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An efficient parallel algorithm for DNS of buoyancy-driven turbulent flows

  • Special Column on 30th NCHD
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Abstract

In this paper, an explicit low-storage simplified M — stage Runge-Kutta (SRK) scheme for high Reynolds-number incompressible flows is presented. In the SRK scheme, the Poisson equation is solved only once in the final substage of each time step. By taking advantage of the SRK scheme and the advanced hybrid MPI+MPI model, we have developed an efficient parallel solver for buoyancy-driven turbulent flow. The spatial and temporal accuracies of the solver are validated with Taylor-Green vortex flow. Both the RK and SRK schemes are implemented for the simulation of turbulent Rayleigh-Bénard convection as well as Rayleigh-Taylor flow. The results show that the SRK scheme can save approximately 20% of the computation time.

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References

  1. Chorin A. J. Numerical solution of the Navier-Stokes equations [J]. Mathematics of Computation, 1968, 22(104): 745–762.

    Article  MathSciNet  MATH  Google Scholar 

  2. Chorin A. J. On the convergence of discrete approximations to the Navier-Stokes equations [J]. Mathematics of Computation, 1969, 23(106): 342–353.

    Article  MathSciNet  Google Scholar 

  3. Guermond J. L., Minev P., Shen J. An overview of projection methods for incompressible flows [J]. Computer Methods in Applied Mechanics and Engineering, 2006, 195(44-47): 6011–6045.

    Article  MathSciNet  MATH  Google Scholar 

  4. Perot J. B. An analysis of the fractional step method [J]. Journal of Computational Physics, 1993, 108(1): 51–58.

    Article  MathSciNet  MATH  Google Scholar 

  5. Kim J., Moin P. Application of a fractional-step method to incompressible Navier-Stokes equations [J]. Journal of Computational Physics, 1985, 59(2): 308–323.

    Article  MathSciNet  MATH  Google Scholar 

  6. Burton T. M., Eaton J. K. Analysis of a fractional-step method on overset grids [J]. Journal of Computational Physics, 2002, 177(2): 336–364.

    Article  MATH  Google Scholar 

  7. Sanderse B., Koren B. Accuracy analysis of explicit Runge-Kutta methods applied to the incompressible Navier-Stokes equations [J]. Journal of Computational Physics, 2012, 231(8): 3041–3063.

    Article  MathSciNet  MATH  Google Scholar 

  8. Meyers J., Sagaut P. Is plane-channel flow a friendly case for the testing of large-eddy simulation subgrid-scale models? [J]. Physics of Fluids, 2007, 19(4): 048105.

    Article  MATH  Google Scholar 

  9. Capuano F., Coppola G., Rández L. et al. Explicit Runge-Kutta schemes for incompressible flow with improved energy-conservation properties [J]. Journal of Computational Physics, 2017, 328: 86–94.

    Article  MathSciNet  MATH  Google Scholar 

  10. Le H., Moin P. An improvement of fractional-step methods for the incompressible Navier-Stokes equations [J]. Journal of Computational Physics, 1990, 89(1): 253.

    Article  Google Scholar 

  11. Ni M. J., Komori S., Morley N. Projection methods for the calculation of incompressible unsteady flows [J]. Numerical Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, 2003, 44(6): 533–551.

    Article  Google Scholar 

  12. Rabenseifner R., Hager G., Jost G. Hybrid MPI and OpenMP Parallel Programming: MPI+OpenMP and other models on clusters of SMP nodes [EB/OL]. 2013, https://www.openmp.org/press-release/sc13-tutorial-hybrid-mpi-openmp-parallel-programming.

  13. Hoefler T., Dinan J., Buntinas D. et al. MPI + MPI: A new hybrid approach to parallel programming with MPI plus shared memory [J]. Computing, 2013, 95(12): 1121–1136.

    Article  Google Scholar 

  14. Zhu X., Zhang J., Yoshii K. et al. Analyzing MPI-3.0 process-level shared memory: A case study with stencil computations [C]. 2015 15th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing, Shenzhen, China, 2015, 1099–1106.

  15. Vreman A. W. The projection method for the incompressible Navier-Stokes equations: The pressure near a no-slip wall [J]. Journal of Computational Physics, 2014, 263: 353–374.

    Article  MathSciNet  MATH  Google Scholar 

  16. JAMESON A. Successes and challenges in computational aerodynamics [C]. American Institute of Aeronautics and Astronautics 8th Computational Fluid Dynamics Conference, Honolulu, HI, USA, 1987.

  17. Van der Poel E. P., Ostilla-Mónico R., Donners J. et al. A pencil distributed finite difference code for strongly turbulent wall-bounded flows [J]. Computers and Fluids, 2015, 116: 10–16.

    Article  MathSciNet  MATH  Google Scholar 

  18. Roache P. J. Verification and validation in computational science and engineering [M]. Albuquerque, USA: N. M. Hermosa publishers, 1998.

    Google Scholar 

  19. Oberkampf W. L., Trucano T. G. Verification and validation in computational fluid dynamics [J]. Progress in Aerospace Sciences, 2002, 38(3): 209–272.

    Article  Google Scholar 

  20. Zhang J., Jackson T. L. A high-order incompressible flow solver with WENO [J]. Journal of Computational Physics, 2009, 228(7): 2426–2442.

    Article  MATH  Google Scholar 

  21. Harlow F. H., Welch J. E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface [J]. Physics of Fluids, 1965, 8(12): 2182.

    Article  MathSciNet  MATH  Google Scholar 

  22. Ahlers G., Grossmann S., Lohse D. Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection [J]. Reviews of Modern Physics, 2009, 81(2): 503–537.

    Article  Google Scholar 

  23. Lohse D., Xia K. Q. Small-scale properties of turbulent Rayleigh-Bénard convection [J]. Annual Review of Fluid Mechanics, 2010, 42(1): 335–364.

    Article  MATH  Google Scholar 

  24. Chillà F., Schumacher J. New perspectives in turbulent Rayleigh-Bénard convection [J]. The European physical Journal E, 2012, 35(7): 58.

    Article  Google Scholar 

  25. SUN C., Zhou Q. Experimental techniques for turbulent Taylor-Couette flow and Rayleigh-Bénard convection [J]. Nonlinearity, 2014, 27(9): R89–R121.

    Article  MATH  Google Scholar 

  26. Zhang Y. Z., SUN C., Bao Y. et al. How surface roughness reduces heat transport for small roughness heights in turbulent Rayleigh-Bénard convection [J]. Journal of Fluid Mechanics, 2018, 836: R2.

    Article  MATH  Google Scholar 

  27. Krug D., Zhu X., Chung D. et al. Transition to ultimate Rayleigh-Bénard turbulence revealed through extended self-similarity scaling analysis of the temperature structure functions [J]. Journal of Fluid Mechanics, 2018, 851: R3.

    Article  Google Scholar 

  28. Wang Y., Xu W., He X. et al. Boundary layer fluctuations in turbulent Rayleigh-Bénard convection [J]. Journal of Fluid Mechanics, 2018, 840: 408–431.

    Article  MATH  Google Scholar 

  29. Doering C. R. Thermal forcing and ‘classical’ and ‘ultimate’ regimes of Rayleigh-Bénard convection [J]. Journal of Fluid Mechanics, 2019, 868: 1–4.

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhu X., Jiang L. F., Zhou Q. et al. Turbulent Rayleigh-Bénard convection in an annular cell [J]. Journal of Fluid Mechanics, 2019, 869: R5.

    Article  MATH  Google Scholar 

  31. Wang Q., Zhou Q., Wan Z. H. et al. Penetrative turbulent Rayleigh-Bénard convection in two and three dimensions [J]. Journal of Fluid Mechanics, 2019, 870: 718–734.

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang Y., He X., Tong P. Turbulent temperature fluctuations in a closed Rayleigh-Bénard convection cell [J]. Journal of Fluid Mechanics, 2019, 874: 263–284.

    Article  MathSciNet  MATH  Google Scholar 

  33. Shishkina O., Stevens R. J. A. M., Grossmann S. et al. Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution [J]. New Journal of Physics, 2010, 12(7): 75022.

    Article  Google Scholar 

  34. Huang Y. X., Zhou Q. Counter-gradient heat transport in two-dimensional turbulent Rayleigh-Bénard convection [J]. Journal of Fluid Mechanics, 2013, 737: R3.

    Article  MATH  Google Scholar 

  35. Van der Poel E. P., Stevens R. J. A. M., Lohse D. Comparison between two- and three-dimensional Rayleigh-Bénard convection [J]. Journal of Fluid Mechanics, 2013, 736: 177–194.

    Article  MATH  Google Scholar 

  36. Zhou Q. Temporal evolution and scaling of mixing in two-dimensional Rayleigh-Taylor turbulence [J]. Physics of Fluids, 2013, 25(8): 85107.

    Article  Google Scholar 

  37. Zhou Q., Jiang L. F. Kinetic and thermal energy dissipation rates in two-dimensional Rayleigh-Taylor turbulence [J]. Physics of Fluids, 2016, 28(4): 045109.

    Article  Google Scholar 

  38. Boffetta G., Mazzino A. Incompressible Rayleigh-Taylor Turbulence [J]. Annual Review of Fluid Mechanics, 2017, 49(1): 119–143.

    Article  MathSciNet  MATH  Google Scholar 

  39. Pillai D. S., Narayanan R. Rayleigh-Taylor stability in an evaporating binary mixture [J]. Journal of Fluid Mechanics, 2018, 848: R1.

    Article  MathSciNet  MATH  Google Scholar 

  40. Scase M. M., Hill R. J. A. Centrifugally forced Rayleigh-Taylor instability [J]. Journal of Fluid Mechanics, 2018, 852: 543–577.

    Article  MathSciNet  MATH  Google Scholar 

  41. Kord A., Capecelatro J. Optimal perturbations for controlling the growth of a Rayleigh-Taylor instability [J]. Journal of Fluid Mechanics, 2019, 876: 150–185.

    Article  MathSciNet  MATH  Google Scholar 

  42. Huneault J., Plant D., Higgins A. J. Rotational stabilisation of the Rayleigh-Taylor instability at the inner surface of an imploding liquid shell [J]. Journal of Fluid Mechanics, 2019, 873: 531–567.

    Article  MathSciNet  MATH  Google Scholar 

  43. Dalziel S. B., Linden P. F., Youngs D. L. Self- similarity and internal structure of turbulence induced by Rayleigh-Taylor instability [J]. Journal of Fluid Mechanics, 1999, 399: 1–48.

    Article  MathSciNet  MATH  Google Scholar 

  44. Boffetta G., De Lillo F., Mazzino A. et al. Bolgiano scale in confined Rayleigh-Taylor turbulence [J]. Journal of Fluid Mechanics, 2012, 690: 426–440.

    Article  MATH  Google Scholar 

  45. Boffetta G., De Lillo F., Mazzino A. et al. The ultimate state of thermal convection in Rayleigh-Taylor turbulence [J]. Physica D: Nonlinear Phenomena, 2012, 241(3): 137–140.

    Article  MathSciNet  Google Scholar 

  46. Boffetta G., Mazzino A., Musacchio S. et al. Statistics of mixing in three-dimensional Rayleigh-Taylor turbulence at low Atwood number and Prandtl number one [J]. Physics of Fluids, 2010, 22(3): 035109.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200072, China

    Yi-zhao Zhang, Shu-ning Xia, Yu-hong Dong, Bo-fu Wang & Quan Zhou

Authors
  1. Yi-zhao Zhang
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  2. Shu-ning Xia
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  3. Yu-hong Dong
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  4. Bo-fu Wang
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  5. Quan Zhou
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Corresponding author

Correspondence to Quan Zhou.

Additional information

Project supported by the Natural Science Foundation of China (Grant Nos. 11825204, 11972220, 91852202, 11732010 and 91852111), the Key Research Projects of Shanghai Science and Technology Commission (Grant No. 18010500500), the Program of Shanghai Academic Research Leader (Grant No. 19XD1421400) and the Program of Shanghai Municipal Education Commission (Grant No. 2019-01-07-00-09-E00018).

Biography: Yi-zhao Zhang (1990-), Male, Ph. D.

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Zhang, Yz., Xia, Sn., Dong, Yh. et al. An efficient parallel algorithm for DNS of buoyancy-driven turbulent flows. J Hydrodyn 31, 1159–1169 (2019). https://doi.org/10.1007/s42241-019-0090-5

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  • DOI: https://doi.org/10.1007/s42241-019-0090-5

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