The instability of a developing thermal front in a porous medium. III Subharmonic instabilities

In this paper we study the instability of the developing thermal boundary layer that is induced by suddenly raising the temperature of the lower horizontal boundary of a uniformly cold semi-infinite region of saturated porous medium. The basic state consists of no flow, but the evolving temperature field may be described by a similarity solution involving the complementary error function. In very recent papers, Selim and Rees (2007a) (Part I) have sought to determine when this evolving thermal boundary layer becomes unstable and then Selim and Rees (2007b) (Part II) followed the subsequent evolution of horizontally periodic disturbances well into the nonlinear regime. In this paper we investigate the secondary instability of the nonlinear cells by introducing subharmonic disturbances into the evolving flow. We consider three different types of subharmonic disturbance, namely, the 2:1, 3:2, and 4:3 types. Cellular disturbances are seeded into the evolving basic state, the primary mode having an amplitude that is greater than that of the subharmonic. In general, we find that the subharmonic decays at first, while the primary mode grows, but at a time that is dependent on the relative initial amplitudes, the subharmonic experiences an extremely rapid growth and quickly establishes itself as the dominant flow pattern. A fairly detailed account of the 2:1 case is given, including an indication of how the time of transition between the primary and the subharmonic varies with wave number and initial amplitudes. The other two types of subharmonic disturbance yield a richer variety of behaviors; therefore, we present some typical cases to indicate some of the ways in which the primary mode may be destabilized.