Vorticity dynamics and numerical Resolution of Navier-Stokes Equations

Abstract

Abstract. We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data for some test cases to which we apply the computational scheme.Mathematics Subject Classification. 35Q30, 65M06, 76D17.

Convergence of a Compact Scheme for the Pure Streamfunction Formulation of the Unsteady Navier–Stokes System

“…At each time step the scheme solves a time implicit version of (1). This leads to a fourth-order biharmonic problem of the form Δψ − νΔ 2 ψ = f, (4) subject to the boundary conditions (2).…”

A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

“…In this paper we continue the study, which was initiated in [10,28,9,7], of the numerical resolution of the pure streamfunction formulation of the timedependent two-dimensional Navier-Stokes equation. Here we focus on enhancing our second-order scheme, introduced in [28,9,7], to fourth order accuracy.…”

“…Their approach was based on a finite-difference method. The application of various compact schemes to the pure streamfunction formulation is fairly recent [10,41,15,28,38]. We mention also [48,20,40,39,34] for works on the stationary Stokes or Navier-Stokes equation.…”

A Compact Scheme for the Streamfunction Formulation of Navier-Stokes Equations

“…In a previous study [2] we introduced a new methodology for tracking vorticity dynamics, which is modeled by the incompressible Navier-Stokes equations. Let Ω ⊆ R 2 be a bounded domain with smooth boundary ∂Ω.…”

“…The methodology presented in [2] is to evolve the vorticity in time according to (1.1), and then project the vorticity onto ∆(H Indeed, (1.3) is a well posed problem, and can be easily approximated by standard numerical schemes. Based on [2], Kupferman [5] introduced a centraldifference scheme for the pure-streamfunction formulation.In this paper we construct a pure-compact scheme for the streamfunction formulation. The advantage of the streamfunction formulation of the Navier-Stokes equations is that there is no need to invoke inter-functions relations.…”