The reason that I’ve referred to this somewhat esoteric phenomenon, is to make a point about the status of turbulence modelling knowledge in 1971. The point is that it was clear there is no way that the k-epsilon model on its own was going to predict the existence of these lateral motions. Indeed, no model based on the “turbulent viscosity” approach would predict them. It can be shown that, whatever turbulent viscosity distribution one might presume, there is no way that a variation of turbulent viscosity alone can lead to the initiation of lateral motions of the kind. In the absence of source terms of some kind in the lateral momentum equation, the only possible (wrong) fully-developed solution is zero lateral velocities.

However, in parallel with the creation of the k-epsilon model, the work by Spalding and Launder (and particularly Launder with his student Kemo Hanjelic) had led to the development of a full Reynolds stress model – in which, rather than representing the turbulent Reynolds stresses via the turbulent viscosity, the stresses (six in all for a 3D flow) are solved for directly, via their own differential transport equations. At this stage it was impracticable to contemplate solving a full 3D flow using such models (the available computer resources were insufficient), but the models had been used in reduced form to address some particular turbulence phenomena – one of which was these lateral motions for fully developed flow in a square duct.

Launder and Ying had solved the 2D situation (ie only perfectly fully developed flow) for flow in a square duct using a hybrid model, in which the k-epsilon model was used in conjunction with an algebraic form of the Reynolds stress model for the stresses affecting the lateral velocities. Here “algebraic” means that, by making presumptions about the convection and diffusion terms, the Reynolds stress equations were algebraic rather than differential – which made solution more practicable. The result of this is that the normal and shear Reynolds stress terms in the lateral momentum equations appear as source terms, which (Ying showed) led to the prediction reasonably well of these odd lateral motions.

So this was the model that I used – what I was effectively doing was (among other things) investigating the development of these lateral motions up to the fully developed regime that Ying had considered. And what I was able to show is that the model did indeed predict the gradual development of these lateral motions as the flow developed – but that it was only pretty far downstream (more than 30 duct widths) that they became really established and had a real influence on axial velocity, and hence on friction or heat transfer. Which is, I believe, why these lateral motions are really only of academic interest. It is only for situations involving perfectly straight ducts of 30-plus duct widths in length, with no other disturbances likely to create lateral motions themselves, for which these turbulence-induced motions are significant – and such cases are relatively rare in practical engineering situations. But they are an interesting – and challenging – test of turbulence modelling.

Looking back on this work, two things stand out for me. Firstly, it is remarkable that by 1971 the basic k-epsilon model had already reached the form in which it is known and widely used today – and was already a “standard model“ ready for adoption by “users“ such as myself. And, secondly, it is perhaps even more remarkable that, for the “special cases” for which the k-epsilon model fails, there was already a fully-formulated Reynolds-stress modelling approach available to be applied.

My square duct calculations may have been the very first 3D application of the k-epsilon model anywhere – certainly all the prior work I referred to in my thesis is 2D – but if anyone has a counter-claim that’s fine by me. What is more important to me is to recall being a very early “early adopter” of the outcome of the pioneering work on understanding and modelling turbulence by Spalding, Launder and Whitelaw during that remarkable period at the end of the 1960s and early 1970s!

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