On the resonant triad interaction in flows over rigid and flexible boundaries

Besonant interactions in flows over rigid and flexible walls are studied. Attention is concentrated on symmetric, three-dimensional wave triads as proposed by Craik (1971). Location of resonant triads and evaluation of interaction coefficients are performed numerically, for a wide range of Reynolds numbers and wavenumbers, considering the temporal stability problem. Good agreement is found with previous work. It is shown that triads comprising various combinations of Tollmien-Schlichting and/or wall modes are possible, and have some interesting features; also, the possibility of interaction with Squire modes must not be overlooked.

[1]  P. Saffman,et al.  Effect of compliant boundaries on weakly nonlinear shear waves in channel flow , 1990 .

[2]  T. Herbert.  Secondary Instability of Boundary Layers , 1988 .

[3]  P. K. Sen,et al.  On the stability of laminar boundary-layer flow over a flat plate with a compliant surface , 1988, Journal of Fluid Mechanics.

[4]  L. Mack,et al.  A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer , 1976, Journal of Fluid Mechanics.

[5]  A. Craik,et al.  Three-wave resonance for free-surface flows over flexible boundaries , 1988 .

[6]  M. Landahl,et al.  On the stability of a laminar incompressible boundary layer over a flexible surface , 1962, Journal of Fluid Mechanics.

[7]  Daniel D. Joseph,et al.  Eigenvalue bounds for the Orr—Sommerfeld equation. Part 2 , 1969, Journal of Fluid Mechanics.

[8]  F. Smith,et al.  The resonant-triad nonlinear interaction in boundary-layer transition , 1987, Journal of Fluid Mechanics.

[9]  A. D. Garrad,et al.  The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities , 1986, Journal of Fluid Mechanics.

[10]  On the Spectra of the Orr-Sommerfeld Equation. , 1977 .

[11]  A. D. Garrad,et al.  The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 1. Tollmien-Schlichting instabilities , 1985, Journal of Fluid Mechanics.

[12]  W. H. Reid,et al.  On the stability of stratified viscous plane Couette flow. Part 2. Variable buoyancy frequency , 1977, Journal of Fluid Mechanics.

[13]  Three-dimensional spatial secondary instability in boundary-layer flows , 1985 .

[14]  L. H. Thomas.  THE STABILITY OF PLANE POISEUILLE FLOW , 1952 .

[15]  H. Squire.  On the Stability for Three-Dimensional Disturbances of Viscous Fluid Flow between Parallel Walls , 1933 .

[16]  T. Brooke Benjamin,et al.  The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows , 1963, Journal of Fluid Mechanics.

[17]  A. Craik,et al.  Non-linear resonant instability in boundary layers , 1971, Journal of Fluid Mechanics.

[18]  Lennart S. Hultgren,et al.  Algebraic growth of disturbances in a laminar boundary layer , 1981 .

[19]  J. Adam,et al.  ‘Explosive’ resonant wave interactions in a three-layer fluid flow , 1979, Journal of Fluid Mechanics.

[20]  A. E. Gill,et al.  Instabilities of a buoyancy-driven system , 1969, Journal of Fluid Mechanics.

[21]  J. Usher,et al.  Nonlinear wave interactions in shear flows. Part 1. A variational formulation , 1974, Journal of Fluid Mechanics.

[22]  V. Ya. Levchenko,et al.  The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer , 1984, Journal of Fluid Mechanics.

[23]  Philip J. Morris,et al.  The effect of anisotropic wall compliance on boundary-layer stability and transition , 1990, Journal of Fluid Mechanics.

[24]  J. Andrzej Domaradzki,et al.  Stabilization of laminar boundary layers by compliant membranes , 1987 .

[25]  Mohamed Gad-el-Hak,et al.  On the interaction of compliant coatings with boundary-layer flows , 1984, Journal of Fluid Mechanics.

[26]  M. Ross,et al.  Higher eigenstates in boundary-layer stability theory , 1976, Journal of Fluid Mechanics.

[27]  Daniel D. Joseph,et al.  Eigenvalue bounds for the Orr-Sommerfeld equation , 1968, Journal of Fluid Mechanics.

[28]  Frank T. Smith,et al.  On the non-parallel flow stability of the Blasius boundary layer , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[29]  L. Gustavsson,et al.  A resonance mechanism in plane Couette flow , 1980, Journal of Fluid Mechanics.

[30]  R. A. Cairns.  The role of negative energy waves in some instabilities of parallel flows , 1979, Journal of Fluid Mechanics.

[31]  A. Volodin,et al.  Three-wave resonance interaction of disturbances in a boundary layer , 1978 .

[32]  Peter W. Carpenter,et al.  A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls , 1990 .

[33]  L. Gustavsson.  Energy growth of three-dimensional disturbances in plane Poiseuille flow , 1981, Journal of Fluid Mechanics.