Explicit form of spatially linear Navier-Stokes velocity fields

TL;DR

This paper characterizes all smooth, spatially linear, unsteady velocity fields that solve the incompressible Navier-Stokes equations, providing explicit conditions on the matrix component for constructing test flows.

Contribution

It derives explicit necessary and sufficient conditions for linear velocity fields to satisfy the Navier-Stokes equations, including formulas for 2D and 3D cases.

Findings

In 2D, A(t) is the sum of a traceless symmetric matrix and a skew-symmetric matrix.

In 3D, A(t) satisfies a specific ordinary differential equation.

The results enable easy construction of unsteady flows for testing numerical methods.

Abstract

We show that a smooth linear unsteady velocity field $u(x,t)=A(t)x+f(t)$ solves the incompressible Navier--Stokes equation if and only if the matrix $A(t)$ has zero trace, and $\dot{{A}}(t)+A^{2}(t)$ is symmetric. In two dimensions, these constraints imply that $A(t)$ is the sum of an arbitrary time-dependent traceless symmetric matrix and an arbitrary constant skew-symmetric matrix. One can, therefore, verify by inspection if an unsteady spatially linear vector field is a Navier--Stokes solution. In three dimensions, we obtain a simple ordinary differential equation that $A(t)$ must solve. Our formulas enable the construction of simple yet unsteady and dynamically consistent flows for testing numerical schemes and verifying coherent structure criteria.