A New Approach to the Analytical and Numerical Solution of the Bidirectional Vortex Flow

Abstract

*† ‡ , The solution for bulk fluid motion of a bidirectional coaxial vortex for application in vortex engine has been derived. The vortex engine is a novel combustion chamber in which swirl motion of reactants are used to maintain the chamber walls cool. The flow field has been considered both analytically and numerically. The model is based on incompressible, steady, axisymmetric, and non-reactive flow conditions. The governing PDEs are reduced to a system of nonlinear ODEs and then, by a coordinate transformation, their singularity has been relaxed. Solution domain has been decomposed into the inner viscous and outer inviscid regions, then, the velocity and pressure fields are obtained analytically. To verify the results, these ODEs are solved numerically via finite element method and by using the Galerkin and modified weighted residual formulations. The analytical and numerical results are in good agreement with each other. In comparison with other studies, the present calculated azimuthal velocity is more supportive of the experimental and numerical results and also confirms the presence of a forced-free vortex flow pattern. Location of the nontranslating vortex layer, mantle, is illuminated as well. Nomenclature i A = inlet area a = chamber radius b = chamber discharge radius Cμ = coefficient for the turbulence viscosity ij K = element stiffness matrix in the finite element formulation L = chamber length e l = an equivalent length scale for the turbulence i N = shape functions P =