Energy stable boundary conditions for the nonlinear incompressible Navier–Stokes equations

Abstract

The nonlinear incompressible Navier-Stokes equations with different types of boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the energy method. Both formulations are implemented in both strong and weak form and lead to an estimate of the velocity field. Equipped with energy bounding boundary conditions, the problem is approximated by using discrete derivative operators on summation-by-parts form and weak boundary and initial conditions. By mimicking the continuous analysis, the resulting semi-discrete as well as fully discrete scheme are shown to be provably stable, divergence free and high-order accurate.

Energy-stable boundary conditions based on a quadratic form: Applications to outflow/open-boundary problems in incompressible flows

“…By formulating the boundary integral term in the energy balance equation into a quadratic form involving a symmetric matrix, we have derived a general form of boundary conditions that ensure the energy dissipation on the open boundary. It should be pointed out that, due to differences in the formulation of the quadratic form and the symmetric matrix involved therein, the energy-stable boundary conditions obtained here are different from those of [29], even though the procedure used for deriving the boundary conditions is similar.…”

“…This roadmap involves three main steps: (i) reformulate the boundary contribution into a quadratic form in terms of a symmetric matrix, (ii) rotate the variables to diagonalize the matrix, and (iii) formulate the boundary condition in the form of the eigen-variables corresponding to the negative eigenvalues expressed in terms of the eigen-variables corresponding to the positive eigenvalues. This procedure is very recently applied in [29] to the incompressible Navier-Stokes equations in two dimensions to investigate the boundary conditions on solid walls and far fields that can bound the energy of the system.…”

“…Inspired by the works of [28,29], we develop in this paper a set of new open/outflow boundary conditions for tackling the backflow instability for incompressible flows in two and three dimensions based on the procedure of [28]. By formulating the boundary integral term in the energy balance equation into a quadratic form involving a symmetric matrix, we have derived a general form of boundary conditions that ensure the energy dissipation on the open boundary.…”

Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations

“…However, transferring such semidiscrete results to fully discrete schemes is not easy in general. Stability/dissipation results for fully discrete schemes have mainly been limited to semidiscretizations including certain amounts of dissipation [31,32,49,71], linear equations [53,54,64,65,68], or fully implicit time integration schemes [7,10,11,26,36,39,48]. For explicit methods and general equations, there are negative experimental and theoretical results concerning energy/entropy stability [37,38,47,50].…”

An energy stable coupling procedure for the compressible and incompressible Navier-Stokes equations

“…In this paper, we will use the energy method as our main analysis tool and derive an equation for the energy-rate. For clarity, we ignore the outer boundary conditions, which have been studied in [23,26,27,28,29,30,31], and focus only on the interface at y = 0.…”