# zbMATH — the first resource for mathematics

Spatial stability of the incompressible corner flow. (English) Zbl 1148.76312
Summary: The linear spatial stability of the incompressible corner flow under pressure gradient has been studied. A self-similar form has been used for the mean flow, which reduces the related problem to the solution of a two-dimensional problem. The stability problem was formulated using the parabolised stability equations (PSE) and results were obtained for the viscous modes at medium and high frequencies. The related $$N$$-factors indicate that the flow is stable at these frequencies, but probably unstable for small frequencies. Furthermore the inviscid mode for each mean flow was obtained and the results indicate that its importance increases considerably with an increase in the adverse pressure gradient. Finally the dependence of the stability characteristics on the extent of the domain is also considered.

##### MSC:
 76E05 Parallel shear flows in hydrodynamic stability 76D10 Boundary-layer theory, separation and reattachment, higher-order effects
##### Keywords:
transition; PSE; boundary layer; instability
eigs
Full Text:
##### References:
 [1] Andersson, P., Henningson, D.S., Hanifi, A.: On a stabilization procedure for the parabolic stability equations. J. Eng. Math. 33(3), 311–332 (1998) · Zbl 0920.76025 [2] Balachandar, S., Malik, M.R.: Inviscid instability of streamwise corner flow. J. Fluid Mech. 282, 187–201 (1995) · Zbl 0831.76014 [3] Barclay, W.H.: Experimental investigation of the laminar flow along a straight 135{$$\deg$$} corner. Aero. Q. 24(2), 147–154 (1973) [4] Barclay, W.H., El-Gamal, H.A.: Streamwise corner flow with wall suction. AIIA J. 21, 31–37 (1983) · Zbl 0569.76074 [5] Barclay, W.H., Ridha, A.H.: Flow in streamwise corners of arbitrary angle. AIAA J. 18(12), 1413–1420 (1980) · Zbl 0453.76040 [6] Bertolotti, F.B., Herbert, Th.: Analysis of the linear stability of compressible boundary layers using the PSE. Theoret. Comput. Fluid Dyn. 3, 117–124 (1991) · Zbl 0748.76050 [7] Bertolotti, F.B., Herbert, Th., Spalart, P.R.: Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474 (1992) · Zbl 0754.76029 [8] Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2000) [9] Dhanak, M.R.: On the instability of flow in a streamwise corner. Proc. R. Soc. Lond. A 441, 201–210 (1993) · Zbl 0797.76024 [10] Dhanak, M.R., Duck, P.W.: The effects of freestream pressure gradient on a corner boundary layer. Proc. R. Soc. London A 453, 1793–1815 (1997) · Zbl 0941.76019 [11] Duck, P.W., Stow, S.R., Dhanak, M.R.: Non-similarity solutions to the corner boundary-layer equations (and the effects of wall transpiration). J. Fluid Mech. 400, 125–162 (1999) · Zbl 0951.76022 [12] El-Hady, N.M.: Nonparallel Stability Theory of Three-Dimensional Compressible Boundary Layers Part I – Stability Analysis. NASA Contractor Report CR-3245 (1980) [13] Gallionis, I.: Instability of three-dimensional flows. Dissertation, London University (2003) [14] Herbert, Th.: Parabolised stability equations. Annu. Rev. Fluid Mech. 29, 245–283 (1997) [15] Janke, E., Balakumar, P.: On the Stability of Three-dimensional Boundary Layers. Part 1: Linear and Nonlinear Stability. NASA/CR-1999-209330, ICASE Report No. 99-16 (1999) [16] Joslin, R.D., Streett, C.L., Chang, C.L.: Spatial direct numerical simulation of boundary-layer transition mechanisms: validation of PSE theory. Theoret. Comput. Fluid Dyn. 4(6), 271–288 (1993) · Zbl 0781.76062 [17] Lakin, W.D., Hussaini, M.Y.: Stability of the laminar boundary layer in a streamwise corner. Proc. R. Soc. Lond. A 393, 101–116 (1984) · Zbl 0547.76053 [18] Li, F., Malik, M.R.: On the nature of PSE approximation. Theoret. Comput. Fluid Dyn. 8, 253–273 (1996) · Zbl 0947.76022 [19] Lin, C.C.: On the stability of two-dimensional parallel flows. Q. Appl. Math. 3, 117–142 (1945); 3, 213–234 (1945); 3, 277–301 (1946) · Zbl 0061.43503 [20] Lin, R.S., Wang, W.P., Malik, K.R.: Linear stability of incompressible flow along a corner. ASME Fluids Engineering Division Summer Meeting (1996) [21] Malik, M.R., Li, F.: Three-Dimensional Boundary Layer Stability and Transition. SAE Technical Paper Series 921991 (1992) [22] Mughal, M.S.: Transition Prediction in Fully 3D Compressible Flows. I.C. Contract Report prepared for QinetiQ/BAe(MAD) (2001) [23] Pal, A., Rubin, S.G.: Asymptotic features of the viscous flow along a corner. Q. Appl. Math. 29, 91–108 (1971) · Zbl 0227.76037 [24] Parker, S.J., Balachandar, S.: Viscous and inviscid instabilities of flow along a streamwise corner. Theoret. Comput. Fluid Dyn. 13, 231–270 (1999) · Zbl 0968.76022 [25] Ridha, A.: Sur la couche limite incompressible laminaire le long d’un dièdre. C. R. Acad. Sci. Paris Sér. II 311, 1123–1128 (1990) · Zbl 0704.76019 [26] Ridha, A.: On the dual solutions associated with boundary-layer equations in a corner. J. Eng. Math. 26, 525–537 (1992) · Zbl 0762.76020 [27] Ridha, A.: On laminar natural convection in a vertical corner. C. R. Acad. Sci. Paris t. 328, Sér. II b, 485–490 (2000) · Zbl 1004.76079 [28] Ridha, A.: Combined free and forced convection in a corner. Int. J. Heat Mass Transf. 45, 2191–2205 (2000) · Zbl 1021.76049 [29] Rubin, S.G.: Incompressible flow along a corner. J. Fluid Mech. 26, 97–110 (1966) · Zbl 0139.44103 [30] Rubin, S.G., Grossman, B.: Viscous flow along a corner: numerical solution of the corner layer equations. Q. Appl. Math. 29, 169–186 (1971) · Zbl 0233.76057 [31] Schubauer, G.B., Skarmstad, H.K.: Laminar boundary-layer oscillations and stability of laminar flow. J. Aero. Sci. 14(2), 69–78 (1947) [32] Sorenson, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–185 (1992) [33] van der Vorst, H.: BICGSTAB: A fast and smoothly converging variant of BiCG for the solution of non-symmetric linear systems. SIAM J. Sci. Statist. Comput. 13, 631–644 (1992) · Zbl 0761.65023 [34] Weinberg, B.C., Rubin, S.G.: Compressible corner flow. J. Fluid Mech. 56, 753–774 (1972) · Zbl 0273.76050 [35] Zamir, M.: Further solution of the corner boundary-layer equations. Aero. Q. 24(3), 219–226 (1973) [36] Zamir, M., Young, A.D.: Pressure gradient and leading edge effects on the corner boundary layer. Aero. Q. 30(3), 471–484 (1979) [37] Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijhkout, V., Pozo, R., Romine, C., Vorst, H.: Templates for the solution of linear systems. Building blocks for iterative methods. SIAM (1994) · Zbl 0814.65030
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.