How Streamwise Thermal Gradients Influence Mack’s Second Mode

The effects of streamwise wall heating and cooling patterns on the growth of the second-mode instability are investigated both analytically and numerically. The study considers 40-cm-long and 1-m-long sharp tip 7° half-angle cone with run conditions corresponding to AFOSR–Notre Dame Large Mach-6 Quiet Tunnel conditions. Using sinusoidal wall thermal configurations, it is found that wavelengths equal to or less than one-half of the length between the neutral points little to no affect on the integrated amplification rate of the second-mode instability. Further, it is shown that linear thermal gradients behave fundamentally different than wall temperature offsets. Growth rate sensitivity studies show that the linear increment and linear decrement in wall temperature patterns destabilize and stabilize second-mode instabilities, respectively. This behavior is consistent with simple inviscid energetics arguments in which the linear gradient terms effectively modulate the second-mode energy source.