Question

Evaluate flow field around a symmetric 2 dimentional body

Answer 1

Vortex methods are essentially a Lagrangian, grid-free approach in which fluid particles are used as basic computational elements to solve the Navier-Stokes equations in the vorticity-velocity formulation. The Navier-Stokes equations can be rewritten in terms of the vorticity. Taking the curl of the Navier-Stokes equation gives the vorticity transport equation (VTE), simplified aswhere the vorticity  is defined as and the kinematic viscosity  is assumed to be constant. This equation governs the evolution of vorticity in an incompressible viscous flow. The velocity  is coupled with  at all times in a self-consistent way through the relation of and . The position of the vorticity is determined from the velocity field and their strengths interact with each other. A fluid particle’s velocity can be evaluated through the Biot-Savart law. It is obvious that if the Biot-Savart law were used to calculate the velocity of each of the  particles in a simulation, then the direct calculation would involve evaluations in a single time step. By contrast, Vortex-In-Cell (VIC) methods [1, 2] are used to determine the velocity field and place it onto a temporary grid, instead of direct integration using the Biot-Savart law, which is clearly inappropriate for large values of . The VIC method has been improved with an immersed boundary method to handle more complex geometries (see [3–6]). The VIC method is a highly efficient hybrid particle-mesh algorithm to simulate an incompressible viscous flow past a solid body in an infinite domain [7]. Immersed boundary approaches, as reviewed in [8], are appropriate in free numerical computations of flows around complex geometries from technically difficult and time-consuming grid generation algorithms. Immersed boundary methods are used to enforce a no-slip condition at a body surface with decoupling between body boundaries and computational grids [3].