Hydrodynamic fluctuations in the kolmogorov flow: nonlinear regime

Area of Science:

• Fluid dynamics
• Statistical physics
• Nonlinear dynamics

Background:

• Previous work analyzed linearized Kolmogorov flow using fluctuating hydrodynamics.
• This study extends the analysis to the nonlinear regime near the first instability.

Purpose of the Study:

• To investigate the statistical properties of Kolmogorov flow in the nonlinear regime.
• To derive and analyze the normal form amplitude equation for incompressible fluids.
• To extend the analysis to compressible fluids near the instability threshold.

Main Methods:

• Derivation of the normal form amplitude equation.
• Construction of the velocity field near instability.
• Application of a perturbative technique to analyze compressible flow.
• Solution of coupled nonlinear Langevin equations in Fourier space.
• Numerical simulations of nonlinear fluctuating hydrodynamic equations.

Main Results:

• The stochastic dynamics near instability are governed by two coupled nonlinear Langevin equations.
• The solution is expressible via a Landau-Ginzburg functional, identical for incompressible and compressible cases.
• Theoretical predictions are validated by numerical simulations.

Conclusions:

• The nonlinear regime of Kolmogorov flow exhibits universal statistical properties near instability.
• The derived Landau-Ginzburg functional provides a unified description for both fluid types.
• Fluctuating hydrodynamics offers a robust framework for analyzing complex fluid systems.