DOI QR코드

DOI QR Code

A hybrid numerical flux for supersonic flows with application to rocket nozzles

  • Received : 2019.12.30
  • Accepted : 2020.04.30
  • Published : 2020.09.25

Abstract

The numerical simulation of shock waves in supersonic flows is challenging because of several instabilities which can affect the solution. Among them, the carbuncle phenomenon can introduce nonphysical perturbations in captured shock waves. In the present work, a hybrid numerical flux is proposed for the evaluation of the convective fluxes that avoids carbuncle and keeps high-accuracy on shocks and boundary layers. In particular, the proposed flux is a combination between an upwind approximate Riemann problem solver and the Local Lax-Friedrichs scheme. A simple strategy to mix the two fluxes is proposed and tested in the framework of a discontinuous Galerkin discretisation. The approach is investigated on the subsonic flow in a channel, on the supersonic flow around a cylinder, on the supersonic flow on a flat plate and on the flow in a overexpanded rocket nozzle.

Keywords

Acknowledgement

Computational resources were provided by HPC@POLITO, a project of Academic Computing within the Department of Control and Computer Engineering at the Politecnico di Torino (http://www.hpc.polito.it).

References

  1. Ampellio, E., Bertini, F., Ferrero, A., Larocca, F. and Vassio, L. (2016), "Turbomachinery design by a swarm-based optimization method coupled with a CFD solver", Adv. Aircraft Spacecraft Sci., 3(2), 149, https://doi.org/10.12989/aas.2016.3.2.149.
  2. Bassi, F., Botti, L., Colombo, A., Di Pietro, D.A. and Tesini, P. (2012), "On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations", J. Comput. Phys., 231(1), 45-65, https://doi.org/10.1016/j.jcp.2011.08.018.
  3. Bassi, F., Cecchi, F., Franchina, N., Rebay, S. and Savini, M. (2011), "High-order discontinuous Galerkin computation of axisymmetric transonic flows in safety relief valves", Comput. Fluids, 49(1), 203-213, https://doi.org/10.1016/j.compfluid.2011.05.015.
  4. Burbeau, A. and Sagaut, P. (2005), "A dynamic p-adaptive discontinuous Galerkin method for viscous flow with shocks", Comput. Fluids, 34(4-5), 401-417, https://doi.org/10.1016/j.compfluid.2003.04.002.
  5. Chalmers, N., Agbaglah, G., Chrust, M. and Mavriplis, C. (2019), "A parallel hp-adaptive high order discontinuous Galerkin method for the incompressible Navier-Stokes equations", J. Comput. Phys. X, 2, 100023, https://doi.org/10.1016/j.jcpx.2019.100023.
  6. Eskilsson, C. (2011), "An hp-adaptive discontinuous Galerkin method for shallow water flows", Int. J. Numer. Methods Fluids, 67(11), 1605-1623, https://doi.org/10.1002/d.2434.
  7. Ferrero, A. and D'Ambrosio, D. (2020), "An Hybrid Numerical Flux for Supersonic Flows with Application to Rocket Nozzles", in "17-th International Conference of Numerical Analysis and Applied Mathematics", AIP Conference Proceedings.
  8. Ferrero, A. and Larocca, F. (2016), "Feedback filtering in discontinuous Galerkin methods for Euler equations", Prog. Comput. Fluid Dyn., 16(1), 14-25, https://doi.org/10.1504/PCFD.2016.074221.
  9. Ferrero, A. and Larocca, F. (2017), "Adaptive CFD schemes for aerospace propulsion", in "J. Phys. Conf. Ser.", volume 841, page 012017, IOP Publishing, https://doi.org/10.1088/1742-6596/841/1/012017.
  10. Ferrero, A., Larocca, F. and Bernaschek, V. (2017), "Unstructured discretisation of a nonlocal transition model for turbomachinery flows", Adv. Aircraft Spacecraft Sci., 4(5), 555-571, https://doi.org/10.12989/aas.2017.4.5.555.
  11. Ferrero, A., Larocca, F. and Puppo, G. (2015), "A robust and adaptive recovery-based discontinuous Galerkin method for the numerical solution of convection{diffusion equations", Int. J. Numer. Methods Fluids, 77(2), 63-91, https://doi.org/10.1002/fld.3972.
  12. Geuzaine, C. and Remacle, J.F. (2009), "Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities", Int. J. Numer. Methods Eng., 79(11), 1309-1331, https://doi.org/10.1002/nme.2579.
  13. Giorgiani, G., Fernandez-Mendez, S. and Huerta, A. (2013), "Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems", Int. J. Numer. Methods Fluids, 72(12), 1244-1262, https://doi.org/10.1002/d.3784.
  14. Godunov, S.K. (1959), "A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations", Math. Sbornik, 47, 271-306.
  15. Guo, S. and Tao, W.Q. (2018), "A hybrid flux splitting method for compressible flow", Numer. Heat Tr. B-Fund., 73(1), 33-47, https://doi.org/10.1080/10407790.2017.1420315.
  16. Hartmann, R. and Houston, P. (2002), "Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations", J. Comput. Phys., 183(2), 508-532, https://doi.org/10.1006/jcph.2002.7206.
  17. Hu, L.J. and Yuan, L. (2014), "A Robust Hybrid HLLC-FORCE Scheme for Curing Numerical Shock Instability", in "Applied Mechanics and Materials", volume 577, pages 749-753, Trans Tech Publ, https://doi.org/10.4028/www.scientific.net/AMM.577.749.
  18. Jaisankar, S. and Sheshadri, T. (2013), "Directional Diffusion Regulator (DDR) for some numerical solvers of hyperbolic conservation laws", J. Comput. Phys., 233, 83-99, https://doi.org/10.1016/j.jcp.2012.07.031.
  19. Kroll, N., Bieler, H., Deconinck, H., Couaillier, V., van der Ven, H. and Sorensen, K. (2010), ADIGMA-A European Initiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications: Results of a Collaborative Research Project Funded by the European Union, 2006-2009, volume 113, Springer Science & Business Media, https://doi.org/10.1007/978-3-642-03707-8.
  20. Kroll, N., Hirsch, C., Bassi, F., Johnston, C. and Hillewaert, K. (2015), IDIHOM: Industrialization of High-Order Methods-A Top-Down Approach: Results of a Collaborative Research Project Funded by the European Union, 2010-2014, volume 128, Springer, https://doi.org/10.1007/978-3-319-12886-3.
  21. Liou, M.S. (1996), "A sequel to ausm: Ausm+", J. Comput. Phys., 129(2), 364-382, https://doi.org/10.1006/jcph.1996.0256.
  22. Liu, X.D., Osher, S. and Chan, T. (1994), "Weighted essentially non-oscillatory schemes", J. Comput. Phys., 115(1), 200-212, https://doi.org/10.1006/jcph.1994.1187.
  23. Nishikawa, H. and Kitamura, K. (2008), "Very simple, carbuncle-free, boundary-layerresolving, rotated-hybrid Riemann solvers", J. Comput. Phys., 227(4), 2560-2581, https://doi.org/10.1016/j.jcp.2007.11.003.
  24. Osher, S. and Solomon, F. (1982), "Upwind difference schemes for hyperbolic systems of conservation laws", Math. Comput., 38(158), 339-374, https://doi.org/10.1090/S0025-5718-1982-0645656-0.
  25. Pandolfi, M. (1984), "A contribution to the numerical prediction of unsteady flows", AIAA J., 22(5), 602-610, https://doi.org/10.2514/3.48491.
  26. Pandolfi, M. and D'Ambrosio, D. (2001), "Numerical instabilities in upwind methods: analysis and cures for the carbuncle phenomenon", J. Comput. Phys., 166(2), 271-301, https://doi.org/10.1006/jcph.2000.6652.
  27. Pandolfi, M. and D'Ambrosio, D. (2002), "Performances of upwind methods in predicting shear-like flows", Comput. Fluids, 31(4-7), 725-744, https://doi.org/10.1016/S0045-7930(01)00071-8.
  28. Remacle, J.F., Flaherty, J.E. and Shephard, M.S. (2003), "An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible ow problems", SIAM Rev., 45(1), 53-72, https://doi.org/10.1137/S00361445023830.
  29. Roe, P.L. (1981), "Approximate Riemann solvers, parameter vectors, and difference schemes", J. Comput. Phys., 43(2), 357-372, https://doi.org/10.1016/0021-9991(81)90128-5.
  30. Rusanov, V.V. (1962), "The calculation of the interaction of non-stationary shock waves and obstacles", USSR Comput. Math. Math. Phys., 1(2), 304-320, https://doi.org/10.1016/0041-5553(62)90062-9.
  31. Van Leer, B. (1997), "Flux-vector splitting for the Euler equation", in "Upwind and High-Resolution Schemes", pages 80-89, Springer, https://doi.org/10.1007/978-3-642-60543-7 5.
  32. Wang, D., Deng, X., Wang, G. and Dong, Y. (2016), "Developing a hybrid flux function suitable for hypersonic ow simulation with high-order methods", Int. J. Numer. Methods Fluids, 81(5), 309-327, https://doi.org/10.1002/fld.4186.
  33. Wang, Z.J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H.T. et al. (2013), "High-order CFD methods: current status and perspective", Int. J. Numer. Methods Fluids, 72(8), 811-845, https://doi.org/10.1002/fld.3767.
  34. Zenoni, G., Leicht, T., Colombo, A. and Botti, L. (2017), "An agglomeration-based adaptive discontinuous Galerkin method for compressible flows", Int. J. Numer. Methods Fluids, 85(8), 465-483, https://doi.org/10.1002/fld.4390.