The onset of Taylor-Gortler vortices in the time-dependent Couette flow induced by an impulsively imposed shear stress

Min Chan Kim; Chang Kyun Choi

Korean Journal of Chemical Engineering, Vol.23, No.6, 874-880, November, 2006

DOI10.1007/s11814-006-0002-6 Export Citation

The onset of Taylor-Gortler vortices in the time-dependent Couette flow induced by an impulsively imposed shear stress

Min Chan Kim^†, and Chang Kyun Choi¹

Department of Chemical Engineering, Cheju National University, Cheju 690-756, Korea
¹School of Chemical and Biological Engineering, Seoul National University, Seoul 151-744, Korea

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The onset of Taylor-Gortler instability induced by an impulsively started rotating cylinder with constant shear stress was analyzed by using propagation theory based on linear theory and momentary instability concept. It is well-known that the primary transient Couette flow is laminar but secondary motion sets in when the inner cylinder velocity exceeds a certain critical value. The dimensionless critical time τc to mark the onset of instability is presented here as a function of the modified Taylor number T. For the deep-pool case of small τ, since the inner cylinder velocity increases as Vi∝√t in the present impulsive shear system, the present system is more stable than impulsive started case (Vi=constant). Based on the present τc and the Foster’s [1969] comment, the manifest stability guideline is suggested.

Keywords:Taylor-Gortler Vortex;Time-Dependent Couette Flow;Constant Shear;Propagation Theory

[References]

Chadrasekhar S, Hydrodynamic and hydromagnetic stability, Oxford University Press (1961)
Carslaw HS, Jaeger JC, Conduction of heat in solids, 2nd ed., Oxford University Press (1959)
Chen CF, Christensen DK, Phys. Fluids, 10, 1845 (1967)
Chen CF, Kirchner RP, J. Fluid Mech., 48, 365 (1971)
Choi CK, Kang KH, Kim MC, Hwang IG, Korean J. Chem. Eng., 15(2), 192 (1998)
Choi CK, Park JH, Park HK, Cho HJ, Chung TJ, Kim MC, Int. J. Therm. Sci., 43, 817 (2004)
Foster TD, Phys. Fluids, 12, 2482 (1969)
Hwang IG, Choi CK, J. Cryst. Growth, 162, 182 (1996)
Kang KH, Choi CK, Phys. Fluids, 9, 7 (1997)
Kang KH, Choi CK, Hwang IG, AIChE J., 46(1), 15 (2000)
Kasagi N, Hirata N, Stability of time-dependent flow around a rotating cylinder, Proc. Joint JSME-ASME Applied Mechanics Conference, pp. 431-438 (1975)
Kim MC, Park HK, Choi CK, Theoret. Comput. Fluid Dynamics, 16, 49 (2002)
Kim MC, Chung TJ, Choi CK, Theoret. Comput. Fluid Dynamics, 18, 105 (2004)
Kim MC, Chung TJ, Choi CK, Korean J. Chem. Eng., 21(1), 69 (2004)
Kim MC, Park JH, Choi CK, Chem. Eng. Sci., 60(19), 5363 (2005)
Kim MC, Choi CK, Korean J. Chem. Eng., 21(4), 767 (2004)
Kim MC, Park JH, Choi CK, Chem. Eng. Sci., 60(19), 5363 (2005)
Kirchner RP, Chen CF, J. Fluid Mech., 40, 39 (1970)
Liu DCS, Physical and numerical experiments on time-dependent rotational Couette flow, Ph.D. Thesis, Rutgers University, New Jersey (1971)
MacKerrell SO, Blennerhassett PJ, Bassom AP, Phys. Fluids, 14, 2948 (2002)
Otto SR, IMA J. Appl. Math., 51, 13 (1993)
Schweizer PM, Scriven LE, Phys. Fluids, 26, 619 (1983)
Shen SF, J. Aerosol Sci., 28, 397 (1961)
Tan KK, Thorpe RB, Chem. Eng. Sci., 58(1), 149 (2003)
Walowit J, Tsao S, DiPrima RC, Trans. ASME: J. Appl. Mech., 31, 585 (1964)
Yang DJ, Choi CK, Phys. Fluids, 14, 930 (2002)
Yeckel A, Derby JJ, J. Cryst. Growth, 209, 734 (2000)
Yoon DY, Choi CK, Korean J. Chem. Eng., 6, 144 (1989)

[Referenced By]

[Related Articles]

[1] Chadrasekhar S, Hydrodynamic and hydromagnetic stability, Oxford University Press (1961)

[2] Carslaw HS, Jaeger JC, Conduction of heat in solids, 2nd ed., Oxford University Press (1959)

[3] Chen CF, Christensen DK, Phys. Fluids, 10, 1845 (1967)

[4] Chen CF, Kirchner RP, J. Fluid Mech., 48, 365 (1971)

[5] Choi CK, Kang KH, Kim MC, Hwang IG, Korean J. Chem. Eng., 15(2), 192 (1998)

[6] Choi CK, Park JH, Park HK, Cho HJ, Chung TJ, Kim MC, Int. J. Therm. Sci., 43, 817 (2004)

[7] Foster TD, Phys. Fluids, 12, 2482 (1969)

[8] Hwang IG, Choi CK, J. Cryst. Growth, 162, 182 (1996)

[9] Kang KH, Choi CK, Phys. Fluids, 9, 7 (1997)

[10] Kang KH, Choi CK, Hwang IG, AIChE J., 46(1), 15 (2000)

[11] Kasagi N, Hirata N, Stability of time-dependent flow around a rotating cylinder, Proc. Joint JSME-ASME Applied Mechanics Conference, pp. 431-438 (1975)

[12] Kim MC, Park HK, Choi CK, Theoret. Comput. Fluid Dynamics, 16, 49 (2002)

[13] Kim MC, Chung TJ, Choi CK, Theoret. Comput. Fluid Dynamics, 18, 105 (2004)

[14] Kim MC, Chung TJ, Choi CK, Korean J. Chem. Eng., 21(1), 69 (2004)

[15] Kim MC, Park JH, Choi CK, Chem. Eng. Sci., 60(19), 5363 (2005)

[16] Kim MC, Choi CK, Korean J. Chem. Eng., 21(4), 767 (2004)

[17] Kim MC, Park JH, Choi CK, Chem. Eng. Sci., 60(19), 5363 (2005)

[18] Kirchner RP, Chen CF, J. Fluid Mech., 40, 39 (1970)

[19] Liu DCS, Physical and numerical experiments on time-dependent rotational Couette flow, Ph.D. Thesis, Rutgers University, New Jersey (1971)

[20] MacKerrell SO, Blennerhassett PJ, Bassom AP, Phys. Fluids, 14, 2948 (2002)

[21] Otto SR, IMA J. Appl. Math., 51, 13 (1993)

[22] Schweizer PM, Scriven LE, Phys. Fluids, 26, 619 (1983)

[23] Shen SF, J. Aerosol Sci., 28, 397 (1961)

[24] Tan KK, Thorpe RB, Chem. Eng. Sci., 58(1), 149 (2003)

[25] Walowit J, Tsao S, DiPrima RC, Trans. ASME: J. Appl. Mech., 31, 585 (1964)

[26] Yang DJ, Choi CK, Phys. Fluids, 14, 930 (2002)

[27] Yeckel A, Derby JJ, J. Cryst. Growth, 209, 734 (2000)

[28] Yoon DY, Choi CK, Korean J. Chem. Eng., 6, 144 (1989)