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Scaling of Disturbance for Turbulent Transition and Turbulence

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Origin of Turbulence

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Abstract

Using the principle of the energy gradient theory, the scaling of the dimensionless amplitude of normal disturbance of streamwise velocity at turbulent transition is obtained and it agrees well with experiments. The relation of the frequency and the amplitude of disturbance at transition is also obtained and is in agreement with experiment. The role of disturbance in turbulent transition is found to promote to produce singularity. The energy spectrum of turbulence accounting of effect of Reynolds number is obtained and scales with the wave number as an exponent of -2 and scales with the Reynolds number as an exponent of 2, which are consistent with experiments.

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References

  • Adrian RJ, Meinhart CD, Tomkins CD (2000) Vortex organization in the outer region of the turbulent boundary layer. J Fluid Mech 422:1–54

    MathSciNet  MATH  Google Scholar 

  • Bailly C, Comte-Bellot G (2015) Turbulence. Springer, Cham

    Google Scholar 

  • Callies J, Ferrari R (2013) Interpreting energy and tracer spectra of upper-ocean turbulence in the submesoscale range (1–200 km). J Phys Oceanogr 43:2456–2474

    Google Scholar 

  • Chapman SJ (2002) Subcritical transition in channel flows. J Fluid Mech 451:35–97

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • Donzis DA, Sreenivasan KR (2010) The bottleneck effect and the Kolmogorov constant in isotropic turbulence. J Fluid Mech 657:171–188

    MATH  Google Scholar 

  • Dou H-S (2004) Energy gradient theory of hydrodynamic instability. In: The third international conference on nonlinear science, 30 June–2 July. 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, The chinese society of theoretical and applied mechanics, Lanzhou, China, Sept 18–21, 2014 (Paper No. CSTAM 2014-A26-B S01040)

    Google Scholar 

  • Dou H-S (2019) Nonlinear finite disturbance in turbulent transition. In: Proceedings of 3rd international symposium cavitation and multiphase flow, April 19–22, Shanghai

    Google Scholar 

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

    Google Scholar 

  • Dou H-S (2020) Energy spectrum of turbulence accounting for effect of Reynolds number, https://www.researchgate.net/publication/343826953

  • 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, Khoo BC (2008) Mathematical singularity behaviour of turbulent transition. In: 61st annual meeting of the APS division of fluid dynamics, vol 53, No 15, Nov. 23–25, 2008. San Antonio, Texas, Paper No. DFD08-2008-000963

    Google Scholar 

  • Dou H-S, Khoo BC (2010a) Criteria of turbulent transition in parallel flows. Modern Phys Lett B 24(13):1437–1440

    MATH  Google Scholar 

  • Dou H-S, Khoo BC (2010b) Energy gradient method for turbulent transition with consideration of effect of disturbance frequency. J Hydrodyn Ser B 22(5), Supplement 1:23–28

    Google Scholar 

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

    MathSciNet  Google Scholar 

  • Dou H-S, Khoo BC (2012) Energy spectrum of disturbance at turbulent transition via energy gradient method. In: Fourth international symposium on physics of fluids (ISPF4). Int J Mod Phys Conf Ser 19:293–303

    Google Scholar 

  • Drazin PG, Reid WH (2004) Hydrodynamic stability, 2nd ed. Cambridge University Press, Cambridge

    Google Scholar 

  • Dugan JP, Morris WD, Okawa BS (1986) Horizontal wave number distribution of potential energy in the ocean. J Geophys Res 91(C11):12993–13000

    Google Scholar 

  • Einstein A (1956) Investigations on the theory of Brownian movement. Dover, New York

    MATH  Google Scholar 

  • Frisch U (1995) Turbulence: the legacy of A. Cambridge University Press, Cambridge, N. Kolmogorov

    MATH  Google Scholar 

  • Govindarajan R, Narasimha R (2004) Scaling with freestream fluctuations in the laminar-turbulent transition process. In: 21st International Congress on Theory Applications Mechanics, August 15–21, 2004, Warsaw

    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

    Google Scholar 

  • Grant HL, Stewart RW, Moilliet A (1962) Turbulence spectra from a tidal channel. J Fluid Mech 12:241–268

    MATH  Google Scholar 

  • Grossmann S (2000) The onset of shear flow turbulence. Rev Modern Phys 72:603–618

    Google Scholar 

  • He GW, Jin GD, Yang Y (2017) Space-time correlations and dynamic coupling in turbulent flows. Annu Rev Fluid Mech 49:51–71

    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

    Google Scholar 

  • Hof B, Juel A, Mullin T (2003) Scaling of the turbulence transition threshold in a pipe. Phys Rev Lett 91, No. 244502

    Google Scholar 

  • Hommema SE, Adrian RJ (2002) Similarity of apparently random structure in the outer region of wall turbulence. Exp Fluids 33:5–12

    Google Scholar 

  • Hou H, Yu F, Nan F, Yang B, Guan S, Zhang Y (2019) Observation of near-inertial oscillations induced by energy transformation during typhoons. Energies 12:99

    Google Scholar 

  • Kaneda Y, Ishihara T, Yokokawa M, Itakura K, Uno A (2003) Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys Fluids 15(2):L21–L24

    MATH  Google Scholar 

  • von Kármán T, Howarth L (1938) On the statistical theory of isotropic turbulence. Proc Roy Soc Lond A 164:192–215

    MATH  Google Scholar 

  • Khan HH, Anwer SF, Hasan N, Sanghi S (2020) The organized motion of characterized turbulent flow at low Reynolds number in a straight square duct. SN Appl Sci 2:763

    Google Scholar 

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

    MATH  Google Scholar 

  • Kolmogorov AN (1941) The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C R Acad Sci URSS 30:301–305

    MathSciNet  Google Scholar 

  • Kolmogorov AN (1962) A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J Fluid Mech 13:82–85

    MathSciNet  MATH  Google Scholar 

  • Kraichnan RH (1959) The structure of isotropic turbulence at very high Reynolds numbers. J Fluid Mech 5(4):497–543

    MathSciNet  MATH  Google Scholar 

  • Kraichnan RH (1974) On Kolmogorov’s inertial-range theories. J Fluid Mech 62:305–330

    MATH  Google Scholar 

  • Kraichnan RH (1991) Turbulent cascade and intermittency growth. Proc R Soc Lond A 434:65–78

    MathSciNet  MATH  Google Scholar 

  • Landau LD, Lifshitz EM (1987) Fluid mechanics, 2nd edn. Pergamon, Oxford

    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), No. 025303

    Google Scholar 

  • Lesieur M (2008) Turbulence in fluids, fourth revised and enlarged edition, Springer, Berlin

    Google Scholar 

  • Libby PA (1996) Introduction to turbulence. Taylor & Francis, Washington, D.C.

    Google Scholar 

  • Liepmann HW, Laufer J, Liepmann K (1951) On the spectrum of isotropic turbulence, NACA TN 2473, Washington, D.C.

    Google Scholar 

  • Lin C-C (1948) Note of the law of decay of isotropic turbulence. Proc Natl Acad Sci 34:540–543

    MathSciNet  MATH  Google Scholar 

  • Lin C-C (1955) The theory of hydrodynamic stability. Cambridge Press, Cambridge

    MATH  Google Scholar 

  • Lohse D, Xia K-Q (2010) Small-scale properties of turbulent Rayleigh-Bénard convection. Annu Rev Fluid Mech 42:335–364

    MATH  Google Scholar 

  • Long RR (2003) Do tidal-channel turbulence measurements support ? Environ Fluid Mech 3:109–127

    Google Scholar 

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

    Google Scholar 

  • Matsubara M, Alfredsson PH (2001) Disturbance growth in boundary layers subjected to free-stream turbulence. J Fluid Mech 430:149–168

    MATH  Google Scholar 

  • Moin P (2009) Revisiting Taylor’s hypothesis J. Fluid Mech 640:1–4

    MathSciNet  MATH  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

    MATH  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

    Google Scholar 

  • Patel VC, Head MR (1969) Some observations on skin friction and velocity profiles in full developed pipe and channel flows. J Fluid Mech 38:181–201

    Google Scholar 

  • Peixinho J, Mullin T (2007) Finite-amplitude thresholds for transition in pipe flow. J Fluid Mech 582:169–178

    MATH  Google Scholar 

  • Pfenninger W (1961) Boundary layer suction experiments with laminar flow at high Reynolds numbers in the inlet length of a tube by various suction methods. In: Lachmann GV (ed) Boundary layer and flow control, vol 2, pp 961–980. Pergamon

    Google Scholar 

  • Phillips OM (1966) The dynamics of the upper ocean. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Reynolds O (1883) An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil Trans R Soc London A 174:935–982

    MATH  Google Scholar 

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

    Google Scholar 

  • Shan H, Zhang Z, Nieuwstadt FTM (1998) Direct numerical simulation of transition in pipe flow under the influence of wall disturbances. Int J Heat Fluid Flow 19:320–325

    Google Scholar 

  • She Z-S, Leveque E (1994) Universal scaling laws in fully developed turbulence. Phys Rev Lett 72(3):336–339

    Google Scholar 

  • Singer BA (1996) Characteristics of a young turbulent spot. Phys Fluids 8(2):509–521

    MathSciNet  MATH  Google Scholar 

  • Singer BA, Joslin RD (1994) Metamorphosis of a hairpin vortex into a young turbulent spot. Phys Fluids 6:3724–3736

    Google Scholar 

  • Slaughter GM (1964) Investigation of the energy spectrum of turbulence in a closed rectangular conduit. PhD thesis, Georgia Institute of Technology

    Google Scholar 

  • Sreenivasan KR (1999) Fluid turbulence. Rev Mod Phys 71(2):S383–S395

    Google Scholar 

  • Sreenivasan KR, Antonia RA (1997) The phenomenology of small-scale turbulence. Annu Rev Fluid Mech 29:435–472

    MathSciNet  Google Scholar 

  • Sreenivasan KR (1995) On of the Kolmogorov constant. Phys Fluids 7(ll):2278–2284

    Google Scholar 

  • Taylor GI (1938) The spectrum of turbulence. Proc R Soc London A164:476–490

    MATH  Google Scholar 

  • Trefethen LN, Trefethen AE, Reddy SC, Driscoll TA (1993) Hydrodynamic stability without eigenvalues. Science 261:578–584

    MathSciNet  MATH  Google Scholar 

  • Tsuji Y (2004) Intermittency effect on energy spectrum in high-Reynolds number turbulence. Phys Fluids 16(5):L43

    MATH  Google Scholar 

  • Tuckerman LS, Chantry M, Barkley D (2020) Patterns in wall-bounded shear flows. Annu Rev Fluid Mech 52:343–367

    MATH  Google Scholar 

  • Waleffe F (1995) Transition in shear flows, nonlinear normality versus nonnormal linearity. Phys Fluids 7:3060–3066

    Article  MathSciNet  MATH  Google Scholar 

  • Wang YX, Choi K-S, Gaster M, Atkin C, Borodulin V, Kachanov Y (2021) Early development of artificially initiated turbulent spots. J Fluid Mech 916:A1

    MathSciNet  MATH  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 

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

    Google Scholar 

  • Willis AP, Peixinho J, Kerswell RR, Mullin T (2008) Experimental and theoretical progress in pipe flow transition. Phil Trans R Soc A 366(1876):2671–2684

    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). Scaling of Disturbance for Turbulent Transition and Turbulence. In: Origin of Turbulence. Springer, Singapore. https://doi.org/10.1007/978-981-19-0087-7_7

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  • DOI: https://doi.org/10.1007/978-981-19-0087-7_7

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