AIAA 95–0048 Multigrid Unsteady Navier-Stokes Calculations with Aeroelastic Applications

An implicit approach to the solution of the unsteady two-dimensional Navier-Stokes equations is presented. After spatial discretization, the resulting set of coupled implicit non-linear equations is solved iteratively. This is accomplished using well proven convergence acceleration techniques for explicit schemes such as multigrid, residual averaging, and local time-stepping in order to achieve large computational efficiency in the calculation. Calculations are performed in parallel using a domain decomposition technique with optimized communication requirements. In addition, particular care is taken to minimize the effect of numerical dissipation with flux-limited dissipation schemes. Results for the unsteady shedding flow behind a circular cylinder and for a pitching NACA 64A010 airfoil are presented with experimental comparisons, showing the feasibility of accurate, efficient, time-dependent viscous calculations. Finally, a two-dimensional structural model of the cylinder is coupled with the unsteady flow solution, and time responses of the deflections of the structure are analyzed. Nomenclature C l coefficient of lift C d coefficient of drag Cx, Cy damping coefficients in the two coordinate directions D cylinder diameter, cylinder drag E total energy (internal plus kinetic) E(wij) convective Euler fluxes f , g Euler flux vectors H total enthalpy Kx, Ky spring constants in the two coordinate directions L airfoil section lift (normal to free stream), positive up m cylinder mass M∞ free stream Mach number n frequency, 1/sec NS(wij) viscous flux residual for cell i,j p static pressure qi heat flux component R(wij) total flux residual for cell i,j R * modified residual R, S viscous flux vectors ReD Reynolds number based on the diameter St Strouhal frequency, St = nD U∞ T static temperature u, v cartesian velocity components U∞ free stream velocity Vij volume of i,j cell w vector of flow variables xt, yt mesh cartesian velocity components ∆α pitching motion forcing amplitude ∆t implicit real time step γ ratio of specific heats, γ = 1.4 ρ air density σij viscous stress tensor components ω f frequency of the forced oscillations Ω, ∂Ω cell element and boundary Introduction U NSTEADY flow solvers are becoming a necessary part of the toolkit of the computational fluid dynamicist. In order to solve problems which are naturally unsteady (such as vortex shedding flows, moving boundary problems, fluid-structure interaction flows, etc.) it is essential to develop numerical schemes which provide accurate solutions at a reasonable cost. Therefore, computational efficiency is of paramount importance for unsteady numerical solutions. As the governing equations …