Bilinear Optimal Control Problem for the Stationary Navier–Stokes Equations with Variable Density and Slip Boundary Condition

2019 ◽  
Vol 50 (4) ◽  
pp. 871-887
Author(s):  
Exequiel Mallea-Zepeda ◽  
Eber Lenes ◽  
Jonnathan Rodríguez Zambrano
Keyword(s):  
Boundary Condition ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Variable Density ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Stationary Navier Stokes Equations

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.

Numerical resolution of optimal control problem for the in-stationary Navier–Stokes equations

2019 ◽  
Vol 27 (1) ◽  
pp. 43-52
Author(s):  
Jamil Satouri
Keyword(s):  
Optimal Control ◽  
Finite Elements ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stokes Equations ◽  
Explicit Scheme ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Numerical Resolution ◽  
Stationary Navier Stokes Equations

Abstract In this paper we present a study of optimal control problem for the unsteady Navier–Stokes equations. We discuss the existence of the solution, adopt a new numerical resolution for this problem and combine Euler explicit scheme in time and both of methods spectral and finite elements in space. Finally, we give some numerical results proving the effectiveness of our approach.

scholarly journals Large time behavior of solutions to the compressible Navier–Stokes equations in an infinite layer under slip boundary condition

2016 ◽  
Vol 26 (14) ◽  
pp. 2617-2649 ◽  
Author(s):  
Abulizi Aihaiti ◽  
Shota Enomoto ◽  
Yoshiyuki Kagei
Keyword(s):  
Boundary Condition ◽  
Large Time ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Large Time Behavior ◽  
Navier Stokes ◽  
Time Behavior ◽  
Navier Stokes Equations ◽  
Infinite Layer ◽  
Slip Boundary

This paper is concerned with large time behavior of solutions to the compressible Navier–Stokes equations in an infinite layer of [Formula: see text] under slip boundary condition. It is shown that if the initial data is sufficiently small, the global solution uniquely exists and the large time behavior of the solution is described by a superposition of one-dimensional diffusion waves.

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