Additional Thoughts on Half-Equation Model of Johnson and King

The Johnson King turbulence model represented a significant advancement in the understanding and modeling of turbulent flows. Introduced amidst the exploration of first and second equation models, the Johnson King model distinguished itself through the innovative concept of a half-equation model, emphasizing the critical role of memory in turbulence phenomena.

The early stages of turbulence model development were characterized by efforts to create predictive models, a task complicated by the limitations of computational power and the scarcity of high-quality experimental data. Unlike its contemporaries, the Johnson King model introduced a new approach by tracking the ratio of non-equilibrium flow through a single ordinary differential equation (ODE). This innovative strategy, which led to the designation of the model as a half-equation model, did not rely on a new closure equation involving a partial differential equation but rather utilized an ODE to describe the evolution of turbulence.

A key strength of the Johnson King model lies in its predictive capabilities, especially when compared to previous models, such as those based on the work of Cebeci and Smith. The model’s inclusion of turbulence history and the consideration of non-equilibrium effects allowed for a more accurate depiction of the amplification or dissipation of turbulent kinetic energy, particularly in the separation in boundary layers due to shock waves. This focus on the history and memory effects in turbulent flows marked a significant departure from equilibrium-based models and contributed to a deeper understanding of turbulence dynamics. Furthermore, it got away from concepts of one-point statistics of algebraic models.

The practical implications of the Johnson King model were underscored by its performance on VAX computer systems (the most popular DEC system of the day), where it demonstrated superior efficiency compared to one or two equation models. The model’s ODE was specifically valid across a line in the axial or streamwise direction through the flow, at points of maximum Reynolds’ stress. This focus on a streamlined computational approach not only enhanced the model’s speed and accuracy but also laid the groundwork for future developments in the field.

The Johnson King turbulence model, with its emphasis on memory and history effects, has played a pivotal role in the evolution of turbulence modeling. By introducing the half-equation concept and highlighting the importance of non-equilibrium effects, this model has contributed to a more nuanced understanding of turbulent flows and their underlying mechanisms in shock separation. The legacy of the Johnson King model continues to influence contemporary turbulence research, underscoring the lasting impact of their innovative approach.