Skip to main content
Springer Nature Link
Log in

Role of Time Integration in Computing Transitional Flows Caused by Wall Excitation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Numerical investigation of receptivity and flow transition in spatio-temporal framework have shown the central role of spatio-temporal wave-front (STWF) created by wall excitation for transition of a two-dimensional (2D) zero pressure gradient boundary layer (ZPGBL) in Sengupta and Bhaumik (Phys Rev Lett 107:154501, 2011). Although the STWF is created by linear mechanism, it is the later nonlinear stage of evolution revealed by the solution of Navier–Stokes equation (NSE), which causes formation of turbulent spots merging together to create fully developed turbulent flow. Thus, computing STWF for ZPGBL from NSE is of prime importance, which has been reported by the present authors following earlier theoretical investigation. Similar computational efforts using NSE by other researchers do not report finding the STWF. In the present investigation we identify the main reason for other researchers to miss STWF, as due to taking a very short computational domain. Secondly, we show that even one takes a long enough domain and detect STWF, use of traditional low accuracy method will not produce the correct dynamics as reported by Sengupta and Bhaumik (2011). The role of time integration plays a very strong role in the dynamics of transitional flows. We have shown here that implicit methods are more error prone, as compared to explicit time integration methods during flow transition. For the present problem, it is noted that the classical Crank–Nicolson method is unstable for 2D NSE. Same error-prone nature will also be noted for hybrid implicit–explicit time integration methods (known as the IMEX methods). One of the main feature of present analysis is to highlight the accuracy of computations by compact schemes used by the present investigators over a significantly longer domain and over unlimited time, as opposed to those reported earlier in the literature for the wall excitation problem. A consequence of taking long streamwise domain enables one to detect special properties of STWF and its nonlinear growth. The main focus of the present research is to highlight the importance of STWF, which is a new class of spatio-temporal solution obtained from the linear receptivity by solving Orr–Sommerfeld equation and nonlinear analysis of NSE.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32, 797–823 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bhaumik, S.: Direct numerical simulation of inhomogeneous transitional and turbulent flows. Ph.D. Thesis, IIT-Kanpur (2013)

  3. Charney, J.G., Fjörtoft, R., von Neumann, J.: Numerical integration of barotropic vorticity equation. Tellus 2(4), 237–254 (1950)

    Article  MathSciNet  Google Scholar 

  4. Crank, J., Nicolson, P.A.: A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proc. Camb. Philos. Soc. 43(50), 50–67 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eiseman, P.R.: Grid generation for fluid mechanics computation. Annu. Rev. Fluid Mech. 17, 487–522 (1985)

    Article  Google Scholar 

  6. Fasel, H., Konzelmann, U.: Non-parallel stability of a flat-plate boundary layer using the complete Navier–Stokes equations. J. Fluid Mech. 221, 311–347 (1990)

    Article  MATH  Google Scholar 

  7. Fasel, H., Rist, U., Konzelmann, U.: Numerical investigation of three dimensional development in boundary layer transition. AIAA J. 28, 29–37 (1990)

    Article  MathSciNet  Google Scholar 

  8. Gaster, M., Grant, I.: An experimental investigation of the formation and development of a wave packet in a laminar boundary layer. Proc. Roy. Soc. Lond. Ser. A. 347, 253–269 (1975)

    Article  Google Scholar 

  9. Gaster, M., Sengupta, T.K.: The generation of disturbance in a boundary layer by wall perturbation: the vibrating ribbon revisited once more. In: Ashpis, D.E., Gatski, T.B., Hirsch, R. (eds.) Instabilities and Turbulence in Engineering Flows. Kluwer, Dordrecht (1993)

    Google Scholar 

  10. Giraldo, F.X., Kelly, J.F., Constantinescu, E.M.: Implicit-explicit formulations of a three dimensional nonhydrostatic unified model of the atmosphere (NUMA). SIAM J. Sci. Comp. (SISC) (in press)

  11. Hama, F.R., Nutant, J.: Detailed flow-field observations in the transition process in a thick boundary layer. In: Proceedings of Heat Transfer and Fluid Mechanics Institute Stanford University Press, pp. 77–93 (1963)

  12. Hirsch, R.S.: Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. J. Comput. Phys. 19, 90–109 (1975)

    Article  Google Scholar 

  13. Kanevsky, A., Carpenter, M.H., Gottlieb, D., Hesthaven, J.S.: Application of implicit-explicit high order Runge–Kutta methods to Discontinuous–Galerkin schemes. J. Comput. Phys. 225, 1753–1781 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier–Stokes equation. J. Comput. Phys. 58, 308–322 (1985)

    Article  MathSciNet  Google Scholar 

  15. Klebanoff, P.S., Tidstrom, K.D., Sargent, L.M.: The three dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1–34 (1962)

    Article  MATH  Google Scholar 

  16. Kovasznay, L.S.G., Komoda, H., Vasudeva, B.R.: Detailed flow-field in transition. In: Proceedings of Heat Transfer and Fluid Mechanics Institute Stanford University Press, pp. 1–26 (1962)

  17. Kreiss, H., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, 199–215 (1972)

    Article  MathSciNet  Google Scholar 

  18. Lomax, H., Pulliam, T.H., Zingg, D.W.: Fundamentals of CFD. Springer, Berlin (2002)

    Google Scholar 

  19. Persson, P.O.: High-order LES simulations using implicit-explicit Runge–Kutta schemes. In: Proceedings of the 49th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2011-684

  20. Pozrikidis, C.: Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press, Oxford (2011)

    MATH  Google Scholar 

  21. Rajpoot, M.K., Sengupta, T.K., Dutt, P.K.: Optimal time advancing dispersion relation preserving schemes. J. Comput. Phys. 229, 3623–3651 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  23. Sayadi, T.: Numerical simulation of controlled transition to developed turbulence in a zero-pressure gradient flat-plate boundary layer. Ph.D. Thesis, Department of Mechanics Engineering, Stanford University (2012)

  24. Sayadi, T., Hamman, C.W., Moin, P.: Direct numerical simulation of complete H-type and K-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech. 724, 480–509 (2013)

    Article  MATH  Google Scholar 

  25. Schubauer, G.B., Skramstad, H.K.: Laminar boundary layer oscillations and the stability of laminar flow. J. Aeronaut. Sci. 14(2), 69–78 (1947)

    Article  Google Scholar 

  26. Sengupta, T.K.: Impulse response of laminar boundary layer and receptivity. In: Taylor, C. (ed.) In: Proceedings of the 7th International Conference Numerical Methods for Laminar and Turbulent Layers (1991)

  27. Sengupta, T.K.: High Accuracy Computing Methods: Fluid flows and Wave Phenomena. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  28. Sengupta, T.K., Ballav, M., Nijhawan, S.: Generation of Tollmien–Schlichting waves by harmonic excitation. Phys. Fluids 6(3), 1213–1222 (1994)

    Article  MATH  Google Scholar 

  29. Sengupta, T.K., Bhaumik, S.: Onset of turbulence from the receptivity stage of fluid flows. Phys. Rev. Lett. 107, 154501 (2011)

    Article  Google Scholar 

  30. Sengupta, T.K., Bhaumik, S., Bhumkar, Y.G.: Direct numerical simulation of two-dimensional wall-bounded turbulent flows from receptivity stage. Phys. Rev. E 85, 026308 (2012)

    Article  Google Scholar 

  31. Sengupta, T.K., Bhaumik, S., Bose, R.: Direct numerical simulation of transitional mixed convection flows: viscous and inviscid instability mechanisms. Phys. Fluids 25, 094102 (2013)

    Article  Google Scholar 

  32. Sengupta, T.K., Bhumkar, Y., Rajpoot, M.K., Suman, V.K., Saurabh, S.: Spurious waves in discrete computation of wave phenomena and flow problems. Appl. Math. Comput. 218, 9035–9065 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sengupta, T.K., De, S., Sarkar, S.: Vortex-induced instability of an incompressible wall-bounded shear layer. J. Fluid Mech. 493, 277–286 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sengupta, T.K., Dipankar, A.: A comparative study of time advancement methods for solving Navier–Stokes equations. J. Sci. Comput. 21, 225–250 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sengupta, T.K., Dipankar, A., Sagaut, P.: Error dynamics: beyond von Neumann analysis. J. Comput. Phys. 226, 1211–1218 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sengupta, T.K., Rajpoot, M.K., Bhumkar, Y.G.: Space-time discretizing optimal DRP schemes for flow and wave propagation problems. Comput. Fluids 47(1), 144–154 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sengupta, T.K., Rao, A.K., Venkatasubbaiah, K.: Spatio-temporal growing wave-fronts in spatially stable boundary layers. Phys. Rev. Lett. 96, 224,504(1)–224,504(4) (2006)

    Article  Google Scholar 

  38. Sengupta, T.K., Rao, A.K., Venkatasubbaiah, K.: Spatio-temporal growth of disturbances in a boundary layer and energy based receptivity analysis. Phys. Fluids 18, 094,101(1)–094,101(9) (2006)

    Article  MathSciNet  Google Scholar 

  39. Swartz, B., Wendroff, B.: The relative efficiency of finite-difference and finite element methods. I: hyperbolic problems and splines. SIAM J. Numer. Anal. 11(5), 979–993 (1974)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. High Performance Computing Laboratory, Department of Aerospace Engineering, I. I. T. Kanpur, Kanpur, 208 016, India

    Tapan K. Sengupta, V. K. Sathyanarayanan, M. Sriramkrishnan & Akhil Mulloth

Authors
  1. Tapan K. Sengupta
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. V. K. Sathyanarayanan
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. M. Sriramkrishnan
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Akhil Mulloth
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Tapan K. Sengupta.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sengupta, T.K., Sathyanarayanan, V.K., Sriramkrishnan, M. et al. Role of Time Integration in Computing Transitional Flows Caused by Wall Excitation. J Sci Comput 65, 224–248 (2015). https://doi.org/10.1007/s10915-014-9967-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9967-1

Keywords