Receptivity of flow in a square lid driven cavity to implicit numerical excitations for high-fidelity HPC

Tapan K. Sengupta, Abhinav Prakash, Aditi Sengupta, Suman K. Vajjala

Abstract

This study aims to investigate the effects of implicit numerical excitation on the receptivity of flow inside a square lid-driven cavity (LDC) leading to bifurcation and instability for a fixed (257 × 257) grid with different temporal resolutions via the solution of the Navier-Stokes equation. Computational results have been provided showing the flow dynamics of the LDC problem as explained with a time series at a representative point near the top corner of the cavity at (0.95, 0.95) for supercritical Reynolds numbers with respect to the bifurcation phenomenon by lowering the time step. As the accuracy of numerical methods plays a vital role in capturing the dynamics at different Reynolds numbers, this vortex-dominated flow is explained for bifurcation and instability. We propose this as a benchmark problem for the direct numerical simulation (DNS) and for machine learning (ML) of fluid flow that will lead to efficient ML algorithms and an understanding of flow receptivity, instability, and transition by DNS.

Keywords

lid-driven cavity flow; direct numerical simulation; high-performance computing; bifurcation analysis; receptivity analysis; global spectral analysis; error dynamics; machine learning of flows

References

1. Ghia U, Ghia KN, Shin CT. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics. 1982; 48(3): 387–411. doi: 10.1016/0021-9991(82)90058-4

2. Sengupta S, Sengupta TK, Puttam JK, et al. Global spectral analysis for convection-diffusion-reaction equation in one and two-dimensions: Effects of numerical anti-diffusion and dispersion. Journal of Computational Physics. 2020; 408. doi: 10.1016/j.jcp.2020.109310

3. Sagaut P, Suman VK, Sundaram P, et al. Global spectral analysis: Review of numerical methods. Computers and Fluids. 2023; 261. doi: 10.1016/j.compfluid.2023.105915

4. Sengupta TK. High Accuracy Computing Methods: Fluid Flows and Wave Phenomena. Cambridge University Press; 2013.

5. Sengupta S, Sreejith NA, Mohanamuraly P, et al. Global spectral analysis of the Lax-Wendroff central difference scheme applied to convection-diffusion equation. Computers and Fluids. 2022; 242. doi: 10.1016/j.compfluid.2022.105508

6. Sengupta TK, Lakshmanan V, Vijay VVSN. A new combined stable and dispersion relation preserving compact scheme for non-periodic problems. Journal of Computational Physics. 2009; 228(8). doi: 10.1016/j.jcp.2009.01.003

7. Ozbenli E, Vedula P. Construction of invariant compact finite difference schemes. Physical Review E. 2020; 101. doi: 10.1103/PhysRevE.101.023303

8. Sengupta TK, Ganerwal G, Dipankar A. High accuracy compact schemes and Gibbs’ phenomenon. Journal of Scientific Computing. 2004; 21(3). doi: 10.1007/s10915-004-1317-2

9. Auteri F, Quartapelle L, Vigevano L. Accurate ω-ψ spectral solution of the singular driven cavity problem. Journal of Computational Physics. 2002; 180(2): 597–615. doi: 10.1006/jcph.2002.7108

10. Botella O, Peyret R. Benchmark spectral results on the lid-driven cavity flow. Computers and Fluids. 1998; 27(4): 421–433. doi: 10.1016/S0045-7930(98)00002-4

11. Suman VK, Viknesh SS, Tekriwal MK, et al. Grid sensitivity and role of error in computing a lid-driven cavity problem. Physical Review E. 2019; 99(1). doi: 10.1103/PhysRevE.99.013305

12. Burggraf OR. Analytical and numerical studies of the structure of steady separated flows. Journal of Fluid Mechanics. 1966; 24(1): 113–151. doi: 10.1017/S0022112066000545

13. Schreiber R, Keller HB. Driven cavity flows by efficient numerical techniques. Journal of Computational Physics. 1983; 49(2): 310–333. doi: 10.1016/0021-9991(83)90129-8

14. Lestandi L, Bhaumik S, Avatar GRKC, et al. Multiple Hopf bifurcations and flow dynamics inside a 2D singular lid driven cavity. Computers and Fluids. 2018; 166: 86–103. doi: 10.1016/j.compfluid.2018.01.038

15. Sengupta TK, Sundaram P, Suman VK, et al. A high accuracy preserving parallel algorithm for compact schemes for DNS. ACM Transactions on Parallel Computing. 2020; 7(4): 1–32. doi: 10.1145/3418073

16. Sundaram P, Sengupta A, Sengupta TK. A non-overlapping high accuracy parallel subdomain closure for compact scheme: Onset of Rayleigh-Taylor instability by ultrasonic waves. Journal of Computational Physics. 2022; 470. doi: 10.1016/j.jcp.2022.111593

17. Ranade R, Hill C, Pathak J. Discretization Net: A machine-learning based solver for Navier-Stokes equations using finite volume discretization. Computer Methods in Applied Mechanics and Engineering. 2021; 378: 113722. doi: 10.1016/j.cma.2021.113722

18. Sahin M, Owens RG. A novel fully implicit finite volume method applied to the lid-driven cavity problem—Part I: High Reynolds number flow calculations. International Journal for Numerical Methods in Fluids. 2003; 42(1): 57–77. doi: 10.1002/fld.442

19. Bruneau CH, Saad M. The 2D lid-driven cavity problem revisited. Computers and Fluids. 2006; 35(3): 326–348. doi: 10.1016/j.compfluid.2004.12.004

20. Beckers M, van Heijst GJF. The observation of a triangular vortex in a rotating fluid. Fluid Dynamics Research. 1998; 22(5). doi: 10.1016/S0169-5983(97)00039-7

21. Carnevale GF, Kloosterziel RC. Emergence and evolution of triangular vortices. Journal of Fluid Mechanics. 1994; 259: 305–331. doi: 10.1017/S0022112094000157

22. Jansson TRN, Haspang MP, Jensen KH, et al. Polygons on a rotating fluid surface. Physical Review Letters. 2006; 96(17). doi: 10.1103/PhysRevLett.96.174502

23. van der Vorst HA. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing. 1992; 13(2). doi: 10.1137/0913035

24. Landau LD, Lifshitz EM. Fluid Mechanics, 2nd ed. Elsevier; 1987.

25. Brunton SL, Noack BR, Komoutsakos P. Machine learning for fluid mechanics. Annual Review of Fluid Mechanics. 2020; 52: 477–508. doi: 10.1146/annurev-fluid-010719-060214

26. Jiang Q, Zhu L, Shu C, et al. Multilayer perceptron neural network activated by adaptive Gaussian radial basis function and its application to predict lid-driven cavity flow. Acta Mechanica Sinica. 2021; 37(12): 1757–1772. doi: 10.1007/s10409-021-01144-5

27. Amalinadhi C, Palar PS, Stevenson R, et al. On physics-informed deep learning for solving Navier-Stokes equations. In: Proceedings of the AIAA SCITECH 2022 Forum; 3–7 January 2022; San Diego, CA, USA.

28. McDevitt C, Fowler E, Roy S. Physics-constrained deep learning of incompressible cavity flow. In: Proceedings of the AIAA SCITECH 2024 Forum; 8–12 January 2024; Orlando, FL, USA.

29. Sengupta TK. Transition to Turbulence: A Dynamical System Approach to Receptivity. Cambridge University Press; UK, 2021

30. Sengupta TK, Vijay VVSN, Singh N. Universal instability modes in internal and external flows. Computers and Fluids. 2011; 40(1): 221–235. doi: 10.1016/j.compfluid.2010.09.006

31. Eckhaus W. Studies in Non-Linear Stability Theory. In: Springer Tracts in Natural Philosophy. Springer; 1965.

32. Drazin PG, Reid WH. Hydrodynamic Stability. Cambridge University Press; 2004.

33. Sengupta TK. Instability of Flows and Transition to Turbulence. CRC Press; 2012

34. Chu PC, Fan C. A three-point combined compact difference scheme. Journal of Computational Physics. 1998; 140(2): 370–399. doi: 10.1006/jcph.1998.5899

35. Sengupta TK, Dipankar A, Sagaut P. Error dynamics: Beyond von Neumann analysis. Journal of Computational Physics. 2007; 226(2): 1211–1218. doi: 10.1016/j.jcp.2007.06.001

36. Tan R, Ooi A, Sandberg RD. Two-dimensional analysis of hybrid spectral/finite difference schemes for linearized compressible Navier-Stokes equations. Journal of Scientific Computing. 2021; 87(42). doi: 10.1007/s10915-021-01442-x

37. Cheviakov AF, Dorodnitsyn V, Kapstov EI. Invariant conservation law-preserving discretizations of linear and nonlinear wave equations. Journal of Mathematical Physics, vol. 61, 081504, 2020, doi: 10.1063/5.0004372

38. Coppola G, Capuano F, de Luca L. Discrete energy-conservation properties in the numerical simulation of the Navier-Stokes equations. Applied Mechanics Review. 2019; 71(1). doi: 10.1115/1.4042820

39. Coppola G, Capuano F, Pirozzoli S, et al. Numerically stable formulations of convective terms for turbulent compressible flows. Journal of Computational Physics. 2019; 382: 86–104. doi: 10.1016/j.jcp.2019.01.007

40. Sjögreen B, Yee HC, Kotov D. Skew-symmetric splitting and stability of high order central schemes. In: Proceedings of the 11th International Conference on Numerical Modeling of Space Plasma Flows: ASTRONUM-2016; 6–10 June 2016; Monterey, CA, USA.

41. Sengupta TK, Singh N, Suman VK. Dynamical system approach to instability of flow past a circular cylinder. Journal of Fluid Mechanics. 2010; 656: 82–115. doi: 10.1017/S0022112010001035

42. Chomaz JM. Global instabilities in spatially developing flows: Non-normality and nonlinearity. Annual Review of Fluid Mechanics. 2005; 37: 357–392. doi: 10.1146/annurev.fluid.37.061903.175810

43. Sengupta TK, Lestandi L, Haider SI, et al. Reduced order model of flows by time-scaling interpolation of DNS data. Advanced Modeling and Simulation in Engineering Sciences. 2018; 5(26). doi: 10.1186/s40323-018-0119-2

44. Drakoulas GI, Goritsas TV, Bourantas GC, et al. Fast SVD-ML-ROM: A reduced order modelling framework based on machine learning for real time application. Computer Methods in Applied Mechanics and Engineering. 2023; 414: 116155. doi: 10.1016/j.cma.2023.116155

45. Lam R, Sanchez-Gonzalez A, Willson M, et al., Learning skillful medium-range global weather forecasting. Science. 2023; 382(6677): 1416–1421. doi: 10.1126/science.adi23

46. Kochkov D, Smith JA, Alieva A, et al., Machine learning-accelerated computational fluid dynamics. PNAS. 2021; 118(21). doi: 10.1073/pnas.2101784118

47. Salim DM, Burkhart B, Sondak D. Extending a physics-informed machine-learning network for superresolution studies of Rayleigh-Benard convection. The Astrophysical Journal. 2024; 964(1). doi: 10.3847/1538-4357/ad1c55

48. Takhirov A, Trenchia C, Waters J. Second order efficient nonlinear filter stabilization for high Reynolds number flows. Numerical Methods for Partial Differential Equations. 2023; 39(1): 90–107. doi: 10.1002/num.22859

49. Bastian P, Altenbernd M, Dreier NA, et al. Exa-Dune-Flexible PDE solvers, numerical methods and applications. Software for Exascale Computing-SPPEXA 2016–2019. Springer; 2020. Volume 136.

50. Rogallo RS. Numerical experiments in homogeneous turbulence. Available online: https://ntrs.nasa.gov/api/citations/19810022965/downloads/19810022965.pdf (accessed on 20 November 2024).

51. Yeung PK, Donzis DA, Sreenivasan KR. Dissipation, enstrophy and pressure statistics in turbulence simulations at high Reynolds numbers. Journal of Fluid Mechanics. 2012; 700: 5–15. doi: 10.1017/jfm.2012.5

52. Buaria D, Sreenivasan KR. Dissipation range of the energy spectrum in high Reynolds number turbulence. Physical Review of Fluids. 2020; 5(9). doi: 10.1103/PhysRevFluids.5.092601

53. Buaria D, Bodenschatz E, Pumir A. Vortex stretching and enstrophy production in high Reynolds number turbulence. Physical Review Fluids. 2020; 5(10). doi: 10.1103/PhysRevFluids.5.104602

54. Buaria D, Pumir A, Bodenschatz E. Self attenuation of extreme events in Navier-Stokes turbulence. Nature Communications. 2020; 11: 5852. doi: 10.1038/s41467-020-19530-1

55. Orszag SA. Numerical methods for the simulation of turbulence. Physics of Fluids. 1969; 12(12). doi: 10.1063/1.1692445

56. Gottlieb D, Orszag SA. Numerical Analysis of Spectral Methods: Theory and Applications. SIAM; 1977.

57. Ishihara T, Gotoh T, Kaneda Y. Study of high-Reynolds number isotropic turbulence by DNS. Annual Review of Fluid Mechanics. 2009; 41: 165–180. doi: 10.1146/annurev.fluid.010908.165203

58. Ishihara T, Morishita K, Yokokawa M, et al. Energy spectrum in high resolution direct numerical simulation of turbulence. Physical Review Fluids. 2016; 1(8). doi: 10.1103/PhysRevFluids.1.082403

59. Doering CR, Gibbon JD. Applied Analysis of the Navier-Stokes Equations. Cambridge University Press; 1995.

60. Kraichnan RH. Inertial range in two-dimensional turbulence. Physics of Fluids. 1967; 10(7): 1417–1423. doi: 10.1063/1.1762301

61. Batchelor GK. Computation of the energy spectrum in homogeneous two-dimensional turbulence. Physics of Fluids. 1969; 12(12): 233–239. doi: 10.1063/1.1692443

62. Brachet ME, Meneguzzi R, Politano H, et al. The dynamics of freely decaying two dimensional turbulence. Journal of Fluid Mechanics. 1988; 194: 333–349. doi: 10.1017/S0022112088003015

63. Sengupta TK, Sarkar A, Joshi B, et al. Direct simulation of vortex dynamics of multicellular Taylor-Green vortex by Pseudospectral method. European Journal of Mechanics/B Fluids. 2024; 108: 226–236. doi: 10.1016/j.euromechflu.2024.08.004

64. Pawar S, San O. Equation free surrogate modelling and geophysical flows at the intersection of machine learning and data assimilation. Journal of Advanced Modelling Earth System. 2022; 14(11). doi: 10.1029/2022MS003170

65. Quaini A, San O, Veneziani A, et al. Bridging large eddy simulation and reduced order modelling of convection dominated flows through spatial filtering: Reviews and perspectives. Fluids. 2024; 9(8): 178. doi: 10.3390/fluids9080178

66. Takhirov A, Trenchia C. Efficient nonlinear filter stabilization of the Leray-α model. Journal of Computational Physics. 2022; 471: 111668. doi: 10.1016/j.jcp.2022.111668

67. Chung WT, Jung KS, Chen JH, et al. BLASTNet: A call for community-involved big data in combustion machine learning. Applications in Energy and Combustion Science. 2022; 12: 100087. doi: 10.1016/j.jaecs.2022.100087

68. Lucor D, Agrawal A, Sergeant A. Simple computational strategies for more effective physics informed neural networks modeling of turbulent natural convection. Journal of Computational Physics. 2022; 456: 111022. doi: 10.1016/j.jcp.2022.111022

69. Sofos F, Drikakis D, Kokkinakis IW, et al. A deep learning super-resolution model for turbulent image upscaling and its application to shock wave-boundary layer interaction. Physics of Fluids. 2024; 36(2). doi: 10.1063/5.0190272

70. Zhou Y. Turbulence theories and statistical closure approaches. Physics Report. 2021; 935: 1–117. doi: 10.1016/j.physrep.2021.07.001

71. Zhou Y, Clark TT, Clark DS, et al. Turbulent mixing and transition criteria of flows induced by hydrodynamic instabilities. Physics of Plasmas. 2019; 26(8). doi: 10.1063/1.5088745

72. Zhou Y, Williams RJR, Ramaprabhu P, et al. Rayleigh-Taylor and Richtmeyer-Meshkov instabilities: A journey through peaks. Physica D: Nonlinear Phenomena. 2021; 423. doi: 10.1016/j.physd.2020.132832

73. Joshi B, Sengupta TK, Sundaram P, et al. Highly resolved Peta-scale DNS: Onset of Kelvin-Helmholtz Rayleigh-Taylor instability via pressure pulses. Computers and Fluids. 2024; 284: 106442. doi: 10.1016/j.compfluid.2024.106442

74. Zhou Y. Hydrodynamic Instabilities and Turbulence: Rayleigh-Taylor, Richtmeyer-Meshkov and Kelvin-Helmholtz Mixing. Cambridge University Press; 2024.

75. Zhou Y, Saddler JD, Hurricane OA. Instabilities and mixing in inertial confinement fusion. Annual Review of Fluid Mechanics. 2025; 57: 197–225. doi: 10.1146/annurev-fluid-022824-110008

76. Pereira FS, Grinstein FF, Israel D. Effect of numerical discretization schemes in shock driven turbulent mixing simulations. Computers and Fluids. 2020; 201: 104487. doi: 10.1016/j.compfluid.2020.104487

77. Sengupta TK, Sharma PK, Sengupta A, et al. Tracking disturbances in transitional and turbulent flows: Coherent structures. Physics of Fluids. 2029; 31(12): 124106. doi: 10.1063/1.5130918