Hostname: page-component-7d8f8d645b-clzrd Total loading time: 0 Render date: 2023-05-29T03:42:57.168Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Investigation of Turbulent Transition in Plane Couette Flows Using Energy Gradient Method

Published online by Cambridge University Press:  03 June 2015

Hua-Shu Dou*
Temasek Laboratories, National University of Singapore, 10 Kent Ridge Crescent, Singapore 117411, Singapore
Boo Cheong Khoo*
Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore
Corresponding author. URL: Email:
Get access


The energy gradient method has been proposed with the aim of better understanding the mechanism of flow transition from laminar flow to turbulent flow. In this method, it is demonstrated that the transition to turbulence depends on the relative magnitudes of the transverse gradient of the total mechanical energy which amplifies the disturbance and the energy loss from viscous friction which damps the disturbance, for given imposed disturbance. For a given flow geometry and fluid properties, when the maximum of the function K (a function standing for the ratio of the gradient of total mechanical energy in the transverse direction to the rate of energy loss due to viscous friction in the streamwise direction) in the flow field is larger than a certain critical value, it is expected that instability would occur for some initial disturbances. In this paper, using the energy gradient analysis, the equation for calculating the energy gradient function K for plane Couette flow is derived. The result indicates that K reaches the maximum at the moving walls. Thus, the fluid layer near the moving wall is the most dangerous position to generate initial oscillation at sufficient high Re for given same level of normalized perturbation in the domain. The critical value of K at turbulent transition, which is observed from experiments, is about 370 for plane Couette flow when two walls move in opposite directions (anti-symmetry). This value is about the same as that for plane Poiseuille flow and pipe Poiseuille flow (385-389). Therefore, it is concluded that the critical value of K at turbulent transition is about 370-389 for wall-bounded parallel shear flows which include both pressure (symmetrical case) and shear driven flows (anti-symmetrical case).

Research Article
Copyright © Global-Science Press 2011

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.)


[1] Schmid, P. J. and Henningson, D. S., Stability and Transition in Shear Flows, New York, Springer-Verlag, 2000.Google Scholar
[2] Lin, C. C., The Theory of Hydrodynamic Stability, Cambridge, Cambridge Press, 1955.Google Scholar
[3] Betchov, R. and Criminale, W. O. J., Stability of Parallel Flows, New York, Academic Press, 1967.Google Scholar
[4] Drazin, P. G. and Reid, W. H., Hydrodynamic Stability, Cambridge University Press, Cambridge, England, 1981.Google Scholar
[5] Joseph, D. D., Stability of Fluid Motions, Vol.1 and 2, Berlin, Springer-Verlag, 1976.Google Scholar
[6] Rayleigh, L., On the stability or instability of certain fluid motions, Proc. Lond. Math. Soc., 11 (1880), pp. 5770.Google Scholar
[7] Orr, W. M., The stability or instability of the steady motions of a perfect liquid and of a viscous liquid, P. Roy. Irish. Acad. A., 27 (1907), pp. 9138.Google Scholar
[8] Stuart, J. T., Nonlinear stability theory, Annu. Rev. Fluid. Mech., 3 (1971), pp. 347370.Google Scholar
[9] Bayly, B. J., Orszag, S. A. and Herbert, T., Instability mechanism in shear-flow transition, Annu. Rev. Fluid. Mech., 20 (1988), pp. 359391.Google Scholar
[10] Reynolds, O., 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. Lond. A., 174 (1883), pp. 935982.Google Scholar
[11] Patel, V. C. and Head, M. R., Some observations on skin friction and velocity profiles in full developed pipe and channel flows, J. Fluid. Mech., 38 (1969), pp. 181201.Google Scholar
[12] Trefethen, L. N., Trefethen, A. E., Reddy, S. C. and Driscoll, T. A., Hydrodynamic stability without eigenvalues, Science., 261 (1993), pp. 578584.Google Scholar
[13] Darbyshire, A. G. and Mullin, T., Transition to turbulence in constant-mass-flux pipe flow, J. Fluid. Mech., 289 (1995), pp. 83114.Google Scholar
[14] Chapman, S. J., Subcritical transition in channel flows, J. Fluid. Mech., 451 (2002), pp. 3597.Google Scholar
[15] Hof, B., Juel, A. and Mullin, T., Scaling of the turbulence transition threshold in a pipe, Phys. Rev. Lett., 91 (2003), 244502.Google Scholar
[16] Dou, H.-S., Mechanism of flow instability and transition to turbulence, Int. J. Non-Linear. Mech., 41 (2006), pp. 512517, Scholar
[17] Dou, H.-S., Physics of flow instability and turbulent transition in shear flows, Technical Report of National University of Singapore, 2006, Also as part of the invited lecture: Dou, H.-S., Secret of Tornado, International Workshop on Geophysical Fluid Dynamics and Scalar Transport in the Tropics, NUS, Singapore, 13 Nov.–8 Dec., 2006.Google Scholar
[18] Nishioka, M., Iida, S. and Ichikawa, Y., An experimental investigation of the stability of plane Poiseuille flow, J. Fluid. Mech., 72 (1975), pp. 731751.Google Scholar
[19] Nishi, M., Unsal, B., Durst, F. and Biswas, G., Laminar-to-turbulent transition of pipe flows through puffs and slugs, J. Fluid. Mech., 614 (2008), pp. 425446.Google Scholar
[20] Dou, H.-S., Khoo, B. C. and Yeo, K.S., Energy loss distribution in the plane Couette Flow and the Taylor-Couette flow between concentric rotating cylinders, Int. J. Thermal. Sci., 46 (2007), pp. 262275, Scholar
[21] Bottin, S., Dauchot, O., Daviaud, F. and Manneville, P., Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow, Phys. Fluids., 10 (1998), pp. 25972607.Google Scholar
[22] Lessen, M. and Cheifetz, M. G., Stability of plane Coutte flow with respect to finite two-dimensional disturbances, Phys. Fluids., 18 (1975), pp. 939944.Google Scholar
[23] Leutheusser, H. J. and Chu, V. H., Experiments on plane Couette flow, J. Hydr. Eng. Div., 9 (1971), pp. 12691284.Google Scholar
[24] Reichardt, V. H., Uber die Geschwindigkeitsverteilung in einer geradlinigen turbulenten Cou-ettestromung, ZAMM, 36 (1956), pp. S26–S29.Google Scholar
[25] Lundbladh, A. and Johansson, A., Direct simulation of turbulent spots in plane Couette flow, J. Fluid. Mech., 229 (1991), pp. 499516.Google Scholar
[26] Hegseth, J., Daviaud, F. and Bergé, P., Intermittent turbulence in plane and circular Couette flow, Ordered and Turbulent Patterns in Taylor-Couette Flow, Eds. Andereck, C. D. and Hayot, F., Plenum Press, New York, 1992, pp. 159166.Google Scholar
[27] Tillmark, N. and Alfredsson, P. H., Experiments on transition in plane Couette flow, J. Fluid. Mech., 235 (1992), pp. 89102.Google Scholar
[28] Daviaud, F., Hegseth, J. and Bergé, R., Subcritical transition to turbulence in plane Couette flow, Phys. Rev. Lett., 69 (1992), pp. 25112514.Google Scholar
[29] Malerud, S., Mölfy, K. J. and Goldburg, W. I., Measurements of turbulent velocity fluctuations in a planar Couette cell, Phys. Fluids., 7 (1995), pp. 19491955.Google Scholar
[30] Dauchot, O. and Daviaud, F., Finite-amplitude perturbation and spots growth mechanism in plane Couette flow, Phys. Fluids., 7 (1995), pp. 335343.Google Scholar
[31] Bottin, S., Daviaud, F., Manneville, P. and Dauchot, O., Discontinuous transition to spatiotemporal intermittency in plane Couette flow, Europhys. Lett., 43 (1998), pp. 171176.Google Scholar
[32] Manneville, P., Spots and turbulent domains in a model of transitional plane Couette flow, Theoret. Comput. Fluid. Dyn., 18 (2004), pp. 169181.Google Scholar
[33] Dou, H.-S., Khoo, B.C. and Yeo, K.S., Instability of Taylor-Couette flow between concentric rotating cylinders, Int. J. Therm. Sci., 47 (2008), pp. 14221435, Scholar
[34] Dou, H.-S. and Khoo, B. C., Mechanism of wall turbulence in boundary layer flows, Mod. Phys. Lett. B., 23(3) (2009), pp. 457460, Scholar
[35] Dou, H.-S., Khoo, B. C. and Tsai, H. M., Determining the critical condition for turbulent transition in a full-developed annulus flow, J. Petrol. Sci. Eng., 73(1-2) (2010), pp. 4147, Scholar
[36] Dou, H.-S. and Khoo, B. C., Criteria of turbulent transition in parallel flows, Mod. Phys. Lett. B., 24 (13) (2010), pp. 14371440, Scholar
[37] Bech, K. H., Tillmark, N., Alfredsson, P. H. and Andersson, H. I., An investigation of turbulent plane Couette flow at low Reynolds numbers, J. Fluid. Mech., 286 (1995), pp. 291325.Google Scholar