The Crouzeix–Raviart element for the Stokes equations with the slip boundary condition on a curved boundary

2021 ◽  
Vol 383 ◽  
pp. 113123
Author(s):  
Guanyu Zhou ◽  
Issei Oikawa ◽  
Takahito Kashiwabara
Keyword(s):  
Boundary Condition ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Curved Boundary ◽  
Slip Boundary

scholarly journals Involving the Navier–Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method

2011 ◽  
Vol 369 (1944) ◽  
pp. 2184-2192 ◽  
Author(s):  
Joris C. G. Verschaeve
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Grid Spacing ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Boltzmann Method

By means of the continuity equation of the incompressible Navier–Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.

A New Partial Slip Boundary Condition for the Lattice-Boltzmann Method

2013 ◽  
Author(s):  
Marc-Florian Uth ◽  
Alf Crüger ◽  
Heinz Herwig
Keyword(s):  
Boundary Condition ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Slip Model ◽  
Navier Stokes Equations ◽  
Navier Slip ◽  
Slip Boundary ◽  
Boltzmann Method

In micro or nano flows a slip boundary condition is often needed to account for the special flow situation that occurs at this level of refinement. A common model used in the Finite Volume Method (FVM) is the Navier-Slip model which is based on the velocity gradient at the wall. It can be implemented very easily for a Navier-Stokes (NS) Solver. Instead of directly solving the Navier-Stokes equations, the Lattice-Boltzmann method (LBM) models the fluid on a particle basis. It models the streaming and interaction of particles statistically. The pressure and the velocity can be calculated at every time step from the current particle distribution functions. The resulting fields are solutions of the Navier-Stokes equations. Boundary conditions in LBM always not only have to define values for the macroscopic variables but also for the particle distribution function. Therefore a slip model cannot be implemented in the same way as in a FVM-NS solver. An additional problem is the structure of the grid. Curved boundaries or boundaries that are non-parallel to the grid have to be approximated by a stair-like step profile. While this is no problem for no-slip boundaries, any other velocity boundary condition such as a slip condition is difficult to implement. In this paper we will present two different implementations of slip boundary conditions for the Lattice-Boltzmann approach. One will be an implementation that takes advantage of the microscopic nature of the method as it works on a particle basis. The other one is based on the Navier-Slip model. We will compare their applicability for different amounts of slip and different shapes of walls relative to the numerical grid. We will also show what limits the slip rate and give an outlook of how this can be avoided.

scholarly journals Penalty method with Crouzeix–Raviart approximation for the Stokes equations under slip boundary condition

2019 ◽  
Vol 53 (3) ◽  
pp. 869-891
Author(s):  
Takahito Kashiwabara ◽  
Issei Oikawa ◽  
Guanyu Zhou
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Trace Operator ◽  
Essential Boundary Condition ◽  
Slip Boundary ◽  
Polyhedral Domain ◽  
Discrete Counterpart ◽  
Original Domain

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω ⊂ ℝN (N = 2,3). We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix–Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition u · n∂Ω = g on ∂Ω. Because the original domain Ω must be approximated by a polygonal (or polyhedral) domain Ωh before applying the finite element method, we need to take into account the errors owing to the discrepancy Ω ≠ Ωh, that is, the issues of domain perturbation. In particular, the approximation of n∂Ω by n∂Ωh makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator H1 (Ω)N → H1/2(∂Ω); u ↦ u⋅n∂Ω. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates O(hα + ε) and O(h2α + ε) for the velocity in the H1- and L2-norms respectively, where α = 1 if N = 2 and α = 1/2 if N = 3. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter ϵ in the estimates.

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