A robust implicit compact scheme for two-dimensional unsteady flows with a biharmonic stream function formulation

• We propose a new compact FD scheme for the biharmonic form of the N–S equations.
• We document the versatility and efficiency of both formulation and the new scheme.
• The scheme is specifically designed for flows in fluid-embedded body interaction.
• For the first time biharmonic formulation is used in non-rectangular domains.
• The scheme effectively simulates flows of varied physical and geometric complexity.

In this paper, we develop a compact, implicit second order temporally and spatially accurate finite difference scheme for unsteady Navier–Stokes (N–S) equations for incompressible viscous flows. The scheme is specifically designed for flows in fluid-embedded body interaction as well as curved regions. As the scheme utilizes the 4th order pure stream function form of the N–S equations, the entire flow field can be described in terms of only one equation and thus avoids the difficulties associated with pressure field and nonphysical vorticity boundary conditions. We carry out a spectral analysis as well as von Neumann stability analysis of this scheme and also provide an algorithm for the flow computation. We use the scheme to simulate the time development of 2D viscous flows of varied physical complexities in different geometrical settings: Taylor-Green decaying vortices, constricted channel, flow past rotating and in-line oscillating circular cylinders, and flow past an elliptic cylinder and symmetric aerofoils with various angles of attack. The results obtained are compared with experimental and numerical results available in the literature. Excellent match is obtained in all cases, establishing the efficiency and the accuracy of the proposed scheme.