Vortex method with turbulence sub-grid scale modelling

Abstract

In this paper a method for the simulation of convection and diffusion of the vorticity, generated on a body surface is presented. A purely Lagrangian scheme is used for the vorticity convection, and thereby avoiding mesh associated problems. The body surface is simulated by straight-line panels, with constant-strength vortex distribution. The strength of the discrete vortices is obtained directly without going through any additional calculation. Using a primary diffusion process this vorticity is replaced by Lamb vortices located nearby the body surface. The diffusion process of the vorticity is simulated using the random walk scheme. Turbulence sub-grid scale modelling employs a Second Order Velocity Structure Function model adapted to the Lagrangian scheme. With a cloud of discrete vortices this velocity structure function model is employed to simulate the micro structures of the flow. The flow over a circular cylinder and a NACA 0012 aerofoil are considered to evaluate the integrated aerodynamic loads. Comparisons are made with previous theoretical and experimental studies

Numerical experiments of the flow around a bluff body with and without roughness model near a moving wall

“…Therefore, Lagrangian vortex method proceeds by discretizing spatially the vorticity field using Lamb-Oseen vortex elements [25,27], which are characterized by a distribution of vorticity 0 (commonly called the cutoff function), a circulation strength Γ, a core size 0 , and a spatial position x. The method amounts to tracking these vortex elements and their circulations, which in incompressible viscous flows is governed by the vorticity transport equation [3].…”

“…The vorticity transport equation is split into two steps: one without viscosity and one with diffusion solution. Alcântara Pereira et al [3] adapted the solution of these two steps to include LES modeling such as, respectively: where is defined as the eddy viscosity coefficient and ̄ = ∇ ×ū represents the vorticity filtered field, which in two dimensions is scalar and becomes the principal variable for computations.…”

“…The viscosity effect computation requires a very large number of vortex elements when using the random walk method. This step includes LES modeling in which the three-dimensional second-order velocity structure function model of the filtered field [28] was originally adapted by Alcântara Pereira et al [3] to take into account turbulence manifestations into the two-dimensional viscous wake with the following equation:…”

“…Finally, after to compute the vorticity and velocity filtered fields, the filtered pressure field can be recuperated. It is important to observe that pressure term is absent of the vorticity transport equation [3]. The pressure filtered field computation is obtained starting with the stagnation pressure definition in the following form:…”

“…Certainly, LES can be viewed as a product of the deterministic movement in that the large energy-containing scales are directly computed (as in direct numerical simulation: DNS). In this scenario, the second-order velocity structure function model of the filtered field [28] was originally adapted by Alcântara Pereira et al [3] to be applied for two-dimensional Lagrangian vortex method with LES modeling.…”