Bifurcation analysis of scramjet unstart

Scramjet unstart occurs when a strong shock initiated from disturbances in the engine propagates upstream and finally spills out of the engine inlet. When unstart occurs, the airflow into the engine is greatly diminished, leading to loss-of-thrust and engine stall. Because the probablility of unstart increases with increasing heat-release, the danger of unstart is an important limiting factor on the performance of scramjet engines. In the Predictive Science Academic Alliance Program (PSAAP), a large computational and experimental effort has been made at Stanford to quantify the likelihood of unstart for various and potentially uncertain scramjet operating conditions as a way to determine safe operability limits for scramjet engines. The investigation of the dynamics of unstart in a scramjet combustor is motivated by observations of hysteresis effects in both the experiment of Wagner et al. (2010) and in steady Reynolds-averaged Navier-Stokes (RANS) calculations of the same geometry (Jang et al. 2010). For these cases, unstart is a catastrophe in the sense that once un-start is initiated, the scramjet system proceeds irreversibly towards the unstarted state. According to the mathematical study of dynamical systems, this occurs because the branch of steady, started solutions undergoes a fold bifurcation at a critical heat-release rate. Above the critical heat-release rate, the branch of steady started solutions no longer exists, and instead the system is attracted to the branch of solutions corresponding to the unstarted state. In addition to providing physical insight into the dynamics of unstart, bifurcation analysis can also help in the quest to quantify the likelihood of unstart for sub-critical heat-release rates. As we will see below, heat-release rates in the hysteresis zone support both the started and unstarted branches of steady solutions, and these two branches are necessarily separated by a third branch of steady but unstable solutions. The third branch of unstable solutions lies exactly on the boundary between the basins of attraction of the strarted and unstarted solutions. In this brief, we argue that in the neighborhood of the critical heat-release rate, the branch of unstable solutions can be used to define a measure of how close a steady, started solution is to the basin of attraction of the unstarted branch. This new metric represents a combination and refinement of the Rayleigh (thermal-choking) and Korkegi (boundary layer separation) limits (Korkegi 1975, 1985) that have been previously used in determining the unstart bound. The new metric based on the unstable solution …