On the Role of Iterative Errors in Unsteady Flow Simulations

Simulation of (statistically) unsteady flows is no longer an exceptional task in Engineering applications. For wall bounded (turbulent) flows at high Reynolds numbers it is usual to apply implicit techniques for time integration, i.e. a non-linear system of partial differential equations must be solved at each time step. This means that a convergence tolerance is needed to decide on when having solved this non-linear system well enough. Any iterative error propagates to the next time step and so the numerical error of a given solution may not be dominated by the numerical error inherent to the space and time discretization.

In the last two years, two Workshops dedicated to Iterative Errors in Unsteady Flow Simulations were held at the ASME V&V Symposiums of 2017 and 2018, Workshop (2017), Workshop (2018). The main goal of these Workshops was to create awareness to the problem and to confirm that different flow solvers, including ANSYS Fluent, ANSYS CFX, OpenFoam, ReFRESCO, SATURNE and STAR-CCM+, exhibited the same trends. The simple two-dimensional, laminar flow of a single-phase, incompressible, Newtonian fluid around a circular cylinder at the Reynolds number of 100 is sufficient to illustrate the challenges posed by iterative errors in unsteady flow simulations.

This paper presents a brief illustration of the outcome of the two Workshops on Iterative Errors in Unsteady Flow Simulations. However, our main goal is to answer the following questions:

• How far do we need to converge at each time step to guarantee a negligible iterative error?
• How do we estimate iterative errors for the quantities of interest?
• What is the impact of the iterative error on the estimation of the discretization error?