A unified theory for turbulent wake flows described by eddy viscosity

• The conserved quantities for the turbulent wake equation are derived.
• A new conserved quantity, other than that of the classical and momentumless wakes, is found.
• Lie symmetry methods are used to generate the invariant solution.
• A mathematical relationship between the solutions of the three types of wakes is deduced.

In this paper, we consider the conservation laws for the far downstream wake equations described by eddy viscosity. A basis of conserved vectors is constructed. The well-known conserved quantities for the turbulent classical wake and the turbulent wake of a self-propelled body are obtained by integrating the corresponding conservation law across the wake and imposing the boundary conditions. For the wake of a self-propelled body the additional condition that the drag on the body is zero and is required to obtain the conserved quantity. A third conservation law, which possibly belongs to another type of wake, is discovered. The Lie point symmetry associated with the conserved vector is used to obtain the invariant solution and a typical velocity profile for this wake is provided. This wake appears to have common properties with the other two well-known wakes. We then analyse the invariant solutions to all three wake problems and prove that a simple mathematical relationship exists between them thus unifying the theory for turbulent wake flows.