Unsteady incompressible flow analysis using C-type grid with a curved branch cut

For an unsteady viscous flow simulation on a two-dimensional body at high angle of attack, the calculation of unsteady aerodynamic forces acting on the body is influenced not only by the unsteady separated flow near the body but also by the unsteady wake behind the body. To resolve the wake flow behind the trailing edge, an orthogonal C-grid topology with a curved branch cut aligned with the inviscid stagnation streamline is generated using a conformal mapping technique. This permits the desired grid clustering in the wake region and leads to better flow results in that region. The conformal mapping technique also provides analytical Jacobian metrics for the coordinate transformation and an inviscid solution which is useful in initiating the viscous flow of the impulsively started motion. The use of analytical metric coefficients facilitates the direct determination of part of the coefficients in the governing equations without introducing numerical errors. The unsteady two-dimensional incompressible Navier-Stokes equations in generalized orthogonal coordinates are solved using a vorticity-stream function formulation. The analysis also requires coupling of flow circulation in the far field. As a result, the vorticity-stream function formulation introduced in the present study contains the spatially elliptic equation for the disturbance stream function coupled with the temporally parabolic vorticity transport equation. An efficient direct Block-Gaussian Elimination (BGE) technique is used to solve the stream function Poisson problem subject to Neumann and Dirichlet boundary conditions. The vorticity transport equation is solved using the Alternating Direct Implicit (ADI) method. In addition, the Jacobian at the grid points along the curved branch cut is multi-valued and the metric coefficients are found to be discontinuous across the branch cut. Hence, a special finite element interpolation is implemented in the governing equations at those points in order to overcome this discontinuity. To achieve the objective stated above, the unsteady flow over a stationary NACA 0015 airfoil at various angles of attack is selected in the present study.