Secondary instability of a temporally growing mixing layer

Abstract

The three-dimensional stability of two-dimensional vortical states of planar mixing layers is studied by direct numerical integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective timescales, extract their energy from the mean flow and exist at moderately low Reynolds numbers. Their growth rates are comparable with the most rapidly growing inviscid instability and with the growth rates of two-dimensional subharmonic (pairing) modes. At high amplitudes, they can evolve into pairs of counter-rotating, streamwise vortices, connecting the primary spanwise vortices, which are very similar to the structures observed in laboratory experiments. The three-dimensional modes do not appear to saturate in quasi-steady states as do the purely two-dimensional fundamental and subharmonic modes in the absence of pairing. The subsequent evolution of the flow depends on the relative amplitudes of the pairing modes. Persistent pairings can inhibit three-dimensional instability and, hence, keep the flow predominantly two-dimensional. Conversely, suppression of the pairing process can drive the three-dimensional modes to more chaotic, turbulent-like states. An analysis of high-resolution simulations of fully turbulent mixing layers confirms the existence of rib-like structures and that their coherence depends strongly on the presence of the two-dimensional pairing modes.

Instability and transition to turbulence in a free shear layer affected by a parallel magnetic field

“…The algorithm was parallelized and typically executed on 8 to 20 CPUs. As a test of the numerical model, the results of Metcalfe et al (1987) for the transition in a free shear layer with Re = 400 and N = 0 were reproduced with good agreement.…”

“…We can assume that the mechanism leading to these waves is the elliptic instability of strained vortices (Kerswell 2002). Vortical structures similar to the 'ribs' in the 'braid' regions (Metcalfe et al 1987) were observed at some stages of the flow evolution. There are also substantial differences between the magnetic and non-magnetic cases because of the suppression and re-orientation of the secondary structures by the magnetic field and the fact that in the magnetic case the rolls are not perpendicular to the mean flow.…”

“…Extensive earlier investigations of the non-magnetic case (see e.g. Metcalfe et al 1987;Rogers & Moser 1992, 1993 identified the mechanisms and showed that their relative importance was largely determined by the composition of the initial perturbations added to the basic flow and the features of the model such as the horizontal size of the computational domain (one or several wavelengths), choice between temporal or spatial evolution of the layer, etc. In real flows with the initial perturbations being some form of noise, the transition is likely to be a result of superposition of several mechanisms.…”

“…High-q x modes are suppressed by the magnetic field and the spectrum takes the anisotropic form typical for MHD turbulence at moderate to high N (Zikanov & Thess 1998;Vorobev et al 2005). The flow evolution in the case of the one-wave initial conditions has some features reminiscent of the non-magnetic scenario (see, for example, Metcalfe et al 1987;Rogers & Moser 1992, 1993. The wavelength of the secondary waves in figure 9(a) is much smaller than the wavelength of the unstable modes of the linear stability problem for the basic flow.…”

Three-dimensional simulations of compressible tearing instability

“…[12][13][14] When moving from two to three dimensions, it is found that many hydrodynamic and MHD configurations may become unstable to the so called secondary instability. [15][16][17][18][19][20][21] The initial equilibrium starts off being unstable to a twodimensional ͑2D͒ instability ͑primary instability͒ that grows, driving the system to a new 2D configuration that is itself unstable to ideal three-dimensional ͑3D͒ modes that can drive the system toward a turbulent state. It has been shown that the 2D nonlinearly saturated configuration of an initial current sheet subjected to the tearing instability develops an ideal instability that has growth rates exceeding the reconnection rate.…”