Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-30T15:35:23.799Z Has data issue: false hasContentIssue false
  • English
  • Français

Oblique-mode breakdown of the vertical buoyancy layer

Published online by Cambridge University Press:  14 December 2022

K.R. Maryada Open the ORCID record for K.R. Maryada [Opens in a new window] ,
S.W. Armfield Open the ORCID record for S.W. Armfield [Opens in a new window] ,
P. Dhopade Open the ORCID record for P. Dhopade [Opens in a new window]  and
S.E. Norris Open the ORCID record for S.E. Norris [Opens in a new window]
K.R. Maryada*
Affiliation:
Department of Mechanical Engineering, The University of Auckland, Auckland 1010, New Zealand
S.W. Armfield
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, New South Wales 2006, Australia
P. Dhopade
Affiliation:
Department of Mechanical Engineering, The University of Auckland, Auckland 1010, New Zealand
S.E. Norris
Affiliation:
Department of Mechanical Engineering, The University of Auckland, Auckland 1010, New Zealand
*
†Email address for correspondence: kmar699@aucklanduni.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

The O-type transition caused by a pair of small-amplitude oblique waves in a vertical buoyancy layer of a fluid with Prandtl number $0.71$ at a Reynolds number of $200$ is investigated using linear stability analysis and three-dimensional direct numerical simulation. The small-amplitude oblique waves experience linear growth and undergo nonlinear interactions to generate streamwise vortices/streaks, two-dimensional streamwise waves and harmonic oblique waves. The streamwise vortices/streaks and two-dimensional streamwise waves have twice the spanwise or streamwise wavenumber of the original perturbation, respectively. Unlike the O-type transition in isothermal flat-plate incompressible and compressible boundary layers where streaks dominate the transition, in the vertical buoyancy layer, either streaks or two-dimensional streamwise waves can dominate the flow field during the early stages of oblique transition. The growth rates of streaks and two-dimensional waves are dependent on the wavenumber of the initial oblique waves. Streaks dominate the flow for high streamwise wavenumbers, while two-dimensional streamwise waves dominate the flow for low streamwise wavenumbers. Analysis of the turbulent kinetic energy production and the Reynolds stresses reveals that the early stages of the transition differ depending on the wavenumber of the oblique waves. An increase in the initial amplitude of the oblique waves causes a faster transition from laminar flow; however, the growth rates of the streaks and two-dimensional streamwise waves are independent of the initial amplitude. Even though different modes are dominant during the early stages of the O-type transition, the onset of chaotic flow is caused by the breakdown of streak modes.

JFM classification

Convection: Buoyant boundary layers Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 953 , 25 December 2022 , A34
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Armfield, S.W., Morgan, P., Norris, S. & Street, R. 2003 A parallel non-staggered Navier–Stokes solver implemented on a workstation cluster. In Computational Fluid Dynamics 2002 (ed. S.W. Armfield, P. Morgan & K. Srinivas), pp. 30–45. Springer.CrossRefGoogle Scholar
Armfield, S. & Street, R. 2002 An analysis and comparison of the time accuracy of fractional-step methods for the Navier–Stokes equations on staggered grids. Intl J. Numer. Meth. Fluids 38 (3), 255–282.CrossRefGoogle Scholar
Berlin, S., Lundbladh, A. & Henningson, D. 1994 Spatial simulations of oblique transition in a boundary layer. Phys. Fluids 6 (6), 1949–1951.CrossRefGoogle Scholar
Berlin, S., Wiegel, M. & Henningson, D.S. 1999 Numerical and experimental investigations of oblique boundary layer transition. J. Fluid Mech. 393, 23–57.CrossRefGoogle Scholar
Bobke, A., Örlü, R. & Schlatter, P. 2016 Simulations of turbulent asymptotic suction boundary layers. J. Turbul. 17 (2), 157–180.CrossRefGoogle Scholar
Chang, C.L. & Malik, M.R. 1994 Oblique-mode breakdown and secondary instability in supersonic boundary layers. J. Fluid Mech. 273, 323–360.CrossRefGoogle Scholar
Chen, Y.C. & Chung, J.N. 2002 A direct numerical simulation of K-and H-type flow transition in a heated vertical channel. Phys. Fluids 14 (9), 3327–3346.CrossRefGoogle Scholar
Deloncle, A., Chomaz, J.M. & Billant, P. 2007 Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid. J. Fluid Mech. 570, 297–305.CrossRefGoogle Scholar
Drazin, P.G. & Reid, W.H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Dudis, J.J. & Davis, S.H. 1971 Energy stability of the buoyancy boundary layer. J. Fluid Mech. 47 (2), 381–403.CrossRefGoogle Scholar
Fan, Y., Zhao, Y., Torres, J.F., Xu, F., Lei, C., Li, Y. & Carmeliet, J. 2021 Natural convection over vertical and horizontal heated flat surfaces: a review of recent progress focusing on underpinnings and implications for heat transfer and environmental applications. Phys. Fluids 33 (10), 101301.CrossRefGoogle Scholar
Fedorovich, E. & Shapiro, A. 2009 Turbulent natural convection along a vertical plate immersed in a stably stratified fluid. J. Fluid Mech. 636, 41–57.CrossRefGoogle Scholar
Gill, A.E. & Davey, A. 1969 Instabilities of a buoyancy-driven system. J. Fluid Mech. 35 (4), 775–798.CrossRefGoogle Scholar
Giometto, M.G., Katul, G.G., Fang, J. & Parlange, M.B. 2017 Direct numerical simulation of turbulent slope flows up to Grashof number $Gr = 2.1 \times 10^{11}$. J. Fluid Mech. 829, 589–620.CrossRefGoogle Scholar
Goldstein, M.E. & Choi, S.-W. 1989 Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech. 207, 97–120.CrossRefGoogle Scholar
Hanifi, A., Schmid, P.J. & Henningson, D.S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8 (3), 826–837.CrossRefGoogle Scholar
Herbert, T. 1991 Exploring transition by computer. Appl. Numer. Maths 7 (1), 3–25.CrossRefGoogle Scholar
Iyer, P.A. & Kelly, R.E. 1978 Supercritical solutions for the buoyancy boundary layer. Trans. ASME J. Heat Transfer 100 (4), 648–652.CrossRefGoogle Scholar
Jaluria, Y. & Gebhart, B. 1973 An experimental study of nonlinear disturbance behaviour in natural convection. J. Fluid Mech. 61 (2), 337–365.CrossRefGoogle Scholar
Jaluria, Y. & Gebhart, B. 1974 Stability and transition of buoyancy-induced flows in a stratified medium. J. Fluid Mech. 66 (3), 593–612.CrossRefGoogle Scholar
Janssen, R. & Armfield, S.W. 1996 Stability properties of the vertical boundary layers in differentially heated cavities. Intl J. Heat Fluid Flow 17 (6), 547–556.CrossRefGoogle Scholar
Kachanov, Y.S. & Levchenko, V.Y. 1984 The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer. J. Fluid Mech. 138, 209–247.CrossRefGoogle Scholar
Ke, J., Williamson, N., Armfield, S.W., McBain, G.D. & Norris, S.E. 2019 Stability of a temporally evolving natural convection boundary layer on an isothermal wall. J. Fluid Mech. 877, 1163–1185.CrossRefGoogle Scholar
Ke, J., Williamson, N., Armfield, S.W., Norris, S.E. & Komiya, A. 2020 Law of the wall for a temporally evolving vertical natural convection boundary layer. J. Fluid Mech. 902, A31.CrossRefGoogle Scholar
Khapko, T., Schlatter, P., Duguet, Y. & Henningson, D.S. 2016 Turbulence collapse in a suction boundary layer. J. Fluid Mech. 795, 356–379.CrossRefGoogle Scholar
Kim, J. & Moser, R.D. 1989 On the secondary instability in plane Poiseuille flow. Phys. Fluids A 1 (5), 775–777.CrossRefGoogle Scholar
Klebanoff, P.S., Tidstrom, K.D. & Sargent, L.M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12 (1), 1–34.CrossRefGoogle Scholar
Kleiser, L. & Zang, T.A. 1991 Numerical simulation of transition in wall-bounded shear flows. Annu. Rev. Fluid Mech. 23 (1), 495–537.CrossRefGoogle Scholar
Knowles, C.P. & Gebhart, B. 1968 The stability of the laminarnatural convection boundary layer . J. Fluid Mech. 34 (4), 657–686.CrossRefGoogle Scholar
Kosinov, A.D., Maslov, A.A. & Shevelkov, S.G. 1990 Experiments on the stability of supersonic laminar boundary layers. J. Fluid Mech. 219, 621–633.CrossRefGoogle Scholar
Krizhevsky, L., Cohen, J. & Tanny, J. 1996 Convective and absolute instabilities of a buoyancy-induced flow in a thermally stratified medium. Phys. Fluids 8 (4), 971–977.CrossRefGoogle Scholar
Kucala, A. & Biringen, S. 2014 Spatial simulation of channel flow instability and control. J. Fluid Mech. 738, 105–123.CrossRefGoogle Scholar
Laible, A.C. & Fasel, H.F. 2016 Continuously forced transient growth in oblique breakdown for supersonic boundary layers. J. Fluid Mech. 804, 323–350.CrossRefGoogle Scholar
Lee, C. & Chen, S. 2019 Recent progress in the study of transition in the hypersonic boundary layer. Natl Sci. Rev. 6 (1), 155–170.CrossRefGoogle Scholar
Lee, C. & Jiang, X. 2019 Flow structures in transitional and turbulent boundary layers. Phys. Fluids 31 (11), 111301.Google Scholar
Levin, O., Davidsson, E.N. & Henningson, D.S. 2005 Transition thresholds in the asymptotic suction boundary layer. Phys. Fluids 17 (11), 114104.CrossRefGoogle Scholar
Maryada, K.R. & Norris, S.E. 2021 Onset of low-frequency shear-driven instability in differentially heated cavities. In Proceedings of CHT-21. 8th International Symposium on Advances in Computational Heat Transfer. Begell House.CrossRefGoogle Scholar
Mayer, C., von Terzi, D. & Fasel, H. 2008 DNS of complete transition to turbulence via oblique breakdown at Mach 3. In 38th Fluid Dynamics Conference and Exhibit. AIAA Paper 2008-4398.CrossRefGoogle Scholar
Mayer, C.S.J., Von Terzi, D.A. & Fasel, H.F. 2011 Direct numerical simulation of complete transition to turbulence via oblique breakdown at Mach 3. J. Fluid Mech. 674, 5–42.CrossRefGoogle Scholar
Mayer, C., Wernz, S. & Fasel, H. 2007 Investigation of oblique breakdown in a supersonic boundary layer at Mach 2 using DNS. In 45th AIAA Aerosapce Sciences Meeting and Exhibit. AIAA Paper 2007-949.CrossRefGoogle Scholar
McBain, G.D., Armfield, S.W. & Desrayaud, G. 2007 Instability of the buoyancy layer on an evenly heated vertical wall. J. Fluid Mech. 587, 453–469.CrossRefGoogle Scholar
Morkovin, M.V. 1969 On the many faces of transition. In Viscous Drag Reduction (ed. C.S. Wells), pp. 1–31. Springer.CrossRefGoogle Scholar
Nachtsheim, P.R. & Swigert, P. 1965 Statisfaction of asymptotic boundary conditions in numerical solution of systems of nonlinear equations of boundary-layer type. Technical Memorandum NACA TN D-3004. National Advisory Committee for Aeronautics.Google Scholar
Norris, S.E. 2000 A parallel Navier–Stokes solver for natural convection and free surface flow. PhD thesis, The University of Sydney.Google Scholar
Prandtl, L. 1952 Essentials of Fluid Dynamics. Blackie & Son.Google Scholar
Reddy, S.C., Schmid, P.J., Baggett, J.S. & Henningson, D.S. 1998 On stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269–303.CrossRefGoogle Scholar
Rist, U. & Fasel, H. 1995 Direct numerical simulation of controlled transition in a flat-plate boundary layer. J. Fluid Mech. 298, 211–248.CrossRefGoogle Scholar
Ryu, S., Marxen, O. & Iaccarino, G. 2015 A comparison of laminar-turbulent boundary-layer transitions induced by deterministic and random oblique waves at Mach 3. Intl J. Heat Fluid Flow 56, 218–232.CrossRefGoogle Scholar
Sayadi, T., Hamman, C.W. & Moin, P. 2013 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.CrossRefGoogle Scholar
Schmid, P.J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 1992 A new mechanism for rapid transition involving a pair of oblique waves. Phys. Fluids A 4 (9), 1986–1989.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows, 1st edn. Springer.CrossRefGoogle Scholar
Singer, B.A., Reed, H.L. & Ferziger, J.H. 1989 The effects of streamwise vortices on transition in the plane channel. Phys. Fluids A 1 (12), 1960–1971.CrossRefGoogle Scholar
Spalart, P.R. & Yang, K.S. 1987 Numerical study of ribbon-induced transition in Blasius flow. J. Fluid Mech. 178, 345–365.CrossRefGoogle Scholar
Squire, H.B. 1933 On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. Lond. A 142 (847), 621–628.Google Scholar
Tao, J. 2006 Nonlinear global instability in buoyancy-driven boundary-layer flows. J. Fluid Mech. 566, 377–388.CrossRefGoogle Scholar
Tao, J. & Busse, F.H. 2009 Oblique roll instability in inclined buoyancy layers. Eur. J. Mech. B/Fluids 28 (4), 532–540.CrossRefGoogle Scholar
Tao, J., Le Quéré, P. & Xin, S. 2004 a Absolute and convective instabilities of natural convection flow in boundary-layer regime. Phys. Rev. E 70 (6), 066311.CrossRefGoogle ScholarPubMed
Tao, J., Le Quéré, P. & Xin, S. 2004 b Spatio-temporal instability of the natural-convection boundary layer in thermally stratified medium. J. Fluid Mech. 518, 363–379.CrossRefGoogle Scholar
Thumm, A., Wolz, W. & Fasel, H. 1990 Numerical simulation of spatially growing three-dimensional disturbance waves in compressible boundary layers. In Laminar-Turbulent Transition. International Union of Theoretical and Applied Mechanics (ed. D. Arnal & R. Michel), pp. 303–308. Springer.CrossRefGoogle Scholar
Weideman, J.A. & Reddy, S.C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465–519.CrossRefGoogle Scholar
Wray, A. & Hussaini, M.Y. 1984 Numerical experiments in boundary-layer stability. Proc. R. Soc. Lond. A 392 (1803), 373–389.Google Scholar
Xiong, X. & Tao, J. 2017 Lower bound for transient growth of inclined buoyancy layer. Appl. Maths Mech. 38 (6), 779–796.CrossRefGoogle Scholar
Yalcin, A., Turkac, Y. & Oberlack, M. 2021 On the temporal linear stability of the asymptotic suction boundary layer. Phys. Fluids 33 (5), 054111.CrossRefGoogle Scholar
Zhao, Y., Lei, C. & Patterson, J.C. 2016 Natural transition in natural convection boundary layers. Intl Commun. Heat Mass Transfer 76, 366–375.CrossRefGoogle Scholar
Zhao, Y., Lei, C. & Patterson, J.C. 2017 The K-type and H-type transitions of natural convection boundary layers. J. Fluid Mech. 824, 352–387.CrossRefGoogle Scholar
Zhong, X. & Wang, X. 2012 Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers. Annu. Rev. Fluid Mech. 44, 527–561.CrossRefGoogle Scholar