On the Formulation of Boundary Conditions When Solving Hydrodynamic Problems in Vorticity-Stream Function Variables

When solving the Navier-Stokes problem in vorticity - current function variables, the scientist always faces the problem of redefined boundary conditions for the current function and their absence for the vorticity function. The classical approach to solving this problem is to construct an additional differential operator for the desired functions on the boundary of the domain, from the solution of which intermediate boundary conditions for the vorticity function can be determined. Despite the significant progress achieved in the implementation of this approach, it has two serious drawbacks. First, it requires the construction of a differential operator normal to the boundary at each boundary node of the computational domain, which significantly complicates algorithms for problems with curved boundaries. Secondly, it generates an additional iterative process even when solving the linear Stokes problem.
The paper proposes a new algorithm that makes it possible to determine boundary conditions in the finite element method when solving problems of hydrodynamics in variables vorticity as a function of current. In the proposed algorithm, the boundary values for vorticity on an arbitrary non-degenerate contour of the boundary of the computational domain are determined from the equation for the current function written in a weak integral form taking into account the Neumann conditions for the current function at the boundary of the domain. The algorithm does not require the construction of additional difference operators on the contour of the area to obtain the boundary conditions of the problem and allows using the vector formulation of the problem to solve the Stokes problem in one iteration. When solving Navier-Stokes problems in a vector formulation, the method makes it possible to obtain consistent vorticity fields and current functions at each time/nonlinearity step, which makes it possible to control the convergence processes of the problem being solved. The stability of the proposed algorithm is confirmed by numerical experiments.

Keywords

convection; weak problem formulation; boundary conditions; stream function – vorticity; finite element method

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