Nitsche’s method for Navier–Stokes equations with slip boundary conditions

Abstract

We formulate Nitsche’s method to implement slip boundary conditions for flow problems in domains with curved boundaries. The slip boundary condition, often referred to as the Navier friction condition, is critical for understanding and simulating a wide range of phenomena such as turbulence, droplet spread and flow through microdevices. In this work, we highlight the role of the approximation of the normal and tangent vector. In particular, we show that using the normal and tangent vectors with respect to the discretized domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript h"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\Omega _h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , denoted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold n Subscript h"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">n</mml:mi> </mml:mrow> <mml:mi>h</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {n}_h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic tau Subscript h"> <mml:semantics> <mml:msub> <mml:mi mathvariant="bold-italic"> τ </mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\boldsymbol {\tau }_h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is suboptimal. Taking instead a projection of the normal and tangent vectors with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , denoted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold n Subscript pi"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">n</mml:mi> </mml:mrow> <mml:mi> π </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {n}_\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic tau Subscript pi"> <mml:semantics> <mml:msub> <mml:mi mathvariant="bold-italic"> τ </mml:mi> <mml:mi> π </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\boldsymbol {\tau }_\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , gives the best convergence rate that can be expected for a polygonal approximation of a curved boundary. Finally we also prove that, if you use instead the exact slip with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold n Subscript h"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">n</mml:mi> </mml:mrow> <mml:mi>h</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {n}_h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic tau Subscript h"> <mml:semantics> <mml:msub> <mml:mi mathvariant="bold-italic"> τ </mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\boldsymbol {\tau }_h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the approximation converges to the wrong solution. This is known as the Babuška-Sapondzhyan Paradox. Thus Nitsche’s method relaxes the slip condition and avoids the lack of convergence.

Topics & Concepts