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Turbulent Transition Through Velocity Discontinuity

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Origin of Turbulence

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Abstract

The mechanism of turbulent transition is uncovered that turbulent transition is induced by the discontinuity of the streamwise velocity. These locations of discontinuity form the singular points of the Navier-Stokes equations. For pressure-driven flows, it is demonstrated that there exist no smooth and physically reasonable solutions of Navier–Stokes equations for transitional flow and turbulence, with both the energy gradient theory and the analysis of Poisson equation. For shear-driven flow, turbulent transition in plane Couette flow is also shown to be resulted from the singularity of the Navier-Stokes equations. The large scale structure of turbulence in the core region of plane Couette flow, which is absent in plane Poiseuille flow, is explained with velocity singularity.

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References

  • Adrian RJ (2007) Hairpin vortex organization in wall turbulence. Phys Fluids 19:041301

    Google Scholar 

  • Alfredsson PH, Johansson AV (1984) On the detection of turbulence-generating events. J Fluid Mech 139:325–345

    Article  Google Scholar 

  • Avila K, Moxey D, de Lozar A, Avila M, Barkley D, Hof B (2011) The onset of turbulence in pipe flow. Science 333:192–196

    Article  MATH  Google Scholar 

  • Avsarkisov V, Hoyas S, Oberlack M, García-Galache JP (2014) Turbulent plane Couette flow at moderately high Reynolds number. J Fluid Mech 751:R1

    Article  Google Scholar 

  • Beale JT, Kato T, Majda AJ (1984) Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun Math Phys 94(1):61–66

    Article  MathSciNet  MATH  Google Scholar 

  • Bech KH, Tillmark N, Alfredsson PH, Andersson HI (1995) An investigation of turbulent plane Couette flow at low Reynolds numbers. J Fluid Mech 286:291–325

    Article  Google Scholar 

  • Bernard PS, Thomas JM, Handler RA (1993) Vortex dynamics and the production of Reynolds stress. J Fluid Mech 253:385–419

    Article  MATH  Google Scholar 

  • Biringen S (1984) Final stages of transition to turbulence in plane channel flow. J Fluid Mech 148:413–442

    Article  MATH  Google Scholar 

  • Borodulin VI, Kachanov YS, Roschektayev AP (2011) Experimental detection of deterministic turbulence. J Turbul 12:N23

    Article  Google Scholar 

  • Cantwell B (1981) Organized motion in turbulent flow. Ann Rev Fluid Mech 13:457–515

    Article  Google Scholar 

  • Casper KM, Beresh SJ, Schneider SP (2014) Pressure fluctuations beneath instability wave packets and turbulent spots in a hypersonic boundary layer. J Fluid Mech 756:1058–1091

    Article  Google Scholar 

  • Cherubini S, De Palma P (2015) Minimal-energy perturbations rapidly approaching the edge state in Couette flow. J Fluid Mech 764:572–598

    Article  MathSciNet  Google Scholar 

  • Couliou M, Monchaux R (2015) Large-scale flows in transitional plane Couette flow: a key ingredient of the spot growth mechanism. Phys Fluids 27:034101

    Google Scholar 

  • Darbyshire AG, Mullin T (1995) Transition to turbulence in constant-mass-flux pipe flow. J Fluid Mech 289:83–114

    Article  Google Scholar 

  • Daviaud F, Hegseth J, Berge P (1992) Subcritical transition to turbulence in plane Couette flow. Phys Rev Lett 69:2511–2514

    Article  Google Scholar 

  • Doering CR (2009) The 3D Navier-Stokes problem. Annu Rev Fluid Mech 41:109–128

    Article  MathSciNet  MATH  Google Scholar 

  • Dou H-S (2004) Energy gradient theory of hydrodynamic instability. In: The third international conference on nonlinear science, Singapore, 30 June–2 July, 2004. http://arxiv.org/abs/nlin.CD/0501049

  • Dou H-S (2006) Mechanism of flow instability and transition to turbulence. Int J Non-Linear Mech 41(4):512–517

    Google Scholar 

  • Dou H-S (2011) Physics of flow instability and turbulent transition in shear flows. Int J Phys Sci 6(6):1411–1425

    Google Scholar 

  • Dou H-S (2014) Secret hidden in Navier-Stokes equations: singularity and criterion of turbulent transition. In: Proceedings of the 8th national conference on fluid mechanics, Lanzhou, China, Sept 18–21, 2014

    Google Scholar 

  • Dou H-S (2021) Singularity of Navier-Stokes equations leading to turbulence. Adv Appl Math Mech 13(3):527–553

    Article  MathSciNet  MATH  Google Scholar 

  • Dou H-S (2022) No existence and smoothness of solution of the Navier-Stokes equation. Entropy 24:339. https://www.mdpi.com/1099-4300/24/3/339

  • Dou H-S, Ben A (2015) Simulation and instability investigation of the flow around a cylinder between two parallel walls. J Therm Sci 24(2):140–148

    Article  Google Scholar 

  • Dou H-S, Khoo BC (2011) Investigation of turbulent transition in plane Couette flows using energy gradient method. Adv Appl Math Mech 3(2):165–180

    Article  MathSciNet  Google Scholar 

  • Dou H-S, Khoo BC (2012) Direct numerical simulation of turbulent transition for plane Couette flows using full Navier-Stokes equations. In: Proceedings of 50th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, 9–12 Jan 2012, Nashville, Tennessee, USA (AIAA 2012-0891)

    Google Scholar 

  • Dou H-S, Khoo BC, Yeo KS (2008) Instability of Taylor-Couette flow between concentric rotating cylinders. Int J Therm Sci 47:1422–1435

    Article  Google Scholar 

  • Dou H-S, Xu W, Khoo BC (2018) Stability of boundary layer flow based on energy gradient theory. Mod Phys Lett B 32(12):1840003

    Google Scholar 

  • Duguet Y, Schlatter P, Henningson DS (2010) Formation of turbulent patterns near the onset of transition in plane Couette flow. J Fluid Mech 650:119–129

    Article  MATH  Google Scholar 

  • Falco RE (1977) Coherent motions in the outer regions of turbulent boundary layers. Phys Fluids 20(10, II):5124–5132

    Google Scholar 

  • Fefferman CL (2000) Existence and smoothness of the Navier-Stokes equation, Clay Mathematics Institute, pp 1–6. http://www.claymath.org/millennium-problems/navier-stokes-equation

  • Fletcher CAJ (1991) Computational techniques for fluid dynamics, vol. 1, Springer, Berlin

    Google Scholar 

  • Foias C, Manley O, Rosa R, Temam R (2004) Navier-Stokes equations and turbulence. Cambridge University Press, Cambridge

    Google Scholar 

  • Gad-El-Hak M, Blackwelder RF, Riley JJ (1981) On the growth of turbulent regions in laminar boundary layers. J Fluid Mech 110:73–95

    Article  Google Scholar 

  • Ghaemi S, Scarano F (2013) Turbulent structure of high-amplitude pressure peaks within the turbulent boundary layer. J Fluid Mech 2013(735):381–426

    Article  MATH  Google Scholar 

  • Govindarajan R, Narasimha R (1991) The role of residual nonturbulent disturbances on transition onset in two-dimensional boundary layers. J Fluids Eng 113(1):147–149

    Article  Google Scholar 

  • Han G, Tumin A, Wygnanski I (2000) Laminar-turbulent transition in Poiseuille pipe flow subjected to periodic perturbation emanating from the wall, part 2. Late stage of transition. J Fluid Mech 419:1–27

    Article  MATH  Google Scholar 

  • Hof B, de Lozar A, Avila M, Tu X, Schneider TM (2010) Eliminating turbulence in spatially intermittent flows. Science 327(5972):1491–1494

    Article  Google Scholar 

  • Hof B, Juel A, Mullin T (2003) Scaling of the turbulence transition threshold in a pipe. Phys Rev Lett 91:244502

    Google Scholar 

  • Hof B, van Doorne CWH, Westerweel J, Nieuwstadt FTM, Faisst H, Eckhardt B, Wedin H, Kerswell RR, Waleffe F (2004) Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305(5690):1594–1598

    Article  Google Scholar 

  • Jiménez J (2012) Cascades in wall-bounded turbulence. Annu Rev Fluid Mech 44:27–45

    Article  MathSciNet  MATH  Google Scholar 

  • Johansson AV, Her J-Y, Haritonidis J (1987) On the generation of high-amplitude wall-pressure peaks in turbulent boundary layers and spots. J Fluid Mech 175:119–142

    Article  Google Scholar 

  • Kachanov YS (1994) Physical mechanisms of laminar-boundary-layer transition. Annu Rev Fluid Mech 26:411–482

    Article  MathSciNet  Google Scholar 

  • Kao TW, Park C (1970) Experimental investigations of the stability of channel flows part 1: flow of a single liquid in a rectangular channel. J Fluid Mech 43(1):145–164

    Article  Google Scholar 

  • Kim HT, Kline SJ, Reynolds WC (1971) The production of turbulence near a smooth wall in a turbulent boundary layer. J Fluid Mech 50(1):133–160

    Article  Google Scholar 

  • Kim J (1989) On the structure of pressure fluctuations in simulated turbulent channel flow. J Fluid Mech 205:421–451

    Article  Google Scholar 

  • Kim J, Moin P, Moser RD (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177:133–166

    Article  MATH  Google Scholar 

  • Kitoh O, Nakabayashi K, Nishimura F (2005) Experimental study on mean velocity and turbulence characteristics of plane Couette flow, low Reynolds number effects and large longitudinal vortical structure. J Fluid Mech 539:199–227

    Article  MATH  Google Scholar 

  • Kitoh O, Umeki M (2008) Experimental study on large-scale streak structure in the core region of turbulent plane Couette flow. Phys Fluids 20(2):025107

    Google Scholar 

  • Klebanoff PS, Tidstrom KD, Sargent LM (1962) The three-dimensional nature of boundary layer instability. J Fluid Mech 12(1):1–34

    Article  MATH  Google Scholar 

  • Kleiser L, Zang TA (1991) Numerical simulation of transition in wall-bpounded shear flows. Annu Rev Fluid Mech 23:495–537

    Article  Google Scholar 

  • Kline SJ, Reynolds WC, Schraub FA, Runstadler PW (1967) The structure of turbulent boundary layers. J Fluid Mech 30:741–773

    Article  MATH  Google Scholar 

  • Komminaho J, Lundbladh A, Johansson AV (1996) Very large structures in plane turbulent Couette flow. J Fluid Mech 320:259–285

    Article  MATH  Google Scholar 

  • Kundu PK, Cohen IM, Dowling DR (2016) Fluid mechanics, 6th edn. Academic Press, London

    MATH  Google Scholar 

  • Landahl MT, Mollo-Christensen E (1992) Turbulence and random processes in fluid mechanics. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Lemoult G, Aider J-L, Wesfreid JE (2012) Experimental scaling law for the subcritical transition to turbulence in plane Poiseuille flow. Phys Rev E 85(2):025303

    Google Scholar 

  • Leray J (1934) Sur le mouvement d’un liquide visquex emplissent l’espace. Acta Math J 63:193–248

    Google Scholar 

  • Luchikt TS, Tiederman WG (1987) Timescale and structure of ejections and bursts in turbulent channel flows. J Fluid Mech 174:524–552

    Google Scholar 

  • Luchikt TS, Tiederman WG (1988) Turbulent structure in low-concentration drag-reducing channel flows. J Fluid Mech 190:241–263

    Article  Google Scholar 

  • Lumley JL, Yaglom AM (2001) A century of turbulence. Flow Turbul Combust 66:241–286

    Article  MATH  Google Scholar 

  • Lundbladh A, Johansson A (1991) Direct simulation of turbulent spots in plane Couette flow. J Fluid Mech 229:499–516

    Article  MATH  Google Scholar 

  • Luo J, Wang X, Zhou H (2005) Inherent mechanism of breakdown in laminar-turbulent transition of plane channel flows. Sci China Ser G Phys Mech Astro 48(2):228–236

    Article  Google Scholar 

  • Malerud S, Mölfy KJ, Goldburg WI (1995) Measurements of turbulent velocity fluctuations in a planar Couette cell. Phys Fluids 7:1949–1955

    Article  Google Scholar 

  • Manneville P (2017) Laminar-turbulent patterning in transitional flows. Entropy 19:316

    Article  Google Scholar 

  • McWilliams JC (2019) A survey of submesoscale currents. Geosci. Lett. 6:3

    Article  Google Scholar 

  • Moffatt HK (2019) Singularities in fluid mechanics. Phys Rev Fluids 4:110502

    Google Scholar 

  • Moffatt HK, Kimura Y (2019) Towards a finite-time singularity of the Navier-Stokes equations. Part 2. Vortex reconnection and singularity evasion. J Fluid Mech 870:R1

    Google Scholar 

  • Moin P, Mahesh K (1998) Direct numerical simulation: a tool in turbulent research. Annu Rev Fluid Mech 30:539–578

    Article  MATH  Google Scholar 

  • Monty JP, Hutchins N, Ng HCH, Marusic I, Chong MS (2009) A comparison of turbulent pipe, channel and boundary layer flows. J Fluid Mech 632:431–442

    Article  MATH  Google Scholar 

  • Nepomuceno HG, Lueptow RM (1997) Pressure and shear stress measurements at the wall in a turbulent boundary layer on a cylinder. Phys Fluids 9(9):2732–2739

    Article  Google Scholar 

  • Nishi M, Unsal B, Dust F, Biswas G (2008) Laminar-to-turbulent transition of pipe flows through puffs and slugs. J Fluid Mech 614:425–446

    Article  MATH  Google Scholar 

  • Nishioka M, Asai M, Iida S (1981) Wall phenomena in the final stage of transition to turbulence. In: Meyer RE (ed) Transition and turbulence. Academic Press, New York, pp 113–126

    Google Scholar 

  • Nishioka M, Iida S, Ichikawa Y (1975) An experimental investigation of the stability of plane Poiseuille flow. J Fluid Mech 72:731–751

    Article  Google Scholar 

  • Orszag SA (1971) Accurate solution of the Orr-Sommerfeld stability equation. J Fluid Mech 50:689–703

    Article  MATH  Google Scholar 

  • Papavassiliou DV, Hanratty TJ (1997) Interpretation of large-scale structures observed in a turbulent plane Couette flow. Int J Heat Fluid Flow 18(1):55–69

    Article  Google Scholar 

  • Rayleigh L (1880) On the stability or instability of certain fluid motions. Proc Lond Maths Soc 11:57–70

    MathSciNet  MATH  Google Scholar 

  • Robinson SK, Kline SJ, Spalart PR (1989) A review of quasi-coherent structures in a numerically simulated turbulent boundary layer, NASA-TM-102191

    Google Scholar 

  • Robinson SK (1991) Coherent motion in the turbulent boundary layer. Annu Rev Fluid Mech 23:601–639

    Article  Google Scholar 

  • Sandham ND, Kleiser L (1992) The late stages of transition to turbulence in channel flow. J Fluid Mech 245:319–348

    Article  MATH  Google Scholar 

  • Schlatter P, Stolz S, Kleiser L (2006) Large-eddy simulation of spatial transition in plane channel flow. J Turbul 7:N33

    Article  MathSciNet  MATH  Google Scholar 

  • Schlichting H (1979) Boundary layer theory, 7th edn. Springer, Berlin

    Google Scholar 

  • Schmid PJ, Henningson DS (2001) Stability and transition in shear flows. Springer, New York

    Book  MATH  Google Scholar 

  • Sengupta TK, Kasliwal A, De S, Nair M (2003) Temporal flow instability for Robins-Magnus effect at high rotation rates. J Fluids Struct 17:941–953

    Article  Google Scholar 

  • Smits AJ, Delo C (2001) Self-sustaining mechanisms of wall turbulence. In: Reguera D, Rubí JM, Bonilla LL (eds) Coherent structures in complex systems. Lecture notes in physics, vol 567, Springer, Berlin, Heidelberg, pp 17–38

    Google Scholar 

  • Sullivan PP, McWilliams JC (2018) Frontogenesis and frontal arrest for a dense filament in the oceanic surface boundary layer. J Fluid Mech 837:341–380

    Article  MathSciNet  MATH  Google Scholar 

  • Swearingen JD, Blackwelder RF, Spalart PR (1987) Inflectional instabilities in the wall region of bounded turbulent shear flows. In: Proceedings of the summer program 1987, Center for Turbulence Research, Stanford University, pp 291–295 (NASA N88-23114)

    Google Scholar 

  • Tillmark N, Alfredsson PH (1992) Experiments on transition in plane Couette flow. J Fluid Mech 235:89–102

    Article  Google Scholar 

  • Tollmien W (1935) Ein allgemeines kriterium der instabilitat laminarer gescgwindigkeitsverteilungen, Nachr. Wiss Fachgruppe, Gottingen. Math Phys 1:1935, 79–114. Translated as, General instability criterion of laminar velocity disturbances, NACA TM-792, 1936

    Google Scholar 

  • Tsukahara T, Kawamura H, Shingai K (2006) DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region. J Turbul 7:N19

    Article  Google Scholar 

  • Tuckerman LS, Barkley D (2011) Patterns and dynamics in transitional plane Couette flow. Phys Fluids 23:041301

    Google Scholar 

  • Wedin H, Kerswell RR (2004) Exact coherent structures in pipe flow: travelling wave solutions. J Fluid Mech 508:333–371

    Article  MathSciNet  MATH  Google Scholar 

  • Wei T, Willmarth WW (1989) Reynolds-number effects on the structure of a turbulent channel flow. J Fluid Mech 204:57–95

    Article  Google Scholar 

  • White FM (1991) Viscous fluid flow, 2nd edn. McGraw-Hill, New York

    Google Scholar 

  • Xia ZH, Shi YP, Cai QD, Wan MP, Chen SY (2018) Multiple states in turbulent plane Couette flow with spanwise rotation. J Fluid Mech 837:477–490

    Article  MathSciNet  MATH  Google Scholar 

  • Yao J, Hussain F (2020) On singularity formation via viscous vortex reconnection. J Fluid Mech 888:R2

    Google Scholar 

  • Yeo KS, Zhao X, Wang ZY, Ng KC (2010) DNS of wave packet evolution in a Blasius boundary layer. J Mech Fluid 652:333–372

    Article  MATH  Google Scholar 

  • Zhou J, Adrian RJ, Balachandar S, Kendall TM (1999) Mechanisms for generating coherent packets of hairpin vortices in channel flow. J Fluid Mech 387:353–396

    Article  MathSciNet  MATH  Google Scholar 

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  1. Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, China

    Hua-Shu Dou

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Dou, HS. (2022). Turbulent Transition Through Velocity Discontinuity. In: Origin of Turbulence. Springer, Singapore. https://doi.org/10.1007/978-981-19-0087-7_5

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