An improved immersed boundary method for curvilinear grids

Abstract

In the present paper we propose an extension of the direct-forcing immersed boundary technique, recently developed and employed by Verzicco and co-authors [Fadlun EA, Verzicco R, Orlandi P, Mohd-Yusof J. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J Comput Phys 2000;161:35–60; Verzicco R, Fatica M, Iaccarino G, Moin P, Khalighi B. Large eddy simulation of a road vehicle with drag-reduction devices. AIAA J 2002;40(12):2447–55; Cristallo A, Verzicco R. Combined immersed boundary/large-eddy-simulations of incompressible three-dimensional complex flows. Flow Turbul Combust 2006;77(1–4):3–26.] and successively improved by Balaras and coauthors [Gilmanov A, Sotiropoulos F, Balaras E. A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids. J Comput Phys 003;191:660–9; Balaras E. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations. Comput Fluids 2004;33:375–404]. We extend the aforementioned technique to curvilinear-coordinate, structured grid, Navier–Stokes solvers. This improved technique allows for more flexibility and efficiency when compared to standard methods in which the technique is coupled with orthogonal-grid solvers. Additional modifications are also proposed with respect to the state-of-art, which allow to deal with general shaped, multiple-body immersed surfaces and to make the interpolation of the velocity field off the body suitable for curvilinear grids. Several tests have been carried out to check the reliability of the proposed technique: first we have considered the three-dimensional Stokes flow around a sphere, and compared the numerical results with the analytical ones. Second we have considered the two-dimensional unsteady flow around a circular cylinder placed between two parallel solid walls and compared the results with those of the database of the Priority Research Program ‘Flow Simulation on High Performance Computers’ of the German Research Association (DFG). Third, we have considered the two-dimensional flow within a S-shaped duct containing an elliptical valve. Finally, we have applied the technique to the study of a practical high-Reynolds number industrial problem. The geometrical configuration of the first two test cases is suited for both Cartesian and curvilinear algorithms. The geometry of the third test case is suited for curvilinear meshes and makes the use of Cartesian grids very inefficient and less accurate than the curvilinear ones. In these cases Cartesian – as well as curvilinear – mesh simulations have been carried out. Finally, the geometry of the high-Reynolds industrial problem is suited for curvilinear grids. The proposed technique has shown to preserve at least the same level of accuracy of its Cartesian counterpart allowing to reduce in a considerable way the computational cost of the simulations. When the geometry is better suited for curvilinear meshes, the reduction of the computational cost is accompanied by an increased accuracy with respect to the Cartesian counterpart. We also propose a simplified direct-forcing, semi-implicit method, allowing reduced computational cost with respect to the literature techniques. We have checked the accuracy of the technique and shown that when the Reynolds number is large enough, the present simplified technique allows the use of time steps much larger than those allowed by the explicit time-advancement scheme, preserving the accuracy of the results.