×

zbMATH — the first resource for mathematics

A mean flow first harmonic theory for hydrodynamic instabilities. (English) Zbl 0668.76056
A general theory is presented for nonlinear instabilities arising in steady hydrodynamic motions. For quasiparallel flows at high values of the Reynolds number it is found that for relatively small disturbance levels the usual ideas concerning the generation of higher harmonics and the subsequent modification of the fundamental may be overwhelmed by three dimensional interactions between the evolving mean flow and the first harmonic wave. The differences from and similarities to existing asymptotic and numerical studies are discussed. The theory developed applies to a variety of flow configurations. Numerical results are given for Poiseuille flow and the Blasius boundary layer. In addition the theory developed here is applied to simulate the instabilities produced in a boundary layer due to the presence of free stream disturbances.

MSC:
76E30 Nonlinear effects in hydrodynamic stability
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lin, The Theory of Hydrodynamic Stability (1955) · Zbl 0068.39202
[2] Drazin, Hydrodynamic Stability (1981)
[3] Squire, Proc. Roy. Soc. London Sero A 142 pp 621– (1933) · JFM 59.1458.02 · doi:10.1098/rspa.1933.0193
[4] Gustavsson, J. Fluid Mech. 112 pp 253– (1981) · Zbl 0471.76065 · doi:10.1017/S0022112081000384
[5] Meksyn, Proc. Roy. Soc. London Ser. A 208 pp 517– (1951) · Zbl 0043.40003 · doi:10.1098/rspa.1951.0177
[6] Benney, Phys. Fluids 3 pp 656– (1960) · doi:10.1063/1.1706101
[7] Klebanoff, J. Fluid Mech. 12 pp 1– (1962) · Zbl 0131.41901 · doi:10.1017/S0022112062000014
[8] Stuart, J. Fluid Mech. 9 pp 353– (1960) · Zbl 0096.21102 · doi:10.1017/S002211206000116X
[9] Stuart, Lominar Boundary Layers (1963)
[10] Raetz, Norair Rpt. NOR-59-383 (1959)
[11] Craik, Wave Interactions and Fluid Flows (1985)
[12] Craik, J. Fluid Mech. 34 pp 531– (1968) · Zbl 0172.56201 · doi:10.1017/S0022112068002065
[13] Herbert, J. Fluid Mech. 126 pp 167– (1983) · Zbl 0517.76050 · doi:10.1017/S0022112083000099
[14] Zhou, Proc. Roy. Soc. London Ser. A 381 pp 407– (1982) · Zbl 0489.76053 · doi:10.1098/rspa.1982.0080
[15] Kachanov, J. Fluid Mech. 138 pp 209– (1984) · doi:10.1017/S0022112084000100
[16] Benney, Stud. Appl. Math. 64 pp 185– (1981) · Zbl 0481.76048 · doi:10.1002/sapm1981643185
[17] Jang, J. Fluid Mech. 169 pp 109– (1986) · Zbl 0624.76069 · doi:10.1017/S0022112086000551
[18] Patera, Proceedings of the 7th International Conference on Numerical Methods in Fluid Dynamics (1980)
[19] Herbert, Proc. IUTAM Symp., Kyoto, in: Turbulence and Chaotic Phenomena in Fluids (1984)
[20] Benney, Stud. Appl. Math. 70 pp 1– (1984) · Zbl 0566.76046 · doi:10.1002/sapm19847011
[21] Benney, Stud. Appl. Math. 74 pp 227– (1986) · Zbl 0607.76033 · doi:10.1002/sapm1986743227
[22] Benney, Stud. Appl. Math. 73 pp 201– (1985) · Zbl 0588.76093 · doi:10.1002/sapm1985733261
[23] Maslowe, Hydrodynamic Instabilities and the Transition to Turbulence (1981) · Zbl 0453.00052
[24] Nishioka, J. Fluid Mech. 72 pp 731– (1975) · doi:10.1017/S0022112075003254
[25] Oikawa, Stud. Appl. Math. 76 pp 69– (1987) · Zbl 0632.76020 · doi:10.1002/sapm198776169
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.