Fundamental Concepts of Turbulence

Abstract

All subsequent chapters deal with turbulence; in fact they aim to provide a small introductory insight into the theory of turbulent flows. The question that immediately arises in this connection is, however, why turbulence is dealt with at all in a book of which large parts are devoted to concepts of continuum mechanics and thermodynamics. The obvious answer is that turbulence theory is as much a topic of continuum physics as is continuum thermodynamics, but more importantly, it turned out during the last few decades of intensive research in turbulence modelling that methods of constructing turbulent closure schemes show strong similarities with techniques of establishing constitutive relations of continuum mechanics. There are differences between the two, and these differences are of fundamental nature, but the similarities are so strong that accessing the topic from a structural concept of continuum thermodynamics facilitates the learning of the subject tremendously — apart from providing new insight. To give hints as to these similarities and differences — they will be outlined in the ensuing chapters — we mention that establishing turbulent closure conditions follows similar lines as postulating constitutive equations for a certain material. The complexity of the material behaviour corresponds to the detailed reproduction of the fluctuating quantities by so-called higher order closure schemes. As for rules, valid in continuum thermomechanics for the constitutive relations, these closure relations must be functional relationships between objective tensor fields (of various ranks). However a principle, such as the rule of material objectivity cannot as strictly be imposed as it is commonly done in continuum thermodynamics. Furthermore, equations describing the evolution of the mean fields intend to describe physical observables and are therefore as much subjected to thermodynamic irreversibility requirements as are such fields in continuum thermodynamics. In other words, postulated turbulent closure conditions, which are the analoga to the constitutive relations in continuum thermodynamics, must be such that the second law of thermodynamics is fulfilled. This requirement will constrain the closure conditions in a similar way as the entropy principle constrains the constitutive functionals of continuum mechanics. A complete satisfaction of the second law of thermodynamics is commonly, however, not demonstrated in turbulence modeling — there are exceptions which we will be dealing with in Chap. 12 — but at least some rules are generally obeyed which often may be interpreted as fulfilling certain dissipation principles. These rules are called realizability conditions. It follows that turbulent fields that are in conformity with the second law of thermodynamics are automatically physically realizable.