×

Inviscid instability of an incompressible boundary layer on a compliant surface. (English. Russian original) Zbl 1450.76010

Comput. Math. Math. Phys. 60, No. 7, 1228-1239 (2020); translation from Zh. Vychisl. Mat. Mat. Fiz. 60, No. 7, 1268-1280 (2020).
Summary: The instability of an incompressible boundary layer on a compliant plate with respect to inviscid perturbations is analyzed on the basis of triple-deck theory. It is shown that unstable inviscid perturbations persist only if the inertia of the plate is taken into account. It is found that an important role is played by the bending stiffness of the plate. Specifically, as it approaches a certain value, the instability can become arbitrarily high, but, with a further increase in the bending stiffness, it vanishes completely as soon as the bending stiffness reaches a threshold value.

MSC:

76E05 Parallel shear flows in hydrodynamic stability
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kramer, M. O., Boundary-layer stabilization by distributed damping, J. Aeronaut. Sci., 24, 458-460 (1957)
[2] Benjamin, T. B., Effects of a flexible boundary on hydrodynamic stability, J. Fluid Mech., 9, 513-532 (1960) · Zbl 0094.40304 · doi:10.1017/S0022112060001286
[3] Benjamin, T. B., The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows, J. Fluid Mech., 16, 436-450 (1963) · Zbl 0116.19103 · doi:10.1017/S0022112063000884
[4] Landahl, M. T., On the stability of a laminar incompressible boundary layer over a flexible surface, J. Fluid Mech., 13, 609-632 (1962) · Zbl 0104.42801 · doi:10.1017/S002211206200097X
[5] Carpenter, P. W.; Garrad, A. D., The hydrodynamic stability of flow over Kramer-type compliant surfaces: Part 1. Tollmien-Schlichting instabilities, J. Fluid Mech., 155, 465-510 (1985) · Zbl 0596.76053 · doi:10.1017/S0022112085001902
[6] Riley, J. J.; Gad-el-Hak, M.; Metcalfe, R. W., Compliant coatings, Annu. Rev. Fluid Mech., 20, 393-420 (1988) · doi:10.1146/annurev.fl.20.010188.002141
[7] Flow Past Highly Compliant Boundaries and in Collapsible Tubes: Proceedings of the IUTAM Symposium, University of Warwick, UK, March 26-30, 2001 (Kluwer Academic, 2003).
[8] Gad-el-Hak, M., Compliant coatings for drag reduction, Progr. Aerospace Sci., 38, 77-99 (2002) · doi:10.1016/S0376-0421(01)00020-3
[9] Carpenter, P. W., “Recent progress in the use of compliant walls for laminar flow control,” Progress in Industrial Mathematics at ECMI 2006 (2008) · Zbl 1308.76107
[10] Neiland, V. Ya., Theory of laminar boundary layer separation in supersonic flow, Fluid Dyn., 4, 33-35 (1969) · Zbl 0256.76041
[11] Stewartson, K.; Williams, P. G., Self-induced separation, Proc. R. Soc. London Ser. A, 312, 181-206 (1969) · Zbl 0184.52903 · doi:10.1098/rspa.1969.0148
[12] Messiter, A. F., Boundary-layer flow near the trailing edge of a flat plate, SIAM J. Appl. Math., 18, 241-257 (1970) · Zbl 0195.27701 · doi:10.1137/0118020
[13] Savenkov, I. V., The suppression of the growth of nonlinear wave packets by the elasticity of the surface around which flow occurs, Comput. Math. Math. Phys., 35, 73-79 (1995) · Zbl 0842.76028
[14] Savenkov, I. V., Absolute instability of incompressible boundary layer over a compliant plate, Comput. Math. Math. Phys., 58, 264-273 (2018) · Zbl 1444.76050 · doi:10.1134/S096554251802015X
[15] Savenkov, I. V., Influence of inertia of a compliant surface on viscous instability of an incompressible boundary layer, Comput. Math. Math. Phys., 59, 667-675 (2019) · Zbl 1448.76079 · doi:10.1134/S0965542519040146
[16] Zhuk, V. I., Tollmien-Schlichting Waves and Solitons (2001), Moscow: Nauka, Moscow
[17] Savenkov, I. V., Unsteady axisymmetric flow through tubes with elastic walls, Comput. Math. Math. Phys., 36, 255-267 (1996) · Zbl 1027.74509
[18] Walker, J. D. A.; Fletcher, A.; Ruban, A. I., Instabilities of a flexible surface in supersonic flow, Q. J. Mech. Appl. Math., 59, 253-276 (2006) · Zbl 1136.76430 · doi:10.1093/qjmam/hbl001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.