In all these statistical and dynamical problems we need to extend to and understand more realistic problems where the turbulence is inhomogeneous and affected by boundaries. A practical goal for DNS modellers and mathematicians might be the generation of an improved near-wall turbulence model. While the first example of models of this type possessed a rather narrow breadth of applicability and was heavily reliant on empirically tuned coefficients, a wide range of newer models is becoming available that is more fundamentally based and to which mathematical analysis can make a major contribution. The most interesting level in practice is that of second-moment closure which leads, using the momentum and scalar transport equations, together with assumptions about tensorial invariance and local homogeneity, to a formally soluble set of equations for the Reynolds stresses and other second moments. Such models pose challenging problems of accounting for inhomogeneous 'pressure-reflection' effects from boundaries while ensuring that the turbulence field is infallibly realisable.
Analogous problems arise in the fine scales near boundaries and also in combustion phenomena.
Such problems need the interplay of the mathematician and the physicist for satisfactory resolution. There are also problems of numerical solution associated with the strong intercoupling of the turbulence equations especially in flows where buoyancy or other force fields are strong or where combustion is present. It is important to see the fundamentals of turbulence in the context of its applications. If, indeed, the nature of the turbulence is strongly affected by the particular system and there is no general modelling framework, then consideration of individual systems becomes even more important. Unsteady turbulent flows are often of importance.