Finite element solution of the Navier-Stokes equations for 3-D turbulent free shear flows

Abstract

A half-equation model of turbulence has been developed to described the eddy viscosity distribution of two and three-dimensional turbulent free shear flows. The model is derived by integrating the parabolized transport equation for the turbulence kinetic energy over the cross section of the flow. The Prandtl-Kolmogrov hypothesis is used to obtain an ordinary differential equation for the eddy viscosity. The model is used in a general purpose finite element procedure using primitive variables. The penalty function method is used, in a generalized Galerkin weak formulation of the Navier-Stokes equations, to enforce the conservation of mass. In this procedure the pressure does not explicitly appear, this significantly reducing the computation time when compared to the velocity-pressure approach.

Numerical solution are obtained for four problems: a round jet issuing from a wall into still surroundings, a three-dimensional square jet issuing from a wall into still surroundings, a uniform flow past a free running propeller, and a shear flow past a free running propeller. An actuator disk with variable radial distribution of thrust and torque is used to model the propeller.

The numerical solution in the far field of the round jet agrees very well with the analytical similar solution. Very good agreement between prediction and experiments is observed for the square jet problem.

A simplified analysis of the flow past a propeller is used to provide the initial value of the eddy viscosity. Numerical experiments on the uniform flow past a thrusting disk confirmed the validity of the analysis and illustrated the effect of the initial value of the initial value of the eddy viscosity. For both propeller flows, agreement between predictions and experiments is excellent for both the axial and swirl velocity components at two stations located at x/D = 0.025 and 0.23. The quality of the swirl prediction is a major improvement over previous analyses. Pressure predictions are obtained for the first time, and are in reasonable agreement with the experiments. The radial velocity prediction is in fair agreement with the experiments at the station x/D = 0.025 .The discrepancy between the finite element solutions and the experiments at the station x/D = 0.23, for the pressure an the radial velocity are attributed to the presence of the body housing the propeller drive train. The body is not included in the present study. The complex three-dimensional nature of the shear flow past the propeller is very well captured in the simulation.