Mack’s second mode has been known to be the dominant disturbance leading to transition to turbulence in traditional hypersonic boundary layer flows at zero angle of attack. Physically, the second mode exists due to trapped acoustic waves within the boundary layer. The second mode has been widely studied and the conditions that stabilize or amplify the second mode are well documented. Predicting the second mode amplification is the basis of contemporary transition prediction techniques such as the eN method. There has been a renewed interest in studying hypersonic boundary layer stability in high-enthalpy flows with highly-cooled walls due to its applicability to experiments and some real flight conditions. One physical phenomenon that occurs in these flows is the creation of a supersonic mode, which is associated with an unstable mode F1 synchronizing with the slow acoustic spectrum. This causes the disturbance to travel upstream supersonically relative to the mean flow outside the boundary layer and radiate sound away from the boundary layer. The supersonic mode has been known to exist for decades, but has until recently been deemed negligible in comparison to the second mode. However, a resurgence in interest in the supersonic mode has shown the supersonic mode to exist in unexpected conditions with considerable peak growth rates compared to the second mode. Namely, recent research in the field has shown the supersonic mode in hot-wall flows, upending the notion that it is an artifact of highly-cooled walls. Additionally, a dominant supersonic mode with significantly larger growth rate than the second mode has been found on very blunt cones. Therefore, because the supersonic mode has not been systematically investigated, the mechanism of its creation and the conditions under which it exists are not yet clear.
The objective of this work is to systematically investigate the supersonic mode using numerical and theoretical tools to simulate hypersonic flow over blunt cones. Specifically, this work aims to (1) Determine the characteristics of the supersonic mode and under what conditions it exists, (2) Explore the effectiveness of Linear Stability Theory (LST) on predicting the supersonic mode, and (3) Examine the impact of the supersonic mode on transition to turbulence under realistic flight or experimental conditions. This work explores the supersonic mode on a 1 mm nose radius cone in various free stream flow configurations with a 5-species, two-temperature nonequilibrium gas model for air. A combined approach of Direct Numerical Simulation (DNS) and Linear Stability Theory (LST) are used to numerically investigate the supersonic mode. New LST equations with linearized Rankine-Hugoniot shock relation boundary conditions are derived and verified. In addition, a theoretical schematic has been developed to aid future experimentalists and those performing DNS in visualizing the supersonic mode. Mach numbers of 5 and 10 are considered with wall-temperature-to-free-stream-temperature ratios (Tw/T∞) between 0.2 and 1.43. Additionally, the impact of thermochemical nonequilibrium on the supersonic mode is assessed. Both LST and DNS results have confirmed the existence of the supersonic mode on a Mach 5 axisymmetric cold-wall (Tw/T∞ = 0.2) cone. On a warmer wall (Tw/T∞ = 0.667) under the same free stream conditions, LST indicated the supersonic mode was stabilized, although some weak sound radiation was still apparent in DNS.
For the Mach 10 case, LST predicted a stable supersonic mode for both wall temperature cases (Tw/T∞ = 1.43, Tw/T∞ = 0.43), however a prominent supersonic mode was observed in DNS. The supersonic mode was determined to be excited via a modal interaction that is ignored in LST due to the independent mode assumption. Furthermore, the supersonic mode in the Mach 10 case with Tw/T∞ = 0.43 exhibited a stronger peak growth rate for the supersonic mode compared to Mack’s traditional second mode. These findings illustrate the need for combined LST and DNS studies of the supersonic mode. Overall, this study has determined that the supersonic mode is destabilized by largely the same factors as Mack’s second mode. Namely, wall cooling is destabilizing, increasing Mach number/stagnation enthalpy is destabilizing, and vibrational nonequilibrium is stabilizing. The impact of chemical nonequilibrium is hypothesized to be slightly destabilizing, although was not able to be confirmed with the cases explored here. Based on the results presented here, transition prediction analyses relying on LST, such as the eN method, should be used with caution when applied to the supersonic mode, as it has been shown that LST may not fully capture the mechanism of the supersonic mode’s creation.