Stefan Heinz

The Terra Incognita (or gray-zone) problem seen in atmospheric flow simulations causes serious consequences: it implies, e.g., significantly incorrect flow predictions and results that often simply depend on flow simulation settings as the computational grid applied. There is definitely the need for a robust gray-zone modeling to ensure that research and technology decisions are based on reliable results. As a matter of fact, solution approaches to deal with this problem in atmospheric and engineering type simulations reveal remarkable differences. In contrast to atmospheric flow simulations, there exists a broad spectrum of solution concepts for engineering applications. Driven by these conceptual differences, the paper presents an analysis of the Terra Incognita problem and corresponding solution concepts. Specifically, the paper presents a modeling approach that overcomes the core problem of currently applied methods. A new method of providing a resolution-aware turbulence length scale (one of the major problems in atmospheric flow simulations) is presented. This approach is capable of seamlessly covering the full range of microscale to mesoscale simulations, and to appropriately deal with mesoscale to microscale couplings.

Usually applied simulation methods for turbulent flows as large eddy simulation (LES), wall-modeled LES (WMLES), and detached eddy simulation (DES) face significant challenges: they are characterized by improper resolution variations and essential practical simulation problems given by huge computational cost, imbalanced resolution transitions, and resolution mismatch. Alternative simulation methods are described here. By using an extremal entropy analysis, it is shown how minimal error simulation methods can be designed. It is shown that these methods can overcome the typical shortcomings of usually applied simulation methods. A crucial ingredient of this analysis is the identification of a mathematically implied general hybridization mechanism, which is missing in existing methods. Applications to several complex high Reynolds number flow simulations reveal essential performance, functionality, and computational cost advantages of minimal error simulation methods.

The simulation of high Reynolds number (Re) separated turbulent flows faces significant problems for decades: large eddy simulation (LES) is computationally too expensive, and Reynolds-averaged Navier-Stokes (RANS) methods and hybrid RANS-LES methods often provide unreliable results. This has serious consequences, we are currently unable to reliably predict very high Re regimes, which hampers applications and our understanding of turbulence structures. The paper reports the advantages of a strict mathematical approach, continuous eddy simulation (CES), to derive partially resolving turbulence models. In contrast to popular hybrid RANS-LES, this minimal error approach includes a dynamic modification of the turbulence model in response to the actual flow resolution: the model can increase (decrease) its contribution to the simulation in dependence of a low (high) flow resolution. This property is the essential requirement to seamlessly cover RANS and LES regimes. The CES modeling approach offers essential advantages regarding its functionality: basically, it is independent of a variety of simulation settings applied in popular hybrid RANS-LES to improve the model performance. In addition, the CES computational cost can be below the cost of other hybrid RANS-LES and LES by orders of magnitude. Essential simulation performance advantages of CES simulations are described here with respect to three complex flow applications: periodic hill flows at high Reynolds number, the NASA wall-mounted hump flow, and the Bachalo & Johnson axisymmetric transonic bump flow.

The discovery of the law of the wall, the log-law including the von Karman constant, is seen to be one of the biggest accomplishments of fluid mechanics. However, after more than ninety years there is still a controversial debate about the validity and universality of the law of the wall. Clarity about this question matters: in absence of alternatives, a reliable and universal theory involving the law of the wall is needed to provide essential guideline for the validation of theory, computational methods, and experimental studies of very high Reynolds number (Re) flows. The paper presents an analysis of concepts used to derive controversial conclusions. It is shown that nonuniversality is a consequence of simplified modeling concepts, which leads to unrealizable models. On the other hand, realizability implies universality: models in consistency with physical requirements do not need to be adjusted to different flows. There are essential advantages of a universal law of the wall: it enables the design of accurate turbulence models and it provides a bridge between finite Re observations and asymptotic structural theories of turbulence.

Feasible and reliable predictions of separated turbulent flows are a requirement to successfully address the majority of aerospace and wind energy problems. Existing computational approaches such as large eddy simulation (LES) or Reynolds-averaged Navier–Stokes (RANS) methods have suffered for decades from well-known computational cost and reliability issues in this regard. One very popular approach to dealing with these questions is the use of machine learning (ML) methods to enable improved RANS predictions. An alternative is the use of minimal error simulation methods (continuous eddy simulation (CES), which may be seen as a dynamic ML method) in the framework of partially or fully resolving simulation methods. Characteristic features of the two approaches are presented here by considering a variety of complex separated flow simulations. The conclusion is that minimal error CES methods perform clearly better than ML-RANS methods. Most importantly and in contrast to ML-RANS methods, CES is demonstrated to be well applicable to cases not involved in the model development. The reason for such superior CES performance is identified here: it is the ability of CES to properly account for causal relationships induced by the structure of separated turbulent flows.