Weak ψ-ω formulation for unsteady flows in 2D multiply connected domains

This paper describes a variational formulation for solving the time-dependent Navier-Stokes equations expressed in terms of the stream function and vorticitN around multiple airfoils. This approach is an extension to the case of multiply connected domains of the weak formulation based on explicit viscous diffusion recently proposed by Guermond and Quartapelle. In their method the momentum equation was interpreted as a dynamical equation governing the evolution of the ( weak) Laplacian of the stream function, while the Poisson equation for the latter was used as an expression to evaluate the distribution of the vorticity. Time discretizations with the viscous term made explicit were used, leading to the viscosity being split from the incompressibility, similarly to the primitive variable fractional-step method. In the present work the multiconnectedness is addressed by introducing an influence matrix to deter-mine the constant values of the stream function on the air-foils in a noniterative fashion. The explicit treatment of the viscous term leads to an influence matrix rooted in the harmonic problem instead of in the biharmonic problem occurring in methods enforcing integral conditions on the vorticity, such as the Glowinski-Pironneau method. The influence matrix changes at each time step or is constant depending on whether a semi-implicit or fully explicit treatment is adopted for the nonlinear term. The resulting split method is implemented using a first-order Euler backward difference or a second-order BDF scheme and linear finite elements. Numerical results are given and compared with the solutions obtained by means of the biharmonic formulation for multiply connected domains. 2002 Elsevier Science (USA).