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Theoretical and Computational Fluid Dynamics

, Volume 33, Issue 6, pp 577–602 | Cite as

Compressibility effects on the transition to turbulence in a spatially developing plane free shear layer

  • Dongru Li
  • Jonathan Komperda
  • Zia Ghiasi
  • Ahmad Peyvan
  • Farzad MashayekEmail author
Original Article
  • 94 Downloads

Abstract

The compressibility effects on the transition to turbulence in a spatially developing, compressible plane free shear layer are investigated via direct numerical simulation using a high-order discontinuous spectral element method for three different convective Mach numbers of 0.3, 0.5, and 0.7. The location of the laminar–turbulent transition zone is predicted by the analyses of vorticities, Reynolds stresses, and the turbulent dissipation rate. In the turbulence transition and self-similar turbulence regions, the effects of compressibility on the flow properties, such as the velocity autocorrelation function, integral time scale, momentum thickness, Reynolds stress, and turbulent kinetic energy budget, are investigated. The compressibility effects on the onset and length of the turbulence transition zone are studied based on the analyses of such flow properties. The mean velocity, momentum thickness, and Reynolds stress profiles compare well with published experimental data. Vorticity contours and iso-surface of the second invariant of velocity gradient tensor identify the characteristic of flow structures. The two-point correlation functions of velocity components, the one-dimensional (1D) spanwise energy spectrum, and the balance of the turbulent kinetic energy transport equation validate the domain size and resolution of the adopted grid for turbulence simulation. An increase in the convective Mach number leads to a reduction in the sizes of the largest-scale structures, resulting in a significant decrease in Reynolds stresses and turbulence production. The onset of turbulence transition and the location where the transition completes shift downstream, while the length of the transition zone increases with increasing convective Mach number.

Keywords

Direct numerical simulation Transition Compressible flow Plane free shear layer Compressibility effect Spatially developing flow 

Notes

Acknowledgements

Part of the computational resources for this study was provided by the ACER at the University of Illinois at Chicago. This research was also in part supported by the Blue Waters sustained-petascale computing project, which is sponsored by the National Science Foundation (Awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. The Program Development Company GridPro provided us with troubleshooting support and license to access its meshing software, which was used to create the meshes for the simulations presented in this study.

References

  1. 1.
    Atoufi, A., Fathali, M., Lessani, B.: Compressibility effects and turbulent kinetic energy exchange in temporal mixing layers. J. Turbul. 16, 676–703 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barre, S., Bonnet, J.P.: Detailed experimental study of a highly compressible supersonic turbulent plane mixing layer and comparison with most recent DNS results: towards an accurate description of compressibility effects in supersonic free shear flows. Int. J. Heat Fluid Flow 51, 324–334 (2015)CrossRefGoogle Scholar
  3. 3.
    Bernal, L.P.: The coherent structure of turbulent mixing layers. I. Similarity of the primary vortex structure. II. Secondary streamwise vortex structure. Ph.D. Thesis, California Institute of Technology, Pasadena (1981)Google Scholar
  4. 4.
    Birch, S.F., Eggers, J.M.: A critical review of the experimental data for developed free turbulent shear layers. NASA SP 321 (1972)Google Scholar
  5. 5.
    Bogdanoff, D.W.: Compressibility effects in turbulent shear layers. AIAA J. 21, 926–927 (1983)CrossRefGoogle Scholar
  6. 6.
    Bradshaw, P.: The effect of initial conditions on the development of a free shear layer. J. Fluid Mech. 26, 225–236 (1966)CrossRefGoogle Scholar
  7. 7.
    Brown, G.L., Roshko, A.: On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816 (1974)zbMATHCrossRefGoogle Scholar
  8. 8.
    Clemens, N.T., Mungal, M.G.: Large-scale structure and entrainment in the supersonic mixing layer. J. Fluid Mech. 284, 171–216 (1995)CrossRefGoogle Scholar
  9. 9.
    Day, M.J., Reynolds, W.C., Mansour, N.N.: The structure of the compressible reacting mixing layer: insights from linear stability analysis. Phys. Fluids 10, 993–1007 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Elliott, G.S., Samimy, M.: Compressibility effects in free shear layers. Phys. Fluids A 2, 1231–1240 (1990)CrossRefGoogle Scholar
  11. 11.
    Foysi, H., Sarkar, S.: The compressible mixing layer: an LES study. Theor. Comput. Fluid Dyn. 24, 565–588 (2010)zbMATHCrossRefGoogle Scholar
  12. 12.
    Freund, J.B., Lele, S.K., Moin, P.: Compressibility effects in a turbulent annular mixing layer. Part 1. Turbulence and growth rate. J. Fluid Mech. 421, 229–267 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fu, S., Li, Q.: Numerical simulation of compressible mixing layers. Int. J. Heat Fluid Flow 27, 895–901 (2006)CrossRefGoogle Scholar
  14. 14.
    Gao, Z., Mashayek, F.: Stochastic model for non-isothermal droplet-laden turbulent flows. AIAA J. 42, 255–260 (2004)CrossRefGoogle Scholar
  15. 15.
    Gatski, T.B., Bonnet, J.P.: Compressibility, Turbulence and High Speed Flow. Academic Press, Berlin (2013)Google Scholar
  16. 16.
    Ghiasi, Z., Komperda, J., Li, D., Mashayek, F.: Simulation of supersonic turbulent non-reactive flow in ramp-cavity combustor using a discontinuous spectral element method. AIAA Paper 2016-0617 (2017)Google Scholar
  17. 17.
    Ghiasi, Z., Komperda, J., Li, D., Peyvan, A., Nicholls, D., Mashayek, F.: Modal explicit filtering for large eddy simulation in discontinuous spectral element method. J. Comput. Phys. X 3, 100024 (2019)Google Scholar
  18. 18.
    Goebel, S.G., Dutton, J.C.: Experimental study of compressible turbulent mixing layers. AIAA J. 29, 538–546 (1991)CrossRefGoogle Scholar
  19. 19.
    Gortler, H.: Berechnung von aufgaben der freien turbulenz auf grund eines neuen naherungsansatzes. Z. Ange Math Mech 22, 244–54 (1942)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Grosch, C.E., Jackson, T.L.: Inviscid spatial stability of a three-dimensional compressible mixing layer. J. Fluid Mech. 231, 35–50 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gruber, M.R., Messersmith, N.L., Dutton, J.C.: Three-dimensional velocity field in a compressible mixing layer. AIAA J. 31, 2061–2067 (1993)CrossRefGoogle Scholar
  22. 22.
    Hall, J.L., Dimotakis, P.E., Rosemann, H.: Experiments in non-reacting compressible shear layers. AIAA J. 31, 2247–2254 (1993)CrossRefGoogle Scholar
  23. 23.
    Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2013)zbMATHGoogle Scholar
  24. 24.
    Ho, C.M., Huang, L.S.: Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443–473 (1982)CrossRefGoogle Scholar
  25. 25.
    Ho, C.M., Huerre, P.: Perturbed free shear layers. Annu. Rev. Fluid Mech. 16, 365–424 (1984)CrossRefGoogle Scholar
  26. 26.
    Huang, P.G., Coleman, G.N., Bradshaw, P.: Compressible turbulent channel flows: DNS results and modelling. J. Fluid Mech. 305, 185–218 (1995)zbMATHCrossRefGoogle Scholar
  27. 27.
    Hussaini, M.Y., Voigt, R.G.: Instability and Transition. Springer, New York (1990)CrossRefGoogle Scholar
  28. 28.
    Jackson, T.L., Grosch, C.E.: Inviscid spatial stability of a compressible mixing layer. J. Fluid Mech. 208, 609–637 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Jackson, T.L., Grosch, C.E.: Absolute/convective instabilities and the convective mach number in a compressible mixing layer. Phys. Fluids A Fluid Dyn. 2, 949–954 (1990)zbMATHCrossRefGoogle Scholar
  30. 30.
    Jacobs, G.B.: Numerical simulation of two-phase turbulent compressible flows with a multidomain spectral method. Ph.D. Thesis, University of Illinois at Chicago, Chicago (2003)Google Scholar
  31. 31.
    Jacobs, G.B., Kopriva, D.A., Mashayek, F.: A comparison of outflow boundary conditions for the multidomain staggered-grid spectral method. Numer. Heat Transf. Part B 44(3), 225–251 (2003)CrossRefGoogle Scholar
  32. 32.
    Jacobs, G.B., Kopriva, D.A., Mashayek, F.: Compressibility effects on the subsonic two-phase flow over a square cylinder. J. Propul. Power 20, 353–359 (2004)CrossRefGoogle Scholar
  33. 33.
    Jacobs, G.B., Kopriva, D.A., Mashayek, F.: Validation study of a multidomain spectral code for simulation of turbulent flows. AIAA J. 43, 1256–1264 (2005)CrossRefGoogle Scholar
  34. 34.
    Javed, A., Rajan, N.K.S., Chakraborty, D.: Effect of side confining walls on the growth rate of compressible mixing layers. Comput. Fluids 86, 500–509 (2013)CrossRefGoogle Scholar
  35. 35.
    Jiménez, J.: Turbulence and vortex dynamics. Notes for the Polytechnic Course on Turbulence (2004)Google Scholar
  36. 36.
    Jiménez, J., Cogollos, M., Bernal, L.P.: A perspective view of the plane mixing layer. J. Fluid Mech. 152, 125–143 (1985)CrossRefGoogle Scholar
  37. 37.
    Karimi, M., Girimaji, S.S.: Suppression mechanism of Kelvin–Helmholtz instability in compressible fluid flows. Phys. Rev. E 93, 041102 (2016)CrossRefGoogle Scholar
  38. 38.
    Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for CFD. Oxford University Press, New York (1999)zbMATHGoogle Scholar
  39. 39.
    Kopriva, D.A.: A staggered-grid multidomain spectral method for the compressible Navier–Stokes equations. J. Comput. Phys. 143, 125–158 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Kopriva, D.A., Kolias, J.H.: A conservative staggerd-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125, 244–261 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Laizet, S., Lamballais, E.: Direct Numerical Simulation of a Spatially Evolving Flow from an Asymmetric Wake to a Mixing Layer. Springer, Poitiers (2006)zbMATHCrossRefGoogle Scholar
  42. 42.
    Laizet, S., Lardeau, S., Lamballais, E.: Direct numerical simulation of a mixing layer downstream a thick splitter plate. Phys. Fluids 22, 015104 (2010)zbMATHCrossRefGoogle Scholar
  43. 43.
    Lele, S.K.: Direct numerical simulation of compressible free shear flows. AIAA Paper 89-0374 (1989)Google Scholar
  44. 44.
    Lele, S.K.: Compressibility effect on turbulence. Annu. Rev. Fluid Mech. 26, 211–254 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Lesieur, M.: Understanding coherent vortices through computational fluid dynamics. Theor. Comput. Fluid Dyn. 5, 177–193 (1993)zbMATHCrossRefGoogle Scholar
  46. 46.
    Li, D., Ghiasi, Z., Komperda, J., Mashayek, F.: The effect of inflow mach number on the reattachment in subsonic flow over a backward-facing step. AIAA Paper 2016-2077 (2017)Google Scholar
  47. 47.
    Li, Q.B., Fu, S.: Numerical simulation of high-speed planar mixing layer. Comput. Fluids 32, 1357–1377 (2003)zbMATHCrossRefGoogle Scholar
  48. 48.
    Li, Z., Jaberi, F.A.: Numerical investigations of shock–turbulence interaction in a planar mixing layer. AIAA Paper 2010-112 (2010)Google Scholar
  49. 49.
    Liepmann, H.W., Laufer, J.: Investigation of free turbulent mixing. Technical Report TN 1257, NACA (1946)Google Scholar
  50. 50.
    Loucks, R.B.: An experimental examination of the streamwise velocity in a plane mixing layer using a single hot-wire sensor. Army Research Lab Adelphi MD, Adelphi (1997)Google Scholar
  51. 51.
    Lui, C., Lele, S.: Direct numerical simulation of spatially developing, compressible, turbulent mixing layers. AIAA Paper 2001-291 (2001)Google Scholar
  52. 52.
    Mashayek, F.: Droplet–turbulence interactions in low-mach-number homogeneous shear two-phase flows. J. Fluid Mech. 367, 163–203 (1998)zbMATHCrossRefGoogle Scholar
  53. 53.
    McMullan, W.A., Gao, S., Coats, C.M.: The effect of inflow conditions on the transition to turbulence in large eddy simulations of spatially developing mixing layers. Int. J. Heat Fluid Flow 30, 1054–1066 (2009)CrossRefGoogle Scholar
  54. 54.
    Monkewitz, P.A., Heurre, P.: Influence of the velocity ratio on the spatial instability of mixing layers. Phys. Fluids 25, 1137–1143 (1982)CrossRefGoogle Scholar
  55. 55.
    Morris, P.J., Giridharan, M.G., Lilley, G.M.: On the turbulent mixing of compressible free shear layer. Proc. R. Soc. Lond. Ser. A 431, 219–243 (1990)zbMATHCrossRefGoogle Scholar
  56. 56.
    Morris, S.C., Foss, J.F.: Turbulent boundary layer to single-stream shear layer: the transition region. J. Fluid Mech. 494, 187–221 (2003)zbMATHCrossRefGoogle Scholar
  57. 57.
    Moser, R.D., Rogers, M.M.: Mixing transition and the cascade of small scales in a plane mixing layer. Phys. Fluids 3, 1128–1134 (1991)CrossRefGoogle Scholar
  58. 58.
    Moser, R.D., Rogers, M.M.: The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence. J. Fluid Mech. 247, 275–320 (1993)zbMATHCrossRefGoogle Scholar
  59. 59.
    Olsen, M.G., Dutton, J.C.: Planar velocity measurements in a weakly compressible mixing layer. J. Fluid Mech. 486, 51–77 (2003)zbMATHCrossRefGoogle Scholar
  60. 60.
    Pantano, C., Sarkar, S.: A study of compressibility effects in the high-speed turbulent shear layer using direct simulation. J. Fluid Mech. 451, 329–371 (2002)zbMATHCrossRefGoogle Scholar
  61. 61.
    Papamoschou, D., Roshko, A.: The compressible turbulent shear layer: an experimental study. J. Fluid Mech. 197, 453–477 (1988)CrossRefGoogle Scholar
  62. 62.
    Pickett, L.M., Ghandhi, J.B.: Passive scalar mixing in a planar shear layer with laminar and turbulent inlet conditions. Phys. Fluids 14(3), 985 (2002)CrossRefGoogle Scholar
  63. 63.
    Pierrehumbert, R.T., Widnall, S.E.: The two and three dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82 (1982)zbMATHCrossRefGoogle Scholar
  64. 64.
    Pirozzoli, S., Bernardini, M., Marié, S., Grasso, F.: Early evolution of the compressible mixing layer issued from two turbulent streams. J. Fluid Mech. 777, 196–218 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Poinsot, T.J., Lele, S.: Boundary-conditions for direct simulations of compressible viscous flows. J. Comput. Phys. 101, 104–129 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)zbMATHCrossRefGoogle Scholar
  67. 67.
    Ragab, S.A., Wu, J.L.: Linear instabilities in two dimensional compressible mixing layer. Phys. Fluids A Fluid Dyn. 1, 957–966 (1989)CrossRefGoogle Scholar
  68. 68.
    Rogers, M.M., Moser, R.D.: The three-dimensional evolution of a plane mixing layer: the Kelvin–Helmholtz rollup. J. Fluid Mech. 243, 183–226 (1992)zbMATHCrossRefGoogle Scholar
  69. 69.
    Rogers, M.M., Moser, R.D.: Direct simulation of a self-similar turbulent mixing layer. Phys. Fluids 6, 903–923 (1994)zbMATHCrossRefGoogle Scholar
  70. 70.
    Sandham, N.D.: A numerical investigation of the compressible mixing layer. Ph.D. Thesis, Stanford University, Stanford (1989)Google Scholar
  71. 71.
    Sandham, N.D., Reynolds, W.C.: Compressible mixing layer: linear theory and direct simulation. AIAA J. 28, 618–624 (1990)CrossRefGoogle Scholar
  72. 72.
    Sandham, N.D., Reynolds, W.C.: Three-dimensional simulations of large eddies in the compressible mixing layer. J. Fluid Mech. 224, 133–158 (1991)zbMATHCrossRefGoogle Scholar
  73. 73.
    Sandham, N.D., Sandberg, R.D.: Direct numerical simulation of the early development of a turbulent mixing layer downstream of a splitter plate. J. Turbul. 10, 1–17 (2009)CrossRefGoogle Scholar
  74. 74.
    Sarkar, S.: The pressure-dilatation correlation in compressible flows. Phys. Fluids 4, 2674–2682 (1992)CrossRefGoogle Scholar
  75. 75.
    Sarkar, S.: The stabilizing effect of compressibility in turbulent shear flow. J. Fluid Mech. 282, 163–186 (1995)zbMATHCrossRefGoogle Scholar
  76. 76.
    Sarkar, S., Erlebacher, G., Hussaini, M.Y.: Direct simulation of compressible turbulence in a shear flow. Theor. Comput. Fluid Dyn. 2, 291–305 (1991)zbMATHCrossRefGoogle Scholar
  77. 77.
    Sharma, A., Bhaskaran, R., Lele, S.K.: Large-eddy simulation of supersonic, turbulent mixing layers downstream of a splitter plate. AIAA Paper 2011-208 (2011)Google Scholar
  78. 78.
    Shi, X., Chen, J., Bi, W., Shu, C., She, Z.: Numerical simulations of compressible mixing layers with a discontinuous galerkin method. Acta Mech. 27, 318–329 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Shyy, W., Krishnamurty, V.S.: Compressibility effects in modelling complex turbulent flows. Prog. Aerosp. Sci. 33, 587–645 (1997)CrossRefGoogle Scholar
  80. 80.
    Smits, A., Dussauge, J.P.: Turbulent Shear Layers in Supersonic Flow. Springer, Berlin (1996)Google Scholar
  81. 81.
    Thompson, K.W.: Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys. 68, 1–24 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Thompson, K.W.: Time dependent boundary conditions for hyperbolic systems 2. J. Comput. Phys. 89, 439–461 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Urban, W.D., Mungal, M.G.: Planar velocity measurements in compressible mixing layers. J. Fluid Mech. 431, 189–222 (2001)zbMATHCrossRefGoogle Scholar
  84. 84.
    Vreman, A.W., Sandham, N.D., Luo, K.H.: Compressible mixing layer growth rate and turbulence characteristics. J. Fluid Mech. 320, 235–258 (1996)zbMATHCrossRefGoogle Scholar
  85. 85.
    Wang, B., Wei, W., Zhang, Y., Zhang, H., Xue, S.: Passive scalar mixing in \({M}_{{\rm c}} < 1\) planar shear layer flows. Comput. Fluids 123, 32–43 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  86. 86.
    Wang, Y., Tanahashi, M., Miyauchi, T.: Coherent fine scale eddies in turbulence transition of spatially-developing mixing layer. Int. J. Heat Fluid Flow 28, 1280–1290 (2007)CrossRefGoogle Scholar
  87. 87.
    Yoder, D.A., DeBonis, J.R., Georgiadis, N.J.: Modeling of turbulent free shear flows. Comput. Fluids 117, 212–232 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Zhang, D., Tan, J., Yao, X.: Direct numerical simulation of spatially developing highly compressible mixing layer: structural evolution and turbulent statistics. Phys. Fluids 31, 036102 (2019)CrossRefGoogle Scholar
  89. 89.
    Zhou, Q., He, F., Shen, M.Y.: Direct numerical simulation of a spatially developing compressible plane mixing layer: flow structures and mean flow properties. J. Fluid Mech. 711, 437–468 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringUniversity of Illinois at ChicagoChicagoUSA

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