Numerical simulation of internal incompressible flows with enhanced variants of dissipative inlet/outlet conditions. Part 1: Mathematical formulations and solution methods

TL;DR

This paper develops and analyzes advanced numerical methods for simulating unsteady incompressible flows with novel dissipative boundary conditions, including extensions to Navier-Stokes equations and optimal control approaches.

Contribution

It introduces new dissipative boundary condition variants with inertia terms, variational formulations, and solution techniques for both Stokes and Navier-Stokes flows, including control-based methods.

Findings

Effective implementation of dissipative boundary conditions with inertia terms.

Extension of methods to Navier-Stokes flows using OIFS technique.

Incorporation of optimal control and adjoint methods for flow problems.

Abstract

The paper presents numerical methods for unsteady flows of a viscous incompressible fluid in internal domains with many inlet/outlet sections. The novel variants of dissipative boundary conditions augmented by the inertia terms are used at the inlets/outlets of the flow domain. Volumetric flow rates or inlet/outlet average pressure are imposed as additional constrains imposed on a fluid motion. The variational formulations of the Stokes problem with such conditions and constrains are presented and the solution methods are proposed. These methods are based on superposition of appropriately defined auxiliary Stokes problems. Extension of the proposed methodology to the Navier-Stokes flows, based on the Operator-Integration-Factor-Splitting (OIFS) technique, is also described. Next, a nonlinear extension of the inertial-dissipative conditions is formulated and incorporated in the solution framework for the Navier-Stokes flows. Finally, an alternative approach to flow problems with such conditions, based on optimal control theory and adjoint evaluation of the goal functional gradient, is presented. A few remarks concerning numerical implementation issues are formulated.