Non-normal origin of modal instabilities in rotating plane shear flows

Area of Science:

• Fluid dynamics
• Hydrodynamic stability
• Non-normal operators

Background:

• Shear flows exhibit dramatic stability changes with small variations.
• Rotation in flow systems significantly lowers the critical Reynolds number for instabilities.
• The fundamental reason for this sensitivity to rotation requires investigation.

Purpose of the Study:

• To determine the fundamental reason for the sensitivity of shear flow stability to rotation.
• To analyze the role of non-normality in the stability operator.
• To connect rotating flow instability to the pseudospectrum of non-rotating flows.

Main Methods:

• Treating rotation as a perturbation on non-rotating shear flows.
• Analyzing the stability operator and its non-normality.
• Examining the pseudospectrum of the non-rotating stability operator.
• Investigating plane Poiseuille and plane Couette flows.

Main Results:

• Non-normality of the stability operator in non-rotating flows is the fundamental cause of sensitivity to rotation.
• Rotating flows become unstable at small rotation rates, particularly at zero streamwise wavenumber.
• The critical rotation number scales inversely with the square of the Reynolds number.
• This scaling matches the perturbation amplitude required for the pseudospectrum to cross the neutral line.

Conclusions:

• The non-normality of the stability operator is key to understanding shear flow sensitivity to rotation.
• Rotating shear flows can exhibit instability even at low rotation rates due to non-normal effects.
• The observed scaling laws provide a unified framework for understanding instability in both rotating and non-rotating shear flows.