A time averaged steady state method for the Navier–Stokes equations

Abstract

This work derives an incompressible variational multiscales time-averaged Navier–Stokes (NS) formulation that aims at obtaining accurate steady state solutions. Rather than using the standard time instantaneous velocity and pressure, the new formulation devises a time averaging procedure based on rewriting and solving the NS equations in terms of the newly defined time-averaged velocity and pressure. Hence, the method could be understood as a convenient change of variable so that the problem is rewritten directly in terms of the steady state quantities. The important advantage of such a point of view is that it can in principle be applied to any other formulation. Such time averaging procedure is complemented by two time step modification strategies in order to accelerate the convergence to the steady state. The guidelines of an integrated framework are presented in the article, starting with the description of the proposed numerical technique applied to general incompressible flows. The explanation is enhanced with a one-dimensional (1D) nonlinear oscillator example. Several results are presented concerning analytical benchmarks, simulation of flows in laminar, transitional and turbulent regimes with and without an inherently steady solution.