Computers and turbulence
Introduction
Although, from its title, this paper may seem to be an appeal for a new and unnecessary journal, it is intended as a review of the role that the rapid development of direct numerical simulations has played over the last decades in the elucidation of turbulence physics, and as a meditation on its possible evolution in the immediate future. It does not address the parallel development of turbulence models and large-eddy simulations, which are the mainstay of industry but which owe less to computational power. As the story unfolds, it should be clear to the reader that many of the themes that we meet in tracing turbulence research can be generalised to the role of computers in other branches of science and technology, some of which have led the way, while others have lagged behind [1], [2], [3], [4]. We may use the former as guides to likely future developments in our field, and the latter as warnings of the pitfalls to be avoided. In addition, and even if a definite answer is well beyond the scope of a paper like the present one, we will briefly discuss what could be the consequences of continuing in the present direction, both for the field and for its practitioners, and leave to the readers the decision on what they think is the right course to follow.
It has become customary to start historical reviews of turbulence with a reference to Leonardo da Vinci. Unfortunately, although he wrote extensively on fluid mechanics and on what later came to be regarded as turbulence, I have been unable to trace any reference to computation or automata applicable to the present review. However, to maintain tradition, I offer a quote which is relevant to the general themes discussed in the paper.
It refers to the relation between theory, experience and data, and can be applied to the hopes of some people, including the author, of what could be expected from computers. It can also be taken as a working definition of a successful theory: “There is nothing in nature without a cause; understand the cause and you will have no need for the experiment” [5, Cod. Atlantico, 147va]. This is clearly a dubious statement, and Leonardo hedges it elsewhere by declaring that: “It is my intention first to cite experience, then to demonstrate through reasoning why experience operates in a given way” [6, Ms E Paris, 55r], or that we should “avoid the teaching of speculators whose judgement is not confirmed by experience” [6, Ms B Paris, 4v]. While aiming for the first of these quotes, this review will try to follow the last two.
The paper is organised in terms of the cost of individual simulations. Section Section 2 deals with simulations which are very expensive, and Section Section 3 discusses the consequences of some simulations becoming inexpensive enough to be considered routine. These two sections can be seen as a short historical survey of the field up to now, but Section Section 4 speculates on how turbulence research might be affected when simulations become cheap enough to be deemed trivial, and offers an example. The paper is intended to promote discussion among researchers, and offers no conclusions, but a summary and a list of possible open questions are collected in Section Section 5.
Section snippets
Turbulence and numerical simulation
Humans must have been aware from very early times that the flow of water is not always smooth, and that crossing a river involves dealing with eddies comparable to the mean drift of the stream. The same applies to the wind, and some of the first recorded uses of the word ‘turbulent’ are applied to the weather. Quantitative study of turbulent flows, even if not recognised as such, must also have been commonplace, because the design of the extensive irrigation canals and aqueducts of antiquity
Overnight simulations and conceptual turbulence
It is a welcome characteristic of fluid mechanics that we know the equations of motion, and that we can simulate the subject of our study to any desired precision. Moreover, we saw in the last section that computers have evolved to the point at which simulations can address what is probably the asymptotically relevant set of parameters in simple flows. It follows that, once computers become fast enough for these simulations to become routine, they can provide answers to any concrete question
Trivial simulations and Monte Carlo research
In the previous section we have given examples of how using computers as ‘universal answering’ machines, performing routine searches and conceptual experiments in no more than a few hours, can advance the study of turbulence. The procedure follows the classical scientific method of designing experiments to address a particular question, with the only difference that they are performed computationally. However, because the time spent in each simulation is short, but not trivial, testing more
Discussion: competitors or colleagues?
As mentioned in the introduction, the goal of this paper is to promote discussion, and it therefore offers no conclusions. However, we can summarise what we have discussed up to now, and highlight some of the questions that, from the point of view of the author, need to be addressed by the community. The paper has been organised in three sections ordered by the increasing capabilities of computers. The first two sections, which deal with computers as expensive experimental apparatus, or as
Acknowledgement
This work was supported by the European Research Council under the Coturb grant ERC-2014.AdG-669505.
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