Epistemic uncertainty in statistical Markovian turbulence models

Diverse arrays of modern applications require accurate predictions of complex turbulent flows. These range from flows across aircraft and inside turbomachinery, over the planet for weather prediction and through the heart for cardiac auscultation, etc. Due to the disparate character of these complex flows, predictive methods must be robust so as to be applicable for most of these cases, yet possessing a high degree of fidelity in each. Furthermore, as the processes of analysis and engineering design involve repeated iterations, the predictive method must be computationally economical. In this light, Reynolds Averaged Navier Stokes (RANS)-based models represent the pragmatic approach for complex engineering flows. However, the RANS-based modeling paradigm has significant shortcomings. In addition to the well recognized issues of fidelity and realizability for select flows, such models have an inherent degree of uncertainty associated with their predictions for all flows. This uncertainty arises due to specific assumptions utilized in the formulation of the closure. This can be expressed in a hierarchical manner as outlined in Figure 1, wherein each step of model formulation introduces additional variability in the predictions. These can be divided into the uncertainty due to: The choice of a Markovian closure: Single-point turbulence closures assume that turbulence and its evolution can be described in terms of finite set of local tensors. For instance, in the two-equation model of Jones & Launder (1972), this set is composed of the turbulent kinetic energy and the dissipation rate. Similarly, in the Reynolds stress closure of Hanjalic & Launder (1972), this is composed of the Reynolds stress tensor and the mean gradients. This assumption does not adequately account for the history of the flow and leads to Markovian models attempting to replicate the Non-Markovian dynamics of turbulence. This introduces variability into the model’s predictions. Successive steps build upon this uncertainty. The form of the closure and the closure expression: This additional level of uncertainty arises due to the choice between different RANS closures (zero-/one-/two-equation models, or, a second moment closure). Thence, model formulation involves the choice of the exact closure expression with terms of different orders based on rational mechanics. This step may add further variability to the model prediction. The uncertainty resulting from these steps is typically designated as structural uncertainty. The nature and the values of coefficients: The appropriation of the best-possible values of the coefficients and their functional form introduces additional uncertainty in the model framework. This would adhere to the classification of parameter uncertainty. As is exhibited schematically in Figure 1, each level of assumptions may add to the inherent variability of the problem. Furthermore, a specific choice of the closure (or closure