Group invariant solutions for turbulent boundary layer flows described by eddy viscosity

Abstract

The study of turbulence is important in a wide variety of applications. Some examples are: the flow of fluid once a sluice gate is opened; the flow of air over the wing of an aircraft; the flow of air from a ventilation system. A very important concept to consider when dealing with turbulence is the boundary layer that develops around an object as air or fluid moves past the object. The boundary layer is a thin layer in which the velocity gradients increase rapidly from zero at the contact surface, to match the mainstream velocity further away from the surface. The velocity is zero at the contact surface due to the viscous nature of the fluid or air molecules moving past the surface. Within the boundary layer either laminar or turbulent flow, or both types of flow, occur. Laminar boundary layers have been analysed rigorously in the past. In this dissertation we analyse turbulent boundary layers and turbulent wall jets. The turbulence is represented by introducing the concept of eddy viscosity. The Reynolds averaged boundary layer equations are used in this dissertation. The group invariant solution of the two–dimensional turbulent boundary layer equations are derived by finding the Lie point symmetries of the equation. The mainstream velocity is allowed to be an arbitrary function. The Lie point symmetries of the equation modelling the wall jet are used to solve for a group invariant solution. The conserved quantity for the wall jet is found which enables us to solve the problem.