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TRANSITION THRESHOLDS IN BOUNDARY LAYER AND CHANNEL FLOWS P.J. SCHMID Department of Applied Mathematics, Box 352420 University of Washington, Seattle, WA 98195, U.S.A. S.C. REDDY Department of Mathematics Oregon State University, Corvallis, OR 97331, U.S.A. AND D.S. HENNINGSON Aeronautical Research Institute of Sweden (FFA) Box 11021, S-16111 Bromma, Sweden

1. Introduction Despite more than a century of work, transition to turbulence in boundary layer and channel ows is still not completely understood. A transition scenario studied in detail is the secondary instability theory (Klebano, Tidstrom, & Sargent, 1962; Orszag & Patera 1983; Bayly, Orszag & Herbert, 1988): TS-wave ! nonlinear 2D-state ! secondary instability of 2D-state (A) The initial disturbance is a nite-amplitude 2D Tollmien-Schlicting wave, the least stable mode of the linearized Navier-Stokes equations, plus noise. This wave evolves into a equilibrium or quasi-equilibrium state, which is unstable to three-dimensional disturbances. The theory agrees qualitatively and quantitately with experiments where a 2D disturbance is created using the vibrating ribbon technique. On the other hand, the theory may not explain natural transition, which is inherently three-dimensional. In recent work on transition in channel and boundary layer ows the following scenarios have been investigated (Schmid & Henningson 1992; Kreiss, Lundbladh & Henningson 1994; Lundbladh, Reddy & Henningson, 1994;

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Berlin, Lundbladh & Henningson 1994): vortices ! streaks ! secondary instability of streaks (B) oblique waves ! vortices ! streaks ! secondary instability of streaks (C) Scenario (B) is initiated by an array of streamwise vortices, periodic in the spanwise direction, plus noise. This disturbance has the greatest potential for linear transient growth in channel and boundary layer ows (Butler & Farrell 1992; Reddy & Henningson 1993). Streamwise vortices generate streaks by the lift-up mechanism. Streaks break down due to a spanwise in ectional instability (Yu & Liu, 1994; Walee 1995). In scenario (C) oblique waves interact nonlinearly to create streamwise vortices. The rest of the scenario is similar to (B). Noise is added to the initial disturbances to break symmetry, and is required for (B) for transition. The purpose of this paper is to compare the threshold energy and time for transition to turbulence for the above three scenarios in the temporal case for plane Poiseuille ow (PPF) and the Blasius boundary layer (BL).

2. Numerical Simulations Computations are done using a spectral Fourier-Chebyshev algorithm (Lundbladh, Henningson & Johansson 1992). The computational domain is assumed to be periodic in the streamwise and spanwise directions. In the boundary layer case, the growth of the boundary layer is modeled by translating the computational domain downstream with a representative velocity and adjusting the boundary layer thicknessRaccordingly. We de ne disturbance energy by E = 21V period(u2 + v 2 + w2)dx, where the region of integration is one period and V is the volume of the periodic box. We de ne the time for transition T as the time at which the friction coecient cf reaches its mean value. If a turbulent state is not achieved, we de ne T = 1. For PPF the Reynolds number is based on the half-channel height and the centerline velocity. For BL ow the Reynolds number is based on the displacement thickness and the freestream velocity. We compute the initial TS waves by solving the Orr-Sommerfeld equation by a spectral method. The initial vortices and obliques waves in (B) and (C) are optimal disturbances (Butler & Farrell 1992; Reddy & Henningson 1993). Noise of 1% of the disturbance energy in the form of Stokes modes is added to the initial disturbance.

3. Results and Discussion The results are shown in Figure 1. For both ows the lowest energy for transition is achieved by (C). For plane Poiseuille ow our results indicate

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TRANSITION THRESHOLDS −1

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Figure 1. Initial disturbance energy versus transition time. (a) PPF with R = 1500. For (A) and (C), the fundamental streamwise and spanwise wavenumbers are 0 = 1 and 0 = 1, respectively, and 0 = 1 and 0 = 2 for (B). The initial disturbance has  = 1 and = 0 for (A),  = 0 and = 2 for (B), and  = 1 and = 1 for (C). The results have been con rmed using three grids (highest resolution 32  81  64.) (b) Temporally growing BL with initial Reynolds number 500 and frequency F = 160: For (B) and (C), the fundamental streamwise and spanwise wavenumbers are 0 = 0:3 and 0 = 0:325; respectively, and 0 = 0:2 and 0 = 0:2 for (A). The initial disturbance has  = 1 and = 0 for (A),  = 0 and = 2 for (B), and  = 1 and = 1 for (C). The resolution has been chosen as 32  65  32; and the advection speed of the computational domain has been 0:4:

that the threshold energy for transition for (A) is about 2 orders of greater for (A) than for (B) and (C). Scenarios (B) and (C) are three-dimensional from the outset and make use of linear transient growth mechanisms. In

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the Blasius boundary layer, the secondary instability scenario is more competitive. The Reynolds number increases in the downstream direction. It is expected that scenario (A) can be even more competitive for lower frequencies since the distance between branches I and II in the stability diagram increases as the frequency decreases. We plan on investigating these scenarios in greater detail in future work.

Acknowledgements PJS has been supported in part by NSF grant DMS-9406636. PJS and SCR gratefully acknowledge the hospitality of the Department of Mechanics, Royal Institute of Technology, Stockholm, Sweden, where part of this work was done. Some computations were done at the Pittsburgh Supercomputing Center.

References

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ows, Phy. Fluids A, 4, pp. 1637-1650. Klebano, P.S., Tidstrom, K.D. & Sargent, L.M. (1962) The three-dimensional nature of boundary-layer instability, J. Fluid Mech., 12, pp. 1-34. Kreiss, G., Lundbladh, A. and Henningson, D.S. (1994) Bounds for threshold amplitudes in subcritical shear ows. J. Fluid Mech., 270, 175-198. Lundbladh, A., Henningson, D.S. and Johansson, A.V. (1992) An ecient spectral integration method for the solution of the Navier-Stokes equations. FFA Technical Report, FFA-TN 1992-28, Aeronautical Research Institute of Sweden, Bromma, Sweden. Lundbladh, A., Henningson, D.S. and Reddy, S.C. (1994) Threshold amplitudes for transition in channel ows, in: Hussaini, M.Y., Gatski, T.B. and Jackson, T.L. (eds.) Transition ,Turbulence, and Combustion, Volume I, pp. 309-318, Kluwer, Dordrecht, Holland. Orszag, S.A. & Patera, A.T. (1983) Secondary instability of wall-bounded shear ows, J. Fluid Mech., 128, pp. 347-385. Reddy, S.C. and Henningson, D.S. (1993) Energy growth in viscous channel ows, J. Fluid Mech., 252, pp. 209-238. Schmid, P.J. and Henningson, D.S. (1992) A new mechanism for rapid transition involving a pair of oblique waves Phys. Fluids A, 4, 1986-1989. Spalart, P.R. and Yang, K.-S. (1987) Numerical study of ribbon-induced transition in Blasius ow, J. Fluid Mech.,178, pp. 345-365. Yu, X & Liu, J.T.C. (1994) On the mechanism of sinuous and varicose modes in threedimensional viscous secondary instability of nonlinear Gorter vortices, Phys. Fluids A, 6, pp. 736-750. Walee, F. (1995) Hydrodynamic stability and turbulence: Beyond transients to a selfsustaining process, Stud. Appl. Math., 95, pp. 319-343.