Flow simulation around 3D bodies by using Lagrangian vortex loops method with boundary condition satisfaction with respect to tangential velocity components

On the Algorithms for Vortex Element Evolution Modelling in 3D Fully Lagrangian Vortex Loops Method

“…It is proven in [7] that the vector integral equations and (2) is equivalent to its projection onto the tangential plane. Such approach (the T -scheme [8,9]) leads to the vector boundary integral equation of the 2-nd kind…”

“…In order to solve it numerically, the original semi-analytic approach, based on the Galerkin technique is developed, and its "keypoint" technique of the above mentioned surface improper integrals computation. This technique is described in details in [8,9,10].…”

“…After the integral equation solution with respect to the vortex sheet intensity, the double layer potential density is reconstructed (as piecewise-linear continuous distribution) and the vortex loops are generated along its level-set lines. The corresponding procedure is also described in [8].…”

“…The necessary formulae, which are required to calculate the velocity field V (r), taking into account the influence of the body with vortex sheet of known intensity on its surface and the vortex wake, consists of vortex loops, are given in [8].…”

Closed vortex lines in fluid and gas

“…In some cases, it is possible a priori (before making calculations) to state that vortex lines and tubes will be closed (for example, a flow behind a detached shock wave [22]) and checking the closure of vortex lines can serve to verify the calculations. In other cases, closed vortex tubes are detected as a result of a calculation (for example, [23][24][25][26][27][28]), and checking the regularities specific to vortex tubes would allow verification of the calculation. This article is devoted to the search for such regularities in the general 3D case for continuous flows of fluid and gas, hereinafter called as fluid flows.…”