Skip to main content
SpringerLink
Log in

Energy Gradient Theory for Parallel Flow Stability

  • Chapter
  • First Online:
Origin of Turbulence

  • 2779 Accesses

Abstract

Motivated from studies of instability of viscoelastic flows, it is discovered that turbulent transition is generated by the variation of the gradient of the total mechanical energy, rather than simply due to the increase of the Reynolds number. Further, based on the Navier-Stokes equations, the energy gradient theory is proposed for flow stability and turbulent transition. The energy gradient function is constituted which is used to express the characteristics of flow stability and features a local Reynolds number of flow field. The theoretical results obtained good agreement with experimental data for various parallel flows, such as pipe Poiseuille flow, plane Poiseuille flow, annulus Poiseuille flow, and plane Couette flow.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Alavyoon F, Henningson DS, Alfredsson PH (1986) Turbulent spots in plane Poiseuille flow-flow visualization. Phys Fluids 29:1328–1331

    Article  Google Scholar 

  • Bottin S, Dauchot O, Daviaud F, Manneville P (1998a) Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow. Phys Fluids 10(10):2597–2607

    Article  Google Scholar 

  • Bottin S, Daviaud F, Manneville P, Dauchot O (1998b) Discontinuous transition to spatiotemporal intermittency in plane Couette flow. Europhys Lett 43(2):171–176

    Article  Google Scholar 

  • Betchov R, Criminale WO Jr (1967) Stability of parallel flows. Academic Press, New York

    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 

  • 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 

  • Bake S, Meyer DGW, Rist U (2002) Turbulence mechanism in Klebanoff transition: a quantitative comparison of experiment and direct numerical simulation. J Fluid Mech 459:217–243

    Article  MATH  Google Scholar 

  • Carlson DR, Widnall SE, Peeters MF (1982) A flow-visualization study of transition in plane Poiseuille flow. J Fluid Mech 121:487–505

    Article  Google Scholar 

  • Claus S, Phillips TN (2013) Viscoelastic flow around a confined cylinder using spectral/hp element methods. J Non-Newt Fluid Mech 200:131–146

    Article  Google Scholar 

  • Dewangan SK (2021) Effect of eccentricity and inner pipe motion on flow instability for flow through annulus. SN Appl Sci 3:513

    Article  Google Scholar 

  • Dou H-S, Khoo BC, Tsai HM (2010) Determining the critical condition for turbulent transition in a full-developed annulus flow. J Petrol Sci Eng 73(1–2):41–47

    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, Yeo KS (2007) Energy loss distribution in the plane Couette flow and the Taylor-Couette flow between concentric rotating cylinders. Int J Therm Sci 46:262–275

    Article  Google Scholar 

  • Dauchot O, Daviaud F (1995) Finite-amplitude perturbation and spots growth mechanism in plane Couette flow. Phys Fluids 7:335–343

    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 

  • Davies SJ, White CM (1928) An experimental study of the flow of water in pipes of rectangular section. Proc Roy Soc A 119:92–107

    MATH  Google Scholar 

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

    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 

  • Dou H-S (2004a) 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 (2004b) Viscous instability of inflectional velocity profile, recent advances in fluid mechanics. In: Zhuang F, Li J (eds) Proceedings of of the forth international conference on fluid mechanics. Tsinghua University Press & Springer, July 20–23, Dalian, China, pp 76–79

    Google Scholar 

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

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

    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. Modern Phys Lett B 32(12):1840003

    Google Scholar 

  • Dou H-S, Phan-Thien N (2001a) Numerical difficulties at high elasticity for the viscoelastic flow past a confined cylinder. Int J Comput Eng Sci 2(2):249–266

    Google Scholar 

  • Dou H-S, Phan-Thien N (2001b) Parallel simulation of viscoelastic flow past an array of cylinders by a unstructured FVM algorithm. In: Dally BB (ed) Proceedings of 14th Australasian fluid mechanics conference, Adelaide, Australia, 10–14 Dec 2001, No. FM010065, pp 873–876

    Google Scholar 

  • Dou H-S, Phan-Thien N (2002) Numerical simulation of viscoelastic flows past a linear array of cylinders by parallel compuation. In: Armfield S, Morgan P, Srinivas K (eds) Computational fluid dynamics 2002. Springer, Berlin, pp 323–328

    Google Scholar 

  • Dou H-S, Phan-Thien N (2007) Viscoelastic flows around a confined cylinder: instability and velocity inflection. Chem Eng Sci 62(15):3909–3929

    Article  Google Scholar 

  • Dou H-S, Phan-Thien N (2008) An instability criterion for viscoelastic flow past a confined cylinder. Korea Australia Rheol 20:15–26

    Google Scholar 

  • Dou H-S, Phan-Thien N (1998) Parallelisation of an unstructured finite volume implementation with PVM: viscoelastic flow around a cylinder. J Non-Newtonian Fluid Mech 77:21–51

    Article  MATH  Google Scholar 

  • Dou H-S, Phan-Thien N (1999) The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity vorticity (DAVSS-omega) formulation. J Non-Newt Fluid Mech 87:47–73

    Article  MATH  Google Scholar 

  • Dou H-S, Phan-Thien N (2003) Negative wake in the uniform flow past a cylinder. Rheol Acta 42:383–409

    Article  Google Scholar 

  • Eliahou S, Tumin A, Wygnanski I (1998) Laminar–turbulent transition in Poiseuille pipe flow subjected to periodic perturbation emanating from the wall. J Fluid Mech 361:333–349

    Article  MathSciNet  MATH  Google Scholar 

  • Emmons HW (1951) The laminar-turbulent transition in a boundary layer—part 1. J Aero Sci 18:490–498

    Article  MATH  Google Scholar 

  • Farahbakhsh I, Nourazar SS, Ghassemi H, Dou H-S (2014) On the instability of plane poiseuille flow of two immiscible fluids using the energy gradient theory. J Mech 30(3):299–305

    Article  Google Scholar 

  • Fu X, Yuan H, Chen C, Chen H, Song Y, Tan S (2014) Prediction of laminar to turbulent transition in 90° curved duct based on energy gradient theory. In: Proceedings of 2014 22nd international conference nuclear engineering, July 7–11, 2014, Prague, Czech Republic, No. ICONE22-30467, pp 1–8

    Google Scholar 

  • Farahbakhsh I, Dou H-S (2018) Derivation of energy gradient function for Rayleigh-Taylor instability. Fluid Dyn Res 50:045509 (15 pp)

    Google Scholar 

  • Groisman A, Steinberg V (2000) Elastic turbulence in a polymer solution flow. Nature 405:53–55

    Article  Google Scholar 

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

    Article  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 

  • Giiciiyener IH, Mehmetoglu T (1996) Characterization of flow regime in concentric annuli and pipes for yield-pseudoplastic fluids. J Petrol Sci Eng 16:45–60

    Google Scholar 

  • Guzel B, Frigaarda I, Martinezc DM (2009) Predicting laminar–turbulent transition in Poiseuille pipe flow for non-Newtonian fluids. Chem Eng Sci 64:254–264

    Article  Google Scholar 

  • Hinze JO (1975) Turbulence. 2nd edn, McGraw-Hill, New York

    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 

  • Hajmohammadi MR, Nourazar SS (2014) On the insertion of a thin gas layer in micro cylindrical Couette flows involving power-law liquids. Int J Heat Mass Trans 75:97–108

    Article  Google Scholar 

  • Hanks RW (1963) The laminar-turbulent transition for flow in pipes, concentric annuli, and parallel plates. AIChE J 9:45–48

    Article  Google Scholar 

  • Hanks RW, Bonner WF (1971) Transitional flow phenomena in concentric annuli. Ind Eng Chem Fundam 10:105–113

    Article  Google Scholar 

  • Heisenberg W (1924) Uber stabilitat und turbulenz von flussigkeitsstromen. Ann Phys Lpz 4:74, 577–627 (on stability and turbulence of fluid flows, NACA TM-1291, 1951)

    Google Scholar 

  • Hegseth J, Daviaud F, Bergé P (1992) Intermittent turbulence in plane and circular Couette flow. In: Andereck CD, Hayot F (eds) Ordered and turbulent patterns in Taylor-Couette flow. Plenum Press, New York, pp 159–166

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

    Article  Google Scholar 

  • Joseph DD (1976) Stability of fluid motions, vols 1 and 2. Springer-Verlag, Berlin

    Google Scholar 

  • Joo YL, Shaqfeh ESG (1992) A purely elastic instability in Dean and Taylor-Dean flow. Phys Fluids A 4(3):524–543

    Article  MATH  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 

  • 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 

  • Kondratiev AS, Ogorodnikov KF (2020) Determining critical flow parameters for the Poiseuille, Couette and Taylor-Couette flows. Eng J Sci Innov 6:1–17 (in Russia)

    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 

  • Lin C-C (1944) On the stability of two-dimensional parallel flows. Proc Nat Acad Sci 30:316–324

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Lin C-C (1946) On the stability of two-dimensional parallel flows: part III. Stab Viscous Fluid Q Appl Math 3(4):277–301

    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 

  • Lessen M, Cheifetz MG (1975) Stability of plane Couette flow with respect to finite two-dimensional disturbances. Phys Fluids 18:939–944

    Article  MATH  Google Scholar 

  • Leutheusser HJ, Chu VH (1971) Experiments on plane Couette flow. J Hydraulics Div 9:1269–1284

    Article  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 

  • Liu C, Yan Y, Lu P (2014) Physics of turbulence generation and sustenance in a boundary layer. Comput Fluids 102:353–384

    Article  Google Scholar 

  • Liepmann HW (1979) The rise and fall of ideas in turbulence. Am Sci 67(2):221–228

    MathSciNet  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

    Article  Google Scholar 

  • Li Y, Jiao Z, Yu T, Shang Y (2020) Viscous loss analysis of the flooded electro-hydrostatic actuator motor under laminar and turbulent flow states. Processes 8:975

    Article  Google Scholar 

  • Lohrenz J, Kurata F (1960) A friction factor plot for smooth circular conduits, concentric annuli, and parallel plates. Ind Eng Chem 52:703–706

    Article  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 

  • Manneville P (2004) Spots and turbulent domains in a model of transitional plane Couette flow. Theor Comput Fluid Dyn 18:169–181

    Article  MATH  Google Scholar 

  • Mukund V, Hof B (2018) The critical point of the transition to turbulence in pipe flow. J Fluid Mech 839:76–94

    Article  MathSciNet  MATH  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 

  • 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 

  • Nishioka M, Asai M (1985) Some observations of the subcritical transition in plane Poiseuille flow. J Fluid Mech 150:441–450

    Article  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 

  • 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 

  • Nowruzi H, Nourazar SS, Ghassemi H (2018) On the instability of two dimensional backward-facing step flow using energy gradient method. J Appl Fluid Mech 11(1):241–256

    Article  Google Scholar 

  • Nowruzi H, Ghassemi H, Nourazar SS (2019) Hydrodynamic stability study in a curved square duct by using the energy gradient method. J Brazilian Soc Mech Sci Eng 41:288

    Google Scholar 

  • Nowruzi H, Ghassemi H, Nourazar SS (2020) Study of the effects of aspect ratio on hydrodynamic stability in curved rectangular ducts using energy gradient method. Eng Sci Tech 23:334–344

    Google Scholar 

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

    Article  MATH  Google Scholar 

  • Papanastasiou TC, Georgiou GC, Alexandrou AN (1999) Viscous fluid flow. CRC Press, London

    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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Prandtl L (1921) Bemerkungen uber die enstehung der turbulenz. ZAMM 1:431–436

    Article  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 Roy Soc London A 174:935–982

    Article  MATH  Google Scholar 

  • Reichardt VH (1956) Uber die Geschwindigkeitsverteilung in einer geradlinigen turbulenten Couettestromung, ZAMM, pp S26–S29

    Google Scholar 

  • Rist U, Fasel H (1995) Direct numerical simulation of controlled transition in a flat-plate boundary layer. J Mech Fluid 298:211–248

    Article  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 

  • 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 

  • Schlichting H, Gersten K (2000) Boundary layer theory, 8th edn. Springer, Berlin

    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 

  • 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 

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

    Book  MATH  Google Scholar 

  • Stuart JT (1971) Nonlinear stability theory. Annu Rev Fluid Mech 3:347–370

    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 

  • Seki D, Matsubara M (2012) Experimental investigation of relaminarizing and transitional channel flows. Phys Fluids 24:124102-1-124102–23

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  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 

  • Wosnik M, Castill L, George WK (2000) A theory for turbulent pipe and channel flows. J Fluid Mech 421:115–145

    Article  MATH  Google Scholar 

  • Walker JE, Whan GA, Rothfus RR (1957) Fluid friction in noncircular ducts. AIChE J 3:484–489

    Article  Google Scholar 

  • Wang C, Gao P, Tan S, Xu C (2012) Effect of aspect ratio on the laminar-to-turbulent transition in rectangular channel. Ann Nucl Energy 46:90–96

    Article  Google Scholar 

  • Wygnanski IJ, Champagne FH (1973) On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J Fluid Mech 59:281–335

    Article  Google Scholar 

  • Xing D, Yan C, Wang C, Sun L (2013) A theoretical analysis about the effect of aspect ratio on single-phase laminar flow in rectangular ducts. Prog Nucl Energy 65:1–7

    Article  Google Scholar 

  • Yan BH (2011) Analysis of laminar to turbulent transition of pulsating flow in ocean environment with energy gradient method. Ann Nucl Energy 38:2779–2786

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Hua-Shu Dou

Authors
  1. Hua-Shu Dou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hua-Shu Dou .

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Dou, HS. (2022). Energy Gradient Theory for Parallel Flow Stability. In: Origin of Turbulence. Springer, Singapore. https://doi.org/10.1007/978-981-19-0087-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-981-19-0087-7_4

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-19-0086-0

  • Online ISBN: 978-981-19-0087-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation