Variational resolution of outflow boundary conditions for incompressible Navier–Stokes

Abstract

This paper focuses on the so-called weighted inertia-dissipation-energy variational approach for the approximation of unsteady Leray–Hopf solutions of the incompressible Navier–Stokes system. Initiated in (Ortiz et al 2018 Nonlinearity 31 5664–82), this variational method is here extended to the case of non-Newtonian fluids with power-law index r ⩾ 11/5 in three space dimension and large nonhomogeneous data. Moreover, boundary conditions are not imposed on some parts of boundaries, representing, e.g., outflows. Correspondingly, natural boundary conditions arise from the minimisation. In particular, at walls we recover boundary conditions of Navier-slip type. At outflows and inflows, we obtain the condition − 1 2 | v | 2 n + T n = 0 . This provides the first theoretical explanation for the onset of such boundary conditions.

Existence of a steady flow through a rotating radial turbine with an arbitrarily large inflow and an artificial boundary condition on the outflow

“…Thus, as the velocity profile on Γ out is not known in advance, it is not logical to prescribe Dirichlet's boundary condition for $$\mathbf {u}$$ on Γ out . What is an appropriate boundary condition on the outflow has been a subject of discussions in literature, see for example, [1–5, 10, 11, 15, 16, 23], and so forth. We use the condition $$$$\begin{equation} -\nu \, \frac{\partial \mathbf {u}}{\partial \mathbf {n}}+p \mathbf {n}\ =\ \mathbf {h}+ \frac{1+\xi }{2}\, (\mathbf {u}\cdot \mathbf {n})_-\, \mathbf {u}\qquad \mbox{on}\ \Gamma _{\rm out}, \end{equation}$$$$where $$\mathbf {n}$$ denotes the outward normal vector, ξ is a positive parameter and the subscript “−” denotes the negative part.…”