An averaging scheme for the efficient approximation of time-periodic flow problems

TL;DR

This paper introduces a simple, low-cost averaging scheme to efficiently approximate time-periodic solutions of the Navier-Stokes equations, reducing computational effort in finding periodic flow states.

Contribution

The paper proposes and analyzes a novel averaging scheme that accelerates the computation of periodic solutions with minimal additional computational cost.

Findings

The scheme is numerically efficient and robust across various test cases.

Theoretical convergence is proven for the linear Stokes problem.

The method significantly reduces the time to identify periodic flow solutions.

Abstract

We study periodic solutions to the Navier-Stokes equations. The transition phase of a dynamic Navier-Stokes solution to the periodic-in-time state can be excessively long and it depends on parameters like the domain size and the viscosity. Several methods for an accelerated identification of the correct initial data that will yield the periodic state exist. They are mostly based on space-time frameworks for directly computing the periodic state or on optimization schemes or shooting methods for quickly finding the correct initial data that yields the periodic solution. They all have a large computational overhead in common. Here we describe and analyze a simple averaging scheme that comes at negligible additional cost. We numerically demonstrate the efficiency and robustness of the scheme for several test-cases and we will theoretically show convergence for the linear Stokes problem.