Nonlinear stability results for plane Couette and Poiseuille flows

Area of Science:

• Fluid Dynamics
• Nonlinear Stability Theory

Background:

• Plane Couette and Poiseuille flows are fundamental in fluid dynamics.
• Understanding their nonlinear stability is crucial for predicting flow behavior.

Purpose of the Study:

• To determine the nonlinear stability criteria for plane Couette and Poiseuille flows under tilted perturbations.
• To improve upon existing theoretical predictions for critical Reynolds numbers.

Main Methods:

• Analysis of nonlinear stability using tilted perturbations.
• Calculation of critical Orr-Reynolds numbers for various wavelengths and inclination angles.
• Comparison with experimental data and numerical simulations.

Main Results:

• Derived critical Reynolds numbers for nonlinear stability: Re_Orr = 44.3/sin(θ) for Couette flow and Re_Orr = 87.6/sin(θ) for Poiseuille flow.
• These results improve upon previous theoretical values for streamwise perturbations.
• Obtained excellent agreement with experimental and numerical data for specific wavelengths.

Conclusions:

• The study provides improved nonlinear stability criteria for plane Couette and Poiseuille flows.
• The findings offer partial resolution to the long-standing Couette-Sommerfeld paradox.
• The theoretical predictions are validated by experimental and numerical evidence.