roundtable, pp. 93–107, Springer, 2014.

  24. D. Zaide and C. Ollivier-Gooch, “Inserting a shock surface into an existing unstructured mesh,” in Shock Fitting: Classical Techniques, Recent Developments, and Memoirs of Gino Moretti (M. Onofri and R. Paciorri, eds.), pp. 151–169, Cham: Springer International Publishing, 2017.

    Chapter  Google Scholar 

  25. J. C. Vassberg and A. Jameson, “In pursuit of grid convergence for two-dimensional Euler solutions,” Journal of Aircraft, vol. 47, pp. 1152–1166, July 2010.

    Article  Google Scholar 

  26. J. Vassberg, “Drag Prediction Workshop repository.” on-line resource, 2009. https://doi.org/cmb24.larc.nasa.gov/outgoing/Vassberg-2D-NACA0012/.

  27. M. C. Hegedus, “Aero Troll NACA 0012 solutions on John Vassberg’s Grids.” [Online; last accessed 10-December-2018], 2011. https://doi.org/www.hegedusaero.com/examples/Vassberg/m080a0000-char-mom-dis010/surface-04097.dat.

  28. M. C. Hegedus, “Hegedus aerodynamics: Aero troll (v0.3.0b).” [Online; last accessed 10-December-2018], 2013. https://doi.org/www.hegedusaero.com/software.html.

  29. M. C. Hegedus, “Hegedus Aerodynamics: Vassberg’s NACA 0012 Grids.” [Online; last accessed 10-December-2018], 2011. https://doi.org/www.hegedusaero.com/examples/Vassberg/Vassberg.html.

  30. A. Dadone and G. Moretti., “Fast euler solver for transonic airfoils. I — theory. II — applications,” AIAA Journal, vol. 26, pp. 409–416, apr 1988.

    Article  MathSciNet  Google Scholar 

  31. J. Zierep, “New results for the normal shock in inviscid flow at a curved surface,” ZAMM — Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, vol. 83, no. 9, pp. 603–610, 2003.

    Article  MathSciNet  Google Scholar 

  32. R. Kiock, F. Lehthaus, N. C. Baines, and C. H. Sieverding, “The transonic flow through a plane turbine cascade as measured in four european wind tunnels,” Journal of Engineering for Gas Turbines and Power, vol. 108, pp. 277–284, Apr. 1986.

    Article  Google Scholar 

  33. A. Arnone, M.-S. Liou, and L. A. Povinelli, “Transonic cascade flow calculations using non-periodic c-type grids,” Technical Report 19910011758, NASA Lewis Research Center; Cleveland, OH, United States, 1991.

    MATH  Google Scholar 

  34. A. Arnone and R. C. Swanson, “A navier-stokes solver for turbomachinery applications,” Journal of Turbomachinery, vol. 115, pp. 305–313, Apr. 1993.

    Article  Google Scholar 

  35. G. Moretti, F. Marconi, and M. Onofri, “Shock-boundary layer interaction by shock fitting,” in Thirteenth International Conference on Numerical Methods in Fluid Dynamics, pp. 345–349, Springer, 1993.

  36. F. Nasuti and M. Onofri, “Analysis of unsteady supersonic viscous flows by a shock-fitting technique,” AIAA journal, vol. 34, no. 7, pp. 1428–1434, 1996.

    Article  Google Scholar 

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Authors and Affiliations

  1. Scuola di Ingegneria, Università degli Studi della Basilicata, Potenza, Italy

    A. Bonfiglioli

  2. Dip. di Ingegneria Meccanica e Aerospaziale, “Sapienza” — Università di Roma, Roma, Italy

    R. Paciorri

Authors
  1. A. Bonfiglioli
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  2. R. Paciorri
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Bonfiglioli, A., Paciorri, R. Unstructured Shock-Fitting Calculations of the Transonic Flow in a Gas Turbine Cascade. Aerotec. Missili Spaz. 97, 189–197 (2018). https://doi.org/10.1007/BF03406053

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