Multiple paths to subharmonic laminar breakdown in a boundary layer.

Numerical simulations demonstrate that laminar breakdown in a boundary layer induced by the secondary instability of two-dimensional Tollmien-Schlichting waves to threedimensional subharmonic disturbances need not take the conventional lambda vortex/highshear layer path. J_ 1This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NASl-18605 while the second author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. The classicalboundary-layer transitionexperiments of the early 1960% [I],[2],[3]led to the perception that the laminar-turbulent transitionprocess in a B1asius boundary layer may be viewed as a sequence of instabilities. The first two instabilities in the sequence, called the primary and the secondary instabilities, are linear at onset and are now very well understood. The primary instability in an incompressible flat-plate boundary layer is known as a Tollmien-Schlichting (TS) wave. The TS wave, as it equilibrates nonlinearly, becomes susceptible to what is known as the secondary instability. This is truly three-dimensional in character and is thoroughly reviewed by Herbert [4]. The known secondary instabilities are essentially of three kinds the fundamental secondary instability, the subharmonic secondary instability, and the detuned secondary instability. In the first case, both the primary and the secondary wave have the same streamwise wavelength, and a streamwise vortex system on the scale of the primary wave (a "peak-valley splitting") ensues in which the so-called lambda vortices are aligned in the streamwise direction. The nonlinear evolution of the fundamental secondary instability leads to detached high-shear layers. This stage suffers a tertiary instability manifested by a kink in the high-shear layer and an associated peak in the streamwise perturbation oscillogram, usually called a "spike". Also associated with this stage are hairpin vortex elements. Thereafter, multiple spikes appear through what is possibly a rapid sequence of instabilities, and turbulent spot formation becomes imminent. This particular scenario, which is now known to have dominated all of the classical experiments, is called the K-type breakdown. The breakdown arising from a subharmonic secondary instability was observed experimentally much later [5]-[6]. In the subharmonic case, the streamwise wavelength of the secondary wave is double that of the primary wave, and the resulting pattern of lambda vortices is staggered. The detuned secondary instability involves so-called combination modes, i.e., a pair of secondary waves whose wavenumbers combine to yield the wavenumber of the primary wave. To date, high-shear layers and their associated spikes have not been observed experimentally for either the subharmonic or the detuned breakdowns. Instead, these instabilities appear to lead to breakdown by a rapid broadening of the spectrum, not just to the shorter wavelengths that are associated with the formation of "spikes", but also to even longer streamwise wavelengths [7]. Numerical simulations of boundary-layer transition have produced the characteristic highshear layers and spikes of the K-type breakdown [s],[g],[10],[ii] and the early secondary instability stage of subharmonic transition [12]. In this paper we report numerical simulations which address the details of the laminar breakdown induced by subharmonic secondary instability. Numerical simulations of boundary-layer transition have been reviewed recently by Zang and Hussaini [9]. In the present simulations the parallel flow approximation is employed and the evolution is tracked in time rather than in space. The focus is on the vicinity of some point z0 and the approximation is that the displacement thickness is constant in x (although not necessarily in t), the base flow is strictly in the streamwise direction and it is given by ub(y,t) = (us(y%/'i'_), 0, 0), where us(r/) is the Blasius velocity profile which follows from the similar boundary-layer equations, and X = zo + eft; c! is the speed of a moving computational frame. Lengths are scaled by the displacement thickness, 6_, at z0, and velocitiesby the free-streamvelocity, U_. In these units, X/zo = 1 + R_--;cft, where d = 1.72. Although the boundary layer is assumed to be parallel, it is permitted to thicken in time [12], with the speed of the computational frame, c1, equal to the phase speed of the 2D wave. The numerical simulations reported here were performed with the boundary-layer version of the algorithm described in [13], with the nonlinear terms treated in skew-symmetric form [14]. If a and fl denote the fundamental wavenumbers in the streamwise and spanwise directions, respectively, then the fundamental wavelengths in these directions are given by L, = 27r/a and L_ = 2_r/19. The imposed periodicity lengths are s_L, and s,L_, where s_ and s_ are integers which specify the number of subharmonics that are permitted in each direction. (In the cases presented in this paper s, = 1 and s, = 1, 2, or 9.) The rational numbers k, = _,/s_ and /c, = _/s, label the Fourier wavenumbers in the numerical representations with respect to the fundamental wavenumbers a and t9. Case Re wave a /3 KKVF 1100 TS 0.250 0.00