This article describes the open-source C++ code VM2D for the simulation of two-dimensional viscous incompressible flows and solving fluid-structure interaction problems. The code is based on the Viscous Vortex Domains (VVD) method developed by Prof. G. Ya. Dynnikova, where the viscosity influence is taken into account by introducing the diffusive velocity. The original VVD method was supplemented by the author’s algorithms for boundary condition satisfaction, which made it possible to increase the accuracy of flow simulation near the airfoil’s surface line and reduce oscillations when calculating hydrodynamic loads. This paper is aimed primarily at assessing the efficiency of the parallelization of the algorithm. OpenMP, MPI, and Nvidia CUDA parallel programming technologies are used in VM2D, which allow performing simulations on computer systems of various architectures, including those equipped with graphics accelerators. Since the VVD method belongs to the particle methods, the efficiency of parallelization with the usage of graphics accelerators turns out to be quite high. It is shown that in a real simulation, one graphics card can replace about 80 nodes, each of which is equipped with 28 CPU cores. The source code of VM2D is available on GitHub under GNU GPL license.

We consider the methodology of numerical schemes development for two-dimensional vortex method. We describe two different approaches to deriving integral equation for unknown vortex sheet intensity. We simulate the velocity of the surface line of an airfoil as the influence of attached vortex and source sheets. We consider a polygonal approximation of the airfoil and assume intensity distributions of free and attached vortex sheets and attached source sheet to be approximated with piecewise constant or piecewise linear (continuous or discontinuous) functions. We describe several specific numerical schemes that provide different accuracy and have a different computational cost. The study shows that a Galerkin-type approach to solving boundary integral equation requires computing several integrals and double integrals over the panels. We obtain exact analytical formulae for all the necessary integrals, which makes it possible to raise significantly the accuracy of vortex sheet intensity computation and improve the quality of velocity and vorticity field representation, especially in proximity to the surface line of the airfoil. All the formulae are written down in the invariant form and depend only on the geometric relationship between the positions of the beginnings and ends of the panels.

The problem of 2D incompressible flow simulation around airfoils using vortex methods is considered. An exact solution for the boundary integral equation with respect to a free vortex sheet intensity at the airfoil surface line that arises in such problems is obtained. The exact solution is constructed for flows around elliptical and Zhukovsky airfoils using the theory of complex potentials and conformal mappings technique. It is possible to take into account the influence of singularities in the flow domain — point vortices which simulate vortex wake. The obtained exact solutions can be used to verify and estimate the accuracy of numerical schemes for the boundary integral equation solution: such procedure is also described in details.

Two coupling strategies for numerical simulation in Fluid-Structure Interaction problems using vortex methods are considered and implemented in new open-source code VM2D developed by the authors for 2D incompressible flows simulations around airfoils using vortex methods. The first strategy called “partitioned approach” implies that hydrodynamical subproblem and mechanical one are solved separately, step-by-step. However, this method has limited range of applicability. In the second strategy called “monolithic approach” these two subproblems are solved together. Both approaches are implemented in the VM2D code and their comparison is carried out for the model problem of a circular cylinder wind resonance (transversal oscillations) simultation.

The Lagrangian vortex particle method for two-dimensional flow simulation around airfoils is considered. Generally, the most time-consuming operation in the vortex method is vortex particles interaction simulation, especially if it is performed directly “point-to-point” for all pairs of particles. The result of such operation is vortex particles convective velocities. This problem is similar to the N-body problem and has squared computational complexity O(N²). The only efficient approach for its solution in a reasonable time for large N is to use approximate fast algorithms. The fast method suggested by Barnes and Hut having logarithmic computational complexity is considered for the mentioned problem of convective velocities computation. It also has been adapted for other operations in the vortex method such as a system of algebraic equations iterative solution and efficient right-hand side computation. As the result, the Barnes — Hut method allows reducing the whole time of calculations by several tens of times.