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In simple flows with mild mean deformation rates, the turbulence has time to come to equilibrium with the mean flow and the Reynolds stresses are determined by the strain rate. On the other hand, when the mean deformation is very rapid, the turbulent structure takes some time to respond and the Reynolds stresses are determined by the amount of total strain. A good turbulence model should exhibit this viscoelastic character of turbulence, matching the two limiting behaviours and providing a reasonable blend in between. Our goal, so far, has been the development of a one-point model with the proper viscoelastic character for engineering use. We have shown that, to achieve this goal, one needs to include structure information in the tensorial base used in the model, because non-equilibrium turbulence is inadequately characterized by the turbulent stresses themselves. We have also argued that the greater challenge in achieving visco-elasticity in a turbulence model is posed by the matching of rapid distortion theory RDT. Given a good RDT model, its extension to flows with mild deformation rates should be relatively straight- forward.
An elegant way to solve numerically the RDT systems is the Particle Representation Model (PRM). This is a set of equations for the evolution of the properties of a hypothetical particle. The basic idea is to follow the evolution of an ensemble of particles, determine its statistics and use these as the model for the one-point statistics of an evolving field. The equations emulate the exact equations for the evolution of the field, especially the one-point statistics. The key innovation in the present PRM approach lies in the recognition that the linearity of the RDT governing equations makes it possible to emulate exactly the RDT for homogeneous turbulence using a PRM without any modelling assumptions. The non-local pressure effects can be evaluated within the framework of the PRM, thus providing closure. This is unlike traditional particle representation approaches employed by the combustion community. When the time scale of the mean deformation is large compared to that of the turbulence, the non-linear turbulence-turbulence interactions become important in the governing field equations. In the context of the PRM, these non-linear processes should be represented by particle-particle interactions. As in the case of the one-point field equations, the non-linear processes cannot be evaluated directly and modelling is required. In so doing, the linear PRM version for the solution of RDT was extended to what we have termed, in this project, the Interacting Particle Representation Method (IPRM), which takes into account the non-linear effects. Due to the introduction of modelling, the emulation of the field equations by the IPRM is no longer exact, which was the case for the PRM emulation of RDT. The interacting particle representation model (IPRM) provides strong support for this position. The IPRM is, in essence, a very good viscoelastic structure-based turbulence model. The IPRM is also a relatively simple model for the non-linear turbulence-turbulence interactions that is able to handle quite successfully a surprising wide range of flows. Some of these flows involve paradoxical effects and the fact that the IPRM is able to reproduce them suggests that the model captures a significant part of the underlying physics.