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 Ω h \Omega _h , denoted n h \mathbf {n}_h and τ h \boldsymbol {\tau }_h , is suboptimal. Taking instead a projection of the normal and tangent vectors with respect to Ω \Omega , denoted n π \mathbf {n}_\pi and τ π \boldsymbol {\tau }_\pi , 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 n h \mathbf {n}_h and τ h \boldsymbol {\tau }_h , 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.

Resolution of d'Alembert's paradox using slip boundary conditions: The effect of the friction parameter on the drag coefficient

“…where ν is a nondimensional parameter related to the kinematic viscosity, together with boundary conditions u = g on ∂Ω\Γ, u • n = 0 on Γ, (2) with n being the outward pointing unit normal of Ω, and Navier's slip condition [24,10] linking tangential velocity and the shear stress on Γ:…”

“…Consider the tangent space T to Γ and the projection P T onto the tangent space. In [10], we show that the variational formulation of ( 1)-( 3) is:…”

“…The variational formulation (10) can then be discretized directly using Taylor-Hood P 2 -P 1 elements for the flux and pressure, respectively. Taking h as the mesh size, this formulation is stable given that the Nitsche parameter γ > 0 is large enough.…”

“…Taking h as the mesh size, this formulation is stable given that the Nitsche parameter γ > 0 is large enough. This method is validated in [10], using, e. g., potential flow as an analytic solution and computing the corresponding error rates.…”

“…We return here to this subject from a different point of view, one that would have been difficult in 1981. We consider the paradox by approximating solutions of the Navier-Stokes equations with a slip boundary condition, using recently developed numerical techniques [10], and of course current computer platforms. In [29], an attempt was made to understand the paradox by seeking a limiting solution for large Reynolds numbers.…”