Spectral analysis of the Navier-Stokes equations using the combination matrix

TL;DR

This paper extends spectral analysis of the Navier-Stokes equations by using the combination matrix to reformulate the equations as intersecting quadratic polynomials, enabling a geometric interpretation of solutions.

Contribution

It introduces a new approach to analyze forced Navier-Stokes equations in spectral space using the combination matrix to express solutions as intersections of conic sections.

Findings

Reformulation of Navier-Stokes as quadratic polynomial systems

Geometric interpretation of solutions as conic intersections

Potential for new solution methods in spectral analysis

Abstract

This work is a continuation of the analysis first presented in Cheung & Zaki (2014). In that study, the combination matrix was introduced as a means to tractably handle the nonlinear terms in the spectral domain. In this work, a different approach is discussed. Rather than analyze solutions to the energy equation, we examine the forced Navier-Stokes equations in spectral space and determine if direct solutions to the momentum equations can be found. This is done by using the combination matrix to rewrite the Navier-Stokes as a system of intersecting quadratic polynomials. Intrepreted geometrically, any solution to the Navier-Stokes can be represented as a the intersection of a multiple conic sections.