Analysis and direct numerical simulation of the flow at a gravity-current head. Part 2. The lobe-and-cleft instability

Results are presented from a linear-stability analysis of the flow at the h ead of two-dimensional gravity-current fronts. The analysis was undertaken in order to clarify the instability mechanism that leads to the formation o f the complex lobe-and-cleft pattern which is commonly observed at the lead ing edge of gravity currents propagating along solid boundaries. The stabil ity analysis concentrates on the foremost part of the front, and is based o n direct numerical simulation data of two-dimensional lock-exchange flows w hich are described in the companion paper, Hartel et al. (2000). High-order compact finite differences are employed to discretize the stability equati ons which results in an algebraic eigenvalue problem for the amplification rate, that is solved in an iterative fashion. The analysis reveals the exis tence of a vigorous linear instability that acts in a localized way at the leading edge of the front and originates in an unstable stratification in t he flow region between the nose and stagnation point. It is shown that the amplification rate of this instability as well as its spanwise length scale depend strongly on Reynolds number. For validation, three-dimensional dire ct numerical simulations of the early stages of the frontal instability are performed, and close agreement with the results from the linear-stability analysis is demonstrated.