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Despite its limitations, Prandtl’s mixing length model is widely applied in modelling turbulent free shear flows. Prandtl’s extended model addresses many of the shortfalls of the original model, but is not so widely used, in part due to additional mathematical complexities that arise in its derivation and implementation. Furthermore, in both models, Prandtl neglects the kinematic viscosity on the basis that it is much smaller in magnitude than the turbulent viscosity. Recent work has shown that including the kinematic viscosity in the original model has both mathematical and physical advantages. In the present work, a novel derivation of the extended model is provided, and it is demonstrated that similar advantages are again obtained when the kinematic viscosity is included. Additionally, through the use of scaling techniques, similarity mean velocity profiles of the extended model are derived, resulting in a single nonlinear ordinary differential equation that is solved numerically with a Hermite spectral method. The computed profiles for the normalized similarity mean velocity and shear stress are compared with experimental observations and shown to be in excellent agreement.

Abstract. A deterministic mathematical model of the black rhino population in South Africa will be discussed. The model is constructed by dividing the black rhino population into multiple patches. The impact of human intervention on different translocation strategies is incorporated into the model. It is shown that, when implemented correctly, translocation can accelerate the growth rate of the total black rhino population. Equilibrium points are shown with their local stability criteria.

A modified version of Prandtl’s mixing length closure model is applied to the two-dimensional turbulent classical far wake with a variable mainstream flow. This modified version of Prandtl’s mixing length model has been previously applied to the two dimensional turbulent classical wake where the mainstream speed is constant. The model is able to address some of the serious criticisms on the applicability of Prandtl’s mixing length model to free shear flows. For instance, the model is complete in that the mixing length is derived in a systematic way and a wake boundary extending to infinity is predicted. In this work, the effect of a variable mainstream flow on the expression for the mixing length and the effective width of the wake is examined. For variable mainstream flows, it is shown that the effective width is not only proportional to the mixing length but also inversely proportional to the slip velocity. For this problem, the power is conserved. The invariant solution corresponding to the conserved quantity is obtained. The flow behavior is analyzed for a constant, a decelerating, and an accelerating ideal slip-flow.