Skip to main content

Abstract

This paper presents details of the solver Aghora for the simulation of unsteady compressible turbulent flows. Different modelling levels are used: Reynolds averaged Navier-Stokes equations coupled with turbulence transport equations, variational multi-scale formulation of large-eddy simulation, and direct numerical simulation. The space discretization is based on a high-order discontinuous Galerkin method with representation of curved boundaries. High-order explicit and implicit Runge-Kutta methods are used for the time integration. The performance of the solver will be assessed in various examples of compressible turbulent flow numerical simulation in three space dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abgrall, R., Shu, C.-W.: Development of residual distribution schemes for the discontinuous Galerkin schemes: the scalar case with continuous elements, Commun. Comput. Phys. 5, 376–390 (2009)

    MathSciNet  Google Scholar 

  2. Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Apoung Kamga, J.-B., Després, B.: CFL condition and boundary conditions for DGM approximation of convection-diffusion. SIAM J. Numer. Anal. 44, 2245–2269 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baboulin, M., Buttari, A., Dongarra, J., Kurzak, J., Langou, J., Langou, J., Luszczek, P., Tomov, S.: Accelerating scientific computations with mixed precision algorithms. Computer Physics Communications 180, 2526–2533 (2009)

    Article  MATH  Google Scholar 

  5. Bassi, F., Botti, L., Colombo, A., Di Pietro, D.A., Tesini, P.: On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231, 45–65 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bijl, H., Carpenter, M.H., Vatsa, V.N., Kennedy, C.A.: Implicit time integration schemes for the unsteady compressible Navier-Stokes equations: laminar flow. J. Comput. Phys. 179, 313–329 (2002)

    Article  MATH  Google Scholar 

  7. Biswas, R., Devine, K., Flaherty, J.E.: Parallel adaptive finite element methods for conservation laws. Appl. Numer. Math. 14, 255–284 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brachet, M., Meiron, D., Orszag, S., Nickel, B., Morf, R., Frisch, U.: Small-scale structure of the TaylorGreen vortex. J. Fluid Mech. 130(41), 1452 (1983)

    Google Scholar 

  9. Brachet, M., Meneguzzi, M., Vincent, A., Politano, H., Sulem, P.: Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows. Phys. Fluids 4, 2845 (1992)

    Article  MATH  Google Scholar 

  10. Bassi, F., Crivellini, A., Rebay, S., Savini, M.: Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k − ω turbulence model equations. Comput. Fluids 34, 507–540 (2005)

    Article  MATH  Google Scholar 

  11. Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., Savini, M.: A High-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In: Decuypere, R., Dibelius, G. (eds.) Proceedings of the 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Antwerpen, Belgium (1997)

    Google Scholar 

  12. Cahen, J., Couaillier, V., Délery, J., Pot, T.: Validation of code using turbulence model applied to three-dimensional transonic channel. AIAA J 33, 671–679 (1995)

    Article  Google Scholar 

  13. Cant, R.S.: Fergus, a user guide. technical report. Cambridge University Engineering Department (1999)

    Google Scholar 

  14. Chapelier, J.-B., de la Llave Plata, M., Renac, F., Lamballais, E.: Evaluation of a high-order discontinuous Galerkin method for the DNS of turbulent flows. Comp. Fluids 95, 206–226 (2014)

    Article  Google Scholar 

  15. Chavent, G., Cockburn, B.: The local projection P0P1-discontinuous Galerkin finite element method for scalar conservative laws. M2AN Math. Model. Anal. Numer. 23, 565–592 (1989)

    MathSciNet  MATH  Google Scholar 

  16. Cockburn, B., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework. Math. Comp. 52, 411–435 (1989)

    MathSciNet  MATH  Google Scholar 

  17. Cockburn, B., Shu, C.W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Computing 16, 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Collis, S.S.: Discontinuous Galerkin methods for turbulence simulation. In: Proceedings of the Summer Program, CTR, pp. 155–167 (2002)

    Google Scholar 

  19. Dunham, J.: CFD Validation for Propulsion System Components. AGARD Advisory Report 355 (1998)

    Google Scholar 

  20. Gottlieb, D., Shu, C.W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39, 644–668 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guermond, J.-L., Pasquetti, R., Popov, B.: Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230, 4248–4267 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hartmann, R., Houston, P.: Adaptive Discontinuous Galerkin Finite Element Methods for the Compressible Euler Equations. J. Comput. Phys. 183, 508–532 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hartmann, R.: Higher-order and adaptive discontinuous Galerkin methods with shock-capturing applied to transonic turbulent delta wing flow. Int. J. Numer. Meth. Fluids. 72, 883–894 (2013)

    Article  Google Scholar 

  24. Houston, P., Süli, E.: hp-adaptive discontinuous Galerkin finite/ element methods for first-order hyperbolic problems. SIAM J. Sci. Comput. 23, 1226–1252 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hughes, T.J.R.: 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, 387–401 (1995)

    Article  MATH  Google Scholar 

  26. Hughes, T.J.R., Stewart, J.R.: A space-time formulation for multiscale phenomena. J. Comput. Appl. Math. 74, 217–229 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hughes, T.J.R., Mazzei, L., Oberai, A.A., Wray, A.A.: The multiscale formulation of large-eddy simulation: Decay of homogeneous isotropic turbulence. Phys. Fluids 13, 505–512 (2001)

    Article  Google Scholar 

  28. Hughes, T.J.R., Oberai, A.A., Mazzei, L.: Large-eddy simulation of turbulent channel flows by the variational multiscale method. Phys. Fluids 13, 1784–1799 (2001)

    Article  Google Scholar 

  29. Jin, H., Jespersen, D., Mehrotra, P., Biswas, R., Huang, L., Chapman, B.: High performance computing using MPI and OpenMP on multi-core parallel systems. Parallel Computing 37, 562–575 (2011)

    Article  Google Scholar 

  30. Jordan, P., Gervais, Y., Valire, J.-C., Foulon, H.: Final results from single point measurements, Project deliverable D3.4, JEAN EU 5th Framework Programme, G4RD-CT-2000-00313 (2002)

    Google Scholar 

  31. Kværnø, A.: Singly diagonally implicit Runge-Kutta methods with an explicit first stage. BIT Numerical Mathematics 44, 489–502 (2004)

    Article  MathSciNet  Google Scholar 

  32. Kroll, N., Bieler, H., Deconinck, H., Couaillier, V., van der Ven, H., Sørensen, K. (eds.): ADIGMA - A European Initiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications. NNFM, vol. 113. Springer, Heidelberg (2010)

    Google Scholar 

  33. Lavallée, P.-F., Wautelet, P.: Hybrid MPI-OpenMP Programming, CNRS IDRIS Lessons (2013)

    Google Scholar 

  34. Lesaint, P., Raviart, P.-A.: On a finite element method for solving the neutron transport equation. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–123. Academic Press, New York (1974)

    Chapter  Google Scholar 

  35. Lhner, R., Baum, J.D.: Handling tens of thousands of cores with industrial/legacy codes: Approaches, implementation and timings. Computers & Fluids 85, 53–62 (2013)

    Article  Google Scholar 

  36. Metais, O., Lesieur, M.: Spectral Large-Eddy Simulation of isotropic and stably stratified turbulence. J. Fluid Mech. 239, 157–194 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. Moser, R.D., Kim, J., Mansour, N.N.: Direct numerical simulation of turbulent channel flow up to Re τ  = 590. Phys. Fluids 11, 943 (1999)

    Article  MATH  Google Scholar 

  38. Nastase, C.R., Mavriplis, D.J.: High-order discontinuous Galerkin methods using an hp-multigrid approach. J. Comput. Phys. 213, 330–357 (2006)

    Article  MATH  Google Scholar 

  39. Pope, S.B.: Turbulent flows. Cambridge University Press (2000)

    Google Scholar 

  40. Ramakrishnan, S., Collis, S.S.: The Local Variational Multi-Scale Method for Turbulence Simulation. Sandia Report SAND2005-2733 (2005)

    Google Scholar 

  41. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, NM (1973)

    Google Scholar 

  42. Remacle, J.F., Flaherty, J.E., Shephard, M.S.: An Adaptive Discontinuous Galerkin Technique with an Orthogonal Basis Applied to Compressible Flow Problems. SIAM Review 45, 55–73 (2003)

    Article  MathSciNet  Google Scholar 

  43. Reid, L., Moore, R.D.: Design and overall performance of four highly loaded, high-speed inlet stages for an advanced high-pressure-ratio core compressor. NASA Technical Report 1337 (1978)

    Google Scholar 

  44. Smagorinsky, J.: General circulation experiments with the primitive equations: I. The basic equations. Mon. Weather Rev. 91, 99–164 (1963)

    Article  Google Scholar 

  45. Solin, P., Segeth, K., Dolezel, I.: Higher-order finite element methods. Chapman & Hall/CRC Press (2003)

    Google Scholar 

  46. Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469–491 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. Spalart, P.R., Allmaras, S.R.: A one-equation turbulence model for aerodynamic flows. La Rech. Aéro. 1, 5–21 (1994)

    Google Scholar 

  48. Saad, Y., Schultz, M.: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal Sci. Stat. Comput. 7, 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  49. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  50. Taylor, G., Green, A.: Mechanism of the production of small eddies from large ones. Proc. R. Soc. A 158(895), 499 (1937)

    Article  MATH  Google Scholar 

  51. Wilcox, D.C.: Reassessment of the Scale-Determining Equation for Advanced Turbulence Models. AIAA J 26, 1299–1310 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Renac, F., de la Llave Plata, M., Martin, E., Chapelier, J.B., Couaillier, V. (2015). Aghora: A High-Order DG Solver for Turbulent Flow Simulations. In: Kroll, N., Hirsch, C., Bassi, F., Johnston, C., Hillewaert, K. (eds) IDIHOM: Industrialization of High-Order Methods - A Top-Down Approach. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-319-12886-3_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-12886-3_15

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12885-6

  • Online ISBN: 978-3-319-12886-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics