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Bounds for threshold amplitudes in subcritical shear flows. (English) Zbl 0813.76024
Summary: A general theory which can be used to derive bounds on solutions to the Navier-Stokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable half-plane is used to bound the energy growth of the full nonlinear problem. Plane Couette flow is used as an example. The norm of the resolvent in plane Couette flow in the unstable half-plane is proportional to the square of the Reynolds number \((R)\). This is now used to predict the asymptotic behaviour of the threshold amplitude below which all disturbances eventually decay. A lower bound is found to be \(R^{-21/4}\). Examples, obtained through direct numerical simulation, give an upper bound on the threshold curve, and predict a threshold of \(R^{-1}\). The discrepancy is discussed in the light of a model problem.

MSC:
76E05 Parallel shear flows in hydrodynamic stability
76E30 Nonlinear effects in hydrodynamic stability
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Grohne, AVA Göttingen Rep. 28 pp 1071– (1969)
[2] Willke, J. Math. Phys. 46 pp 151– (1967) · Zbl 0168.46702 · doi:10.1002/sapm1967461151
[3] DOI: 10.1017/S0022112060001171 · Zbl 0096.21103 · doi:10.1017/S0022112060001171
[4] DOI: 10.1137/0728057 · Zbl 0731.65064 · doi:10.1137/0728057
[5] Gor’kov, Zh. Eksp. Teor. Fiz. 33 pp 402– (1957)
[6] DOI: 10.1007/BF00276872 · Zbl 0622.76061 · doi:10.1007/BF00276872
[7] Trefethen, Science 261 pp 578– (1993)
[8] DOI: 10.1063/1.861156 · Zbl 0308.76030 · doi:10.1063/1.861156
[9] Tillmark, J. Fluid Mech. 235 pp 89– (1992)
[10] Ehrenstein, J. Fluid Mech. 228 pp 111– (1991)
[11] DOI: 10.1017/S002211206000116X · Zbl 0096.21102 · doi:10.1017/S002211206000116X
[12] DOI: 10.1007/BF00281139 · Zbl 0181.54703 · doi:10.1007/BF00281139
[13] DOI: 10.1007/BF00284160 · doi:10.1007/BF00284160
[14] DOI: 10.1017/S0022112069002217 · Zbl 0214.25404 · doi:10.1017/S0022112069002217
[15] DOI: 10.1063/1.858367 · doi:10.1063/1.858367
[16] DOI: 10.1017/S0022112085002002 · Zbl 0586.76058 · doi:10.1017/S0022112085002002
[17] Romanov, Funkcional Anal. i Proložen. 7 pp 72– (1973)
[18] Butler, Phys. Fluids A 4 pp 1637– (1992) · doi:10.1063/1.858386
[19] Reynolds, Phil. Trans. R. Lond. Soc. A 186 pp 123– (1895)
[20] DOI: 10.1137/0153002 · Zbl 0778.34060 · doi:10.1137/0153002
[21] Reddy, J. Fluid Mech. 252 pp 209– (1993)
[22] Orr, Proc. R. Irish Acad. A 27 pp 9– (1907)
[23] Nagata, J. Fluid Mech. 217 pp 519– (1990)
[24] Meksyn, Proc. R. Soc. Lond. A 208 pp 517– (1951)
[25] DOI: 10.1017/S0022112058000410 · Zbl 0082.39603 · doi:10.1017/S0022112058000410
[26] Lundbladh, J. Fluid Mech. 229 pp 499– (1991)
[27] DOI: 10.1017/S0022112087000892 · Zbl 0616.76071 · doi:10.1017/S0022112087000892
[28] DOI: 10.1007/BF00266474 · Zbl 0141.43803 · doi:10.1007/BF00266474
[29] DOI: 10.1007/BF00250190 · Zbl 0136.23402 · doi:10.1007/BF00250190
[30] Herron, Stud. Appl. Maths 85 pp 269– (1991) · Zbl 0737.76023 · doi:10.1002/sapm1991853269
[31] DOI: 10.1017/S0022112083000099 · Zbl 0517.76050 · doi:10.1017/S0022112083000099
[32] Henningson, Stud. Appl. Maths 87 pp 15– (1992) · Zbl 0751.76032 · doi:10.1002/sapm199287115
[33] Henningson, J. Fluid Mech. 250 pp 169– (1993)
[34] DOI: 10.1017/S0022112080000079 · Zbl 0432.76049 · doi:10.1017/S0022112080000079
[35] DOI: 10.1017/S0022112074002424 · doi:10.1017/S0022112074002424
[36] Gustavsson, J. Fluid Mech. 224 pp 241– (1991)
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