Localized disturbances in parallel shear flows
Tóm tắt
The development of localized disturbances in parallel shear flows is reviewed. The inviscid case is considered, first for a general velocity profile and then in the special case of plane Couette flow so as to bring out the key asymptotic results in an explicit form. In this context, the distinctive differences between the wave-packet associated with the asymptotic behavior of eigenmodes and the non-dispersive (inviscid) continuous spectrum is highlighted. The largest growth is found for three-dimensional disturbances and occurs in the normal vorticity component. It is due to an algebraic instability associated with the lift-up effect. Comparison is also made between the analytical results and some numerical calculations. Next the viscous case is treated, where the complete solution to the initial value problem is presented for bounded flows using eigenfunction expansions. The asymptotic, wave-packet type behaviour is analyzed using the method of steepest descent and kinematic wave theory. For short times, on the other hand, transient growth can be large, particularly for three-dimensional disturbances. This growth is associated with cancelation of non-orthogonal modes and is the viscous equivalent of the algebraic instability. The maximum transient growth possible to obtain from this mechanism is also presented, the so called optimal growth. Lastly the application of the dynamics of three dimensional disturbances in modeling of coherent structures in turbulent flows is discussed.
Tài liệu tham khảo
Abramowitz, M. and Stegun, I.A.,Handbook of Mathematical Functions. Dover (1965).
Bark, F., On the wave structure of the wall region of a turbulent boundary layer.J. Fluid Mech. 70 (1975) 229–250.
Benjamin, T.B., The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane.J. Fluid Mech. 10 (1961) 401–419.
Blackwelder, R.F. and Kaplan, R.E., On the wall structure of the turbulent boundary layer.J. Fluid Mech. 76 (1976) 89–112.
Breuer, K.S. and Haritonidis, J.H., The evolution of a localized disturbance in a laminar boundary layer. Part I: Weak disturbances.J. Fluid Mech. 220 (1990) 569–594.
Butler, K.M. and Farrell, B.F., Three-dimensional optimal perturbations in viscous shear flow.Phys. Fluids A. 4 (1992) 1637–1650.
Case, K.M., Stability of inviscid plane Couette flow.Phys. Fluids 3 (1960) 143–148.
Chin, W.C.,Physics of slowly varying wavetrains in continuous systems. PhD thesis, Massachusetss Institute of Technology, Department of Aeronautics and Astronautics, Cambridge, Mass. (1976).
Corke, T.C., Bar-Sever, A. and Morkovin, M.V., Experiments on transition enhancement by distributed roughness.Phys. Fluids 29 (1986) 3199–3213.
Davis, S.H. and Reid, W.J., On the stability of stratified viscous plane Couette flow.J. Fluid Mech. 80 (1977) 50.
Dikii, L.A., The stability of plane-parallel flows of an ideal fluid.Sov. Phys. Doc. 135 (1960) 1179–1182.
DiPrima, R.C. and Habetler, G.J., A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stability.Arch. Rat. Mech. Anal. 32 (1969) 218–227.
Drazin, P.G. and Reid, W.H.,Hydrodynamic Stability. Cambridge University Press (1981).
Ellingsen, T. and Palm, E., Stability of linear flow.Phys. Fluids. 18 (1975) 487–488.
Farrell, B.F., Optimal excitation of perturbations in viscous shear flow.Phys. Fluids. 31 (1988) 2093–2102.
Gaster, M. 1968, The development of three-dimensional wave packets in a boundary layer.J. Fluid Mech. 32 (1968) 173–184.
Gaster, M., Estimates of the errors incurred in various asymptotic representations of wave packets.J. Fluid Mech. 121 (1982) 365–377.
Gaster, M. and Grant, I., An experimental investigation of the formation and development of a wave packet in a laminar boundary layer.Proc. Roy. Soc. Lond. Ser. A. 347 (1975) 253–269.
Golub, G.H. and van Loan, C.F.,Matrix Computations. Johns Hopkins University Press (1983).
Grosch, C.E. and Salwen, H., The continuous spectrum of the Orr-Sommerfeld equation. Part 1. The spectrum and the eigenfunctions.J. Fluid Mech. 87 (1978) 33–54.
Gustavsson, L.H.,On the evolution of disturbances in boundary layer flows. PhD thesis, Royal Institute of Technology, Stockholm, Sweden (TRITA-MEK 78-02) (1978).
Gustavsson, L.H., Excitation of direct resonances in plane Poiseuille flow.Stud. Appl. Math. 75 (1986) 227–248.
Gustavsson, L.H. 1991. Energy growth of three-dimensional disturbances in plane Poiseuille flow.J. Fluid Mech. 224 (1991) 241–260.
Henningson, D.S., Solution of the three-dimensional Euler equations for the development of a flat eddy in a boundary layer. Master's thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA (1985).
Henningson, D.S., The inviscid initial value problem for a piecewise linear mean flow.Stu. Appl. Math. 78 (1988) 31–56.
Henningson, D.S., Wave growth and spreading of a turbulent spot in plane Poiseuille flow.Phys. Fluids A. 1 (1989) 1876–1882.
Henningson, D.S., An eigenfunction expansion of localized disturbances. In Johansson, A.V. and Alfredsson, P.H. (eds.),Advances in Turbulence 3. Stockholm: Springer-Verlag (1991) pp. 162–169.
Henningson, D.S., Lundbladh, A. and Johansson, A.V., A mechanism for bypass transition from localized disturbances in wall bounded shear flows.J. Fluid Mech. 250 (1993) 169–207.
Henningson, D.S. and Schmid, P.J., Vector eigenfunction expansions for plane channel flows.Stud. Appl. Math 87 (1992) 15–43.
Herbert, T., Secondary instability of boundary layers.Ann. Rev. Fluid Mech. 20 (1988) 487–526.
Hultgren, L.S. and Gustavsson, L.H., Algebraic growth of disturbances in a laminar boundary layer.Phys. Fluids 24 (1981) 1000–1004.
Jimenez, J. and Whitham, G.B., An averaged Lagrangian method for dissipative wavetrains.Proc. Roy. Soc. Lond. Ser. A 349 (1976) 277–287.
Johansson, A.V., Alfredsson, P.H. and Kim, J., Shear layer structure in near wall turbulence. InProceedings of the 1987 summer program, pp. 237–251. NASA / Stanford Center for turbulence research (1987).
Johansson, A.V., Alfredsson, P.H. and Kim, J., Evolution and dynamics of shear-layer structures in near-wall turbulence.J. Fluid Mech. 224 (1991) 579–599.
Joseph, D.D., Eigenvalue bounds for the Orr-Sommerfeld equation.J. Fluid Mech 33 (1968) 617–621.
Joseph, D.D., Eigenvalue bounds for the Orr-Sommerfeld equation. Part 2.J. Fluid Mech. 36 (1969) 721–734.
Kim, H.T., Kline, S.J. and Reynolds, W.C., The production of turbulence near a smooth wall in a turbulent boundary layer.J. Fluid Mech. 50 (1971) 133.
Klingmann, B.G.B., On transition due to three-dimensional disturbances in plane Poiseuille flow.J. Fluid Mech. 240 (1992) 167–195.
Kreiss, G., Lundbladh, A. and Henningson, D.S., Bounds for threshold amplitudes in subcritical shear flows. TRITA-NA 9307, Royal Institute of Technology, Stockholm, Sweden (1993).
Landahl, M.T., A wave-guide model for turbulent shear flow.J. Fluid Mech. 29 (1967) 441–459.
Landahl, M.T., Wave mechanics of breakdown.J. Fluid Mech. 56 (1972) 775–802.
Landahl, M.T., Wave breakdown and turbulence.SIAM J. Appl. Math. 28 (1975) 735–756.
Landahl, M.T., Modeling of coherent structures in boundary layer turbulence. Workshop on coherent structure of turbulent boundary layers, Lehigh University (1979).
Landahl, M.T., A note on an algebraic instability of inviscid parallel shear flows.J. Fluid Mech. 98 (1980) 243–251.
Landahl, M.T., The application of kinematic wave theory to wave trains and packets with small dissipation.Phys. Fluids 25 (1982) 1512–1516.
Landahl, M.T., Theoretical modeling of coherent structures in wall bounded shear flows. Eighth Biennial Symposium on Turbulence, University of Missouri-Rolla (1983).
Landahl, M.T., On sublayer streaks.J. Fluid Mech. 212 (1990) 593–614.
Lin, C.C.,Theory of Hydrodynamic Stability. Cambridge University Press (1955).
Lundbladh, A., Henningson, D.S. and Reddy, S.C., Threshold amplitudes for transition in channel flows. Proceedings from the 1993 ICASE/NASA Langley Workshop on the Transition to Turbulence (1993).
Mack, L.M., A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer.J. Fluid Mech. 73 (1976) 497–520.
Morkovin, M.V., The many faces of transition. In Wells, C.S. (ed.),Viscous Drag Reduction, Plenum Press (1969).
Orr, W.M.F., The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: A perfect liquid. Part II: A viscous liquid.Proc. R. Irish Acad. A 27 (1907) 9–138.
Prandtl, L., Bericht über Untersuchungen zur ausgebildeten Turbulenz.,Zeitschrift für Angewandte Mathematik und Mechanik 5 (1925) 136.
Rayleigh, J.W.S., On the stability, or instability, of certain fluid motions.Proc. Math. Soc. Lond. 11 (1880) 57–70.
Reddy, S.C.,Pseudospectra of operators and discretization matrices and an application to the method of liens. PhD thesis, Massachusetts Institute of Technology, Cambridge (1991).
Reddy, S.C. and Henningson, D.S., Energy growth in viscous channel flows.J. Fluid Mech. 252 (1993) 209–238.
Reddy, S.C., Schmid, P.J. and Henningson, D.S., Pseudospectra of the Orr-Sommerfeld operator.SIAM J. Appl. Math. 53 (1993) 15–47.
Russell, J.M. and Landahl, M.T., The evolution of a flat eddy near a wall in an inviscid shear flow.Phys. Fluids 27 (1984) 557–570.
Schensted, I.V.,Contribution to the theory of hydrodynamic stability. PhD thesis, University of Michigan, Ann Arbor (1961).
Schmid, P.J. and Henningson, D.S., A new mechanism for rapid transition involving a pair of oblique waves.Phys. Fluids A 4 (1992) 1986–1989.
Sommerfeld, A., Ein Beitrag zur hydrodynamischen Erklärung der turbulenten Flüssigkeitbewegungen. InAtti. del 4. Congr. Internat. dei Mat. III pp. 116–124, Roma (1908).
Squire, H.B., On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls.Proc. Roy. Soc. London Ser. A. 142 (1933) 621–628.
Stewartson, K., Some aspects of nonlinear stability theory.Fluid Dyn. Trans. 7 (1974) 101–128.
Trefethen, L.N., Pseudospectra of matrices. InNumerical Analysis 1991 (1992) pp. 234–266. Longman.
Trefethen, L.N., Trefethen, A.E., Reddy, S.C. and Driscoll, T.A., Hydrodynamic stability without eigenvalues.Science 261 (1993) 578–584.
Whitham, G.B.,Linear and Nonlinear waves. Wiley-Interscience (1974).
Wygnanski, I., Haritonidis, J.H. and Kaplan, R.E., On a Tollmien-Schlichting wave packet produced by a turbulent spot.J. Fluid Mech. 92 (1979) 505–528.