Nonlinear fluid dynamics is essential in understanding the complex phenomena fundamental to science and engineering. The Sequence of Bifurcations approach is one method to analyze transitions from simple fluid flow to turbulence by introducing three-dimensional infinitesimal disturbances in a system with homogeneous external conditions in two spatial dimensions and time. Nonlinearity is initially introduced as a small perturbation to laminar flow, using Fourier expansions to construct and solve the Navier-Stokes equations numerically, producing higher-order states. This process continues until achieving aperiodic motion or turbulence.

However, a limitation arises as nonlinear terms may not remain finite at the bifurcation point, causing divergences and failing to identify the sequential hierarchy of higher-order solutions. This issue can be mitigated by Poincare’s discrete iterative approach, mapping the space of nonlinear solutions to itself and examining the full nonlinear Navier-Stokes equations numerically through phase space. Poincare’s method, though, assumes repetitive and similar solution patterns, limiting its ability to fully approach turbulence.

Hilbert’s analysis, contrastingly, relies on empirical analysis through new methods like empirical mode decomposition and Hilbert spectral analysis. This approach has many applications across various dynamical problems in science and engineering, offering a practical means to analyze complex data.

This Research Topic aims to address the theoretical and simulative verification of the Fourier, Poincare, and Hilbert analysis, or combinations thereof, in various dynamical systems that describe nonlinear phenomena based on the Navier-Stokes equations or their variations. This involves identifying unique ways to prove the validity of theoretical and numerically derived solutions using these three types of analyses (or their combinations) to address problems in science, mathematics, or engineering. By doing so, we can enhance our understanding and insight into the complex phenomena observed in nature. Additionally, comparing the results of theoretical approaches for turbulent flows based on group and spatiotemporal symmetry analysis using Fourier, Poincare, and Hilbert analysis will further solidify our grasp of these complex systems.

We welcome submissions that apply the above analysis individually or in a combined way to analyze the spectrum of non-linear solutions obtained via simulations based on Navier-Stokes equations.

Potential topics include but are not limited to the following:

• Bifurcation theory
• Weakly or strongly nonlinear flow
• Incompressible flow
• Stability analysis
• Direct simulations
• Floquet theory
• Couette flow
• Poiseulle flow
• Laterally Heated flow
• Homogenously heated flow
• Open or closed channel flow

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Frontiers in Physics

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