Journal of Scientific Computing

, Volume 26, Issue 2, pp 151–193 | Cite as

High Accuracy Schemes for DNS and Acoustics

  • T. K. SenguptaEmail author
  • S. K. Sircar
  • A. Dipankar


High-accuracy schemes have been proposed here to solve computational acoustics and DNS problems. This is made possible for spatial discretization by optimizing explicit and compact differencing procedures that minimize numerical error in the spectral plane. While zero-diffusion nine point explicit scheme has been proposed for the interior, additional high accuracy one-sided stencils have also been developed for ghost cells near the boundary. A new compact scheme has also been proposed for non-periodic problems—obtained by using multivariate optimization technique. Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise and that rules out dissipation that is otherwise introduced via spatial and temporal discretizations. Acoustics problems are wave propagation problems and hence require Dispersion Relation Preservation (DRP) schemes that simultaneously meet high accuracy requirements and keeping numerical and physical dispersion relation identical. Emphasis is on high accuracy than high order for both DNS and acoustics. While higher order implies higher accuracy for spatial discretization, it is shown here not to be the same for time discretization. Specifically it is shown that the 2nd order accurate Adams-Bashforth (AB)—scheme produces unphysical results compared to first order accurate Euler scheme. This occurs, as the AB-scheme introduces a spurious computational mode in addition to the physical mode that apportions to itself a significant part of the initial condition that is subsequently heavily damped. Additionally, AB-scheme has poor DRP property making it a poor method for DNS and acoustics. These issues are highlighted here with the help of a solution for (a) Navier–Stokes equation for the temporal instability problem of flow past a rotating cylinder and (b) the inviscid response of a fluid dynamical system excited by simultaneous application of acoustic, vortical and entropic pulses in an uniform flow. The last problem admits analytic solution for small amplitude pulses and can be used to calibrate different methods for the treatment of non-reflecting boundary conditions as well.


Computational acoustics DNS compact difference schemes dispersion relation preservation schemes time discretization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lighthill, M.J. 1993Computational AeroacousticsSpringer VerlagNew YorkGoogle Scholar
  2. 2.
    Hirsch, R.S. 1975J. Comput. Phys1990Google Scholar
  3. 3.
    Adam, Y. 1977J. Comput. Phys.2419MathSciNetCrossRefGoogle Scholar
  4. 4.
    Lele, S.K. 1992J. Comput. Phys.10316zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Zhong, X. 1998J. Comput. Phys.144662zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Adams, N.A., Shariff, K. 1996J. Comput. Phys.12727MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sengupta, T.K., Ganeriwal, G., De, S. 2003J. Comp. Phys.192677Google Scholar
  8. 8.
    Visbal, M.R., Gaitonde, D.V. 1999AIAA J.371231Google Scholar
  9. 9.
    Tam, C.K.W., Webb, J.C. 1993J. Comput. Phys.107262MathSciNetCrossRefGoogle Scholar
  10. 10.
    Haras, Z., Ta’asan, S. 1994J. Comput. Phys.114265MathSciNetCrossRefGoogle Scholar
  11. 11.
    Oran, E.S., Boris, J.P. 2001Numerical Simulation of Reactive Flows2Cambridge University PressCambridge, U.KGoogle Scholar
  12. 12.
    Durran, D.R. 1999Numerical Methods for Wave Equation in Geophysical Fluid DynamicsSpringer VerlagNew YorkGoogle Scholar
  13. 13.
    Haltiner, G.J., Williams, R.T. 1980Numerical Prediction and Dynamic MeteorologyWileyNew YorkGoogle Scholar
  14. 14.
    Lilly, D.K. 1965Monthly Weather Review13811Google Scholar
  15. 15.
    Kim, J., Moin, P. 1985J. Comput. Phys.58308MathSciNetGoogle Scholar
  16. 16.
    Kawamura, H., Abe, H., Matsuo, Y. 1999Int. J. Heat Fluid Flow.20196CrossRefGoogle Scholar
  17. 17.
    Maass, C. and Schumann, U. (1994). In Direct and Large Eddy Simulation I, (Voke, Kleiser and Chollet), 287.Google Scholar
  18. 18.
    Boersma, B.J., Brethonwer, G., Nieuwstadt, F.T.M. 1998Phys. Fluids10899CrossRefGoogle Scholar
  19. 19.
    Serre, E., Bontoux, P., Kotorba, R. 2001Int. J. Fluid Dynamics517Google Scholar
  20. 20.
    Shi, J., Thomas, T.G., Williams, J.J.R. 2000J. Hydraulics Res.38465Google Scholar
  21. 21.
    Karamanos, G.S., Karniadakis, G.E. 2000J. Comput. Phys.16322MathSciNetCrossRefGoogle Scholar
  22. 22.
    Clerex, H.J.H. 1997J. Comput. Phys.136186Google Scholar
  23. 23.
    Box, G.E.P., Draper, N.R. 1969Evolutionary OperationWileyNew YorkGoogle Scholar
  24. 24.
    Sengupta, T.K., Nair, M.T. 1999Int. J. Numer. Meth. Fluids31879CrossRefGoogle Scholar
  25. 25.
    Werle, H. 1984La Recherche Aérospatiale1984-439Google Scholar
  26. 26.
    Sengupta, T.K., Kasliwal, A., De, S., Nair, M.T. 2003J Fluids Struct.17941CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringI.I.T.KanpurIndia
  2. 2.Department of MathematicsI.I.T.KanpurIndia

Personalised recommendations