Analytic investigation of the compatibility condition and the initial evolution of a smooth velocity field for the Navier-Stokes equation in a channel configuration

TL;DR

This paper investigates the compatibility conditions for initial velocity fields in Navier-Stokes equations within channel flows, providing methods to check and generate initial conditions that ensure solution regularity at initial time.

Contribution

It introduces a practical calculation method for verifying compatibility conditions and demonstrates how to generate initial velocity fields that violate these conditions, impacting solution regularity.

Findings

Compatibility condition is always fulfilled without wall-normal velocity.

Non-zero wall-normal velocity often violates the condition.

Counter-examples can be generated using optimization methods.

Abstract

A partial differential equation has usually a regular solution at the initial time if the initial condition is smooth in space, fulfills the governing equations and is compatible with the boundary condition. In the case of Navier-Stokes equation, the initial velocity field must also be divergence--free. It is common belief that the initial condition is compatible with the boundary condition if the initial condition fulfills the boundary condition but this is not sufficient. Such a field does not necessarily fulfill the full compatibility condition of the Navier-Stokes equation. If the condition is violated, the solution is not regular at the initial time. This issue has been known for a while but not in the full breadth of the fluid dynamics community. In this paper, a practical calculation method is presented for checking the compatibility condition. Furthermore, a smooth initial condition is presented in a periodic channel flow that violates the condition and has no spatially smooth solution at initial time. The calculations were were performed in an analytical framework. The results for a channel configuration show that in the absence of wall--normal velocity the condition is always fulfilled and the problem has a regular solution. If the wall--normal velocity component is non--zero, the condition is usually not fulfilled but with optimization methods counter-examples can be generated. The generation procedure of such a field is useful to provide correct initial conditions for sensitive time-dependent numerical simulations. The presented methods can provide insight about the applicability of the chosen initial conditions.