# zbMATH — the first resource for mathematics

Linear stability analysis in compressible, flat-plate boundary-layers. (English) Zbl 1161.76471
Summary: The stability problem of two-dimensional compressible flat-plate boundary layers is handled using the linear stability theory. The stability equations obtained from three-dimensional compressible Navier-Stokes equations are solved simultaneously with two-dimensional mean flow equations, using an efficient shoot-search technique for adiabatic wall condition. In the analysis, a wide range of Mach numbers extending well into the hypersonic range are considered for the mean flow, whereas both two- and three-dimensional disturbances are taken into account for the perturbation flow. All fluid properties, including the Prandtl number, are taken as temperature-dependent. The results of the analysis ascertain the presence of the second mode of instability (Mack mode), in addition to the first mode related to the Tollmien-Schlichting mode present in incompressible flows. The effect of reference temperature on stability characteristics is also studied. The results of the analysis reveal that the stability characteristics remain almost unchanged for the most unstable wave direction for Mach numbers above $$4.0$$. The obtained results are compared with existing numerical and experimental data in the literature, yielding encouraging agreement both qualitatively and quantitatively.

##### MSC:
 76E05 Parallel shear flows in hydrodynamic stability 76N20 Boundary-layer theory for compressible fluids and gas dynamics 76N17 Viscous-inviscid interaction for compressible fluids and gas dynamics
Matlab
Full Text:
##### References:
 [1] Lees, L., Lin, C.C.: Investigations of the stability of the laminar boundary layer in a compressible fluid, NACA TN 1115 (1946) · Zbl 0063.03471 [2] Mack L.M. (1975). Linear stability theory and the problem of supersonic boundary-layer transition. AIAA J. 13(3): 278–289 [3] Mack, L.M.: Boundary-layer linear stability theory. AGARD R 709 (1984) · Zbl 0542.76086 [4] Malik M.R. (1989). Prediction and control of transition in supersonic and hypersonic boundary layers. AIAA J. 27(11): 1487–1493 [5] Arnal, D.: Boundary-layer transition: predictions based on linear theory. AGARD R 793 (1994) [6] Masad J.A. and Abid R. (1995). On transition in supersonic and hypersonic boundary-layers. Int. J. Eng. Sci. 33(13): 1893–1919 · Zbl 0899.76149 [7] Malik M.R. and Anderson E.C. (1991). Real gas effects on hypersonic boundary-layer stability. Phys. Fluids A 3(5): 803–821 · Zbl 0732.76040 [8] Malik M.R. (2003). Hypersonic flight transition data analysis using parabolized stability equations with chemistry effects. J. Spacecr. Rockets 40(3): 332–344 [9] Laufer J. and Vrebalovich T. (1960). Stability and transition of a supersonic laminar boundary-layer on a flat-plate. J. Fluid Mech. 9: 257–299 · Zbl 0097.20501 [10] Kendall, J.M.: Supersonic boundary-layer experiments. In: McCauley, W.D. (ed.) Proceedings of Boundary Layer Transition Study Group Meeting II, Aerospace Corp. (1967) [11] Kendall J.M. (1975). Wind tunnel experiments relating to supersonic and hypersonic boundary-layer transition. AIAA J. 13: 290–299 [12] Coles D. (1954). Measurements of turbulent friction on a smooth flat-plate in supersonic flow. J. Aeronaut. Sci. 21(7): 433–448 [13] Deem, R.E., Murphy, J.S.: Flat-plate boundary-layer transition at hypersonic speeds. AIAA Paper 65–128 (1965) [14] Laufer, J., Marte, J.E.: Results and a critical discussion of transition-Reynolds number measurements on insulated cones and flat-plates in supersonic wind tunnels. Jet Propulsion Lab, Pasedena, CA, Rept. 20-96 (1955) [15] Chen F.-J., Malik M.R. and Beckwith I.E. (1989). Boundary-layer transition on a cone and flat-plate at Mach 3.5. AIAA J. 27(6): 687–693 [16] Schlichting H. (1979). Boundary-layer theory, vol. 7. McGraw-Hill, Inc., New York · Zbl 0434.76027 [17] The Mathworks, Inc., MATLAB: the language of technical computing (2004) [18] Mathews J.H. (1987). Numerical methods. Prentice-Hall, London [19] Mack L.M. (1976). A numerical study of the temporal eigenvalue spectrum of the Blasius boundary-layer. J. Fluid Mech. 73(3): 497–520 · Zbl 0339.76030 [20] Press W.H., Teukolsky S.A., Vetterling W.T. and Flannery B.P. (1992). Numerical recipes in Fortran 77. Cambridge University Press, Cambridge · Zbl 0778.65002 [21] Pruett C.D. and Streett C.L. (1991). A spectral collocation method for compressible, non-similar boundary-layers. Int. J. Numer. Methods Fluids 13: 713–737 · Zbl 0739.76054 [22] Obremski, H.J., Morkovin, M.V., Landahl, M.: A portfolio of stability characreristics of incompressible boundary-layers. AGARDograph 136 (1969) [23] Özgen S. (2004). Effect of Heat Transfer on stability and transition characteristics of boundary-layers. Int. J. Heat Mass Transf. 47: 4697–4712 · Zbl 1112.76342 [24] Hayes W.D. and Probstein R.F. (1959). Hypersonic flow theory. Academic, New York · Zbl 0084.42202 [25] Anderson J.D. (1990). Modern compressible flow. McGraw-Hill, New York [26] Laufer J. (1964). Some statistical properties of the pressure field radiated by a turbulent boundary layer. Phys. Fluids 7(8): 1191–1197 · Zbl 0126.22303 [27] Schubauer, G.B., Skramstad, H.K.: Laminar boundary-layer oscillations and transition on a flat-plate, NACA Report 909 (1946)
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.