Abstract
A novel technique based on conformal mapping and the circle theorem has been developed to tackle the boundary penetration issue, in which vortex blobs leak into structures in two-dimensional discrete vortex simulations, as an alternative to the traditional method in which the blobs crossing the boundary are simply removed from the fluid field or reflected back to their mirror-image positions outside the structure. The present algorithm introduces an identical vortex blob outside the body using the mapping method to avoid circulation loss caused by the vortex blob penetrating the body. This can keep the body surface streamlined and guarantees that the total circulation will be constant at any time step. The model was validated using cases of viscous incompressible flow passing elliptic cylinders with various thickness-to-chord ratios at Reynolds numbers greater than Re = 1 × 105. The force and velocity fields revealed that this boundary scheme converged, and the resultant time-averaged surface pressure distributions were all in excellent agreement with wind tunnel tests. Furthermore, a flow around a symmetrical Joukowski foil at Reynolds number Re = 4.62 × 104, without considering the trailing cusp, was investigated, and a close agreement with the experimental data was obtained.
Highlights
The mesh-free discrete vortex method (DVM) developed by Chorin [1] is generally used to simulate incompressible viscous flows past structures under high Reynolds numbers
We first consider an elliptic cylinder with a thickness-to-chord ratio e = 0.16 at the Reynolds number Re = 3 × 105 to investigate the sensitivity of the numerical parameters, including the number of nascent elements and the time-step size
We first numerically simulated the flow past a circular cylinder based on the instantaneous vorticity conserved boundary conditions (IVCBC) method
Summary
The mesh-free discrete vortex method (DVM) developed by Chorin [1] is generally used to simulate incompressible viscous flows past structures under high Reynolds numbers. Another boundary method has been employed by Chorin [29] and Ghoniem et al [30], where elements leaking into the structure need to be mapped across the nearest wall to their mirror-image locations in the fluid domain, without changing their strengths This scheme cannot keep the surface streamlined for common bodies with large curvature. Instead of the algorithms mentioned above, Pang et al [31,32] have presented instantaneous vorticity conserved boundary conditions (IVCBC) to address elements inside structures using the circle theorem technique [4] This algorithm ensures that the total circulation of vortex blobs is conserved at any time step, and it can keep the streamlined surface non-destructive. The present method can be used in flows over arbitrary sectional cylinders, difficulties will be encountered in the approach when trying to obtain conformal transformations for complicated geometries
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