A 50-Year Retrospective and the Future

Abstract

Fifty years ago in the spring of 1965 I went to work as an undergraduate research assistant at the Johns Hopkins University in the wave laboratory of Professor Owen Phillips. The goal was to examine how internal waves generated at a region of sharp density gradient were affected by an adverse current. As the waves steepened they broke into turbulence, then propagated in a different mode. A rather elegant theory could explain the steepening and what happened after the waves broke, but not the actual process. It was clear from even my very first study that turbulence was at the heart of most of the things we didn’t understand about fluid motions, and especially waves. Thus began my life-long love affair with turbulence, and my quest to learn more about it, especially about the equations which predict its behavior.In this paper I try to summarize and illustrate by example a major problem in our field; namely, the lack of exact solutions for turbulence, or even a set of basis functions to analyze our efforts. I contrast our efforts with those in wave mechanics and optics where exact solutions permit careful analysis of the effects of boundaries, solution methods, and even the role of non-linearities. Then I try to show how an idea by Lumley can be used to begin to fill the gap. And why previous attempts have largely failed to enhance our understanding because they did not include enough dimensions in the application. Finally I show how more recent efforts involving four dimensions have moved us toward at least the goal of having a working set of basis functions for particular problems.

Appendices

Appendix 1: A Brief History of Lumley’s Projection and My Involvement with It

History of science is always difficult, since the published literature often does not correspond to the way things happened, and especially since order of publication is often unrelated to the actual chronology. The “first results” are sometimes not published at all, or published much later. As a result, newcomers frequently are misled about who did what and when. This is especially true when an area of research is suppressed, as was the case with Lumley’s early work on this subject as well as my own work with my students. Since I joined this area of research just about the time Lumley was losing his enthusiasm for it, and we played some small role in both keeping it alive and re-inspiring him, the following account might be of some value to those struggling in their own isolated corners of turbulence.

With just a few exceptions, Lumley’s ideas beginning in the early 1960s were not received with great enthusiasm by the turbulence community. Nils Busch, a colleague and friend of Lumley’s in Meteorology at Penn State returned to his native Denmark to set up the Meteorology group at the Danish National Lab (RISOE) and carried Lumley’s ideas with him, most notably the Ph.D. thesis of Eric Lundtang Petersen (of Wind Atlas fame) who applied it to turbulent gusts. And Rex Reed at U. Missouri at Rolla (in Switzerland at the time) also saw the advantages, and for years tried to carry out experiments to obtain enough information to apply them. The group at Poitiers under the leadership of Jean-Paul Bonnet embraced it somewhat later, and collaborated extensively with Glauser starting in the 1980s (e.g., [5, 38]). So this particular respect for Lumley’s work was very much a part of the reason that the U. Poitiers gave him an honorary doctorate. But most others in the world, and especially the USA, ignored this part of Lumley’s work. In part, this was because they didn’t understand it. But that alone cannot explain why a few were so openly very hostile to his idea.

Most of this hostility I believe was based on misinformation, in part a consequence of the early applications of Lumley and his students themselves. In particular, Bakewell [2] carried out experiments in the viscous sublayer of the glycerine tunnel at Penn State built especially to apply the decomposition to near-wall turbulence, and Payne [31] applied it to existing measurements of a turbulent plane wake using Grant’s measurements [19]. Both had the tremendous disadvantage of working with very little data. Bakewell’s near-wall measurements were taken with a single hot-film probe along a single line perpendicular to the wall, with only a single component of velocity and only out to y ⁺ = 40. By using a series of “tricks” to fill in some of the missing component and cross-stream data, they inferred that the near-wall structure might be counter-rotating vortices, and created the schematic seen in many publications. Payne had even less data to work with, but managed to produce a pair of counter-rotating structures which spanned the wake. Unfortunately both of these efforts, instead of stimulating more work, just increased the criticism and the cynicism.

I don’t have any accounts of the Moscow meeting where Lumley presented his now famous paper, other than the paper which got my attention as a first year grad student. But I do know a bit about what happened later—mostly from Lumley himself. Lumley’s problems with the fluids community’s lack of acceptance, or even interest, began with his presentation at the 1967 APS meeting in Hawaii. He felt humiliated by the comments after his presentation, especially by the public criticism from Otto Laporte, and often cited this bad experience as one of his reasons for avoiding such meetings as much as possible—at least in his earlier years. Ironically Laporte did not criticize the ideas themselves, but instead castigated (Lumley’s words) him for using dots (“⋅ ”) instead of symbols for the independent variables (x, t), even though these were quite commonly used by mathematicians.

But the bigger problem with acceptance came from the emerging coherent structures community, and from Lumley’s own misunderstanding of what he had done. The coherent structure people led by the Cal Tech and Stanford groups were observing very active events, but Lumley was still thinking like Townsend’s big and mostly passive eddies. The attacks at meetings were fierce, so that by the time I arrived on the scene in the late 1960s Lumley had already lost interest (at least in the fight). His unwillingness to respond to unsolicited attacks often left me in the 1970s as his lone defender at APS and coherent structure meetings, even when he was present. He often thanked me for my efforts to defend his ideas in public, but clearly was disheartened by the need to do so.

By contrast with Townsend’s “passive large eddies,” and probably thanks to my wave background, I had never thought of Lumley’s decomposition as passive. From the very first I saw the dynamic possibilities, and went to Penn State to work with Lumley—in part because of my enthusiasm for working on the decomposition with him. (The other reason was to avoid being sent to Vietnam. Thanks to being hired at Penn State on a Navy contract I was able to avoid a war I did not approve of. And still finish my dissertation at Hopkins, but under Lumley’s supervision, thanks to Stan Corrsin’s intervention on my behalf.)

Once at Penn State, and somewhat to my disappointment, I instead ended up taking over the non-Newtonian drag-reduction experiments which were underway. And this in turn led to my early work with polymer drag reduction and the LDA. But immediately upon finishing my Ph.D. I set about to reactivate the glycerine tunnel to look for dynamic near-wall events. My very first proposal was to the GHR program of ONR and it was funded in 1972 with Lumley as co-PI. The idea was to include time in the measurements as well as multiple velocity components in multiple planes so we could see how things changed in time, not just space. Unfortunately I left for Buffalo before the facility modifications were finished, and the work was ultimately taken over by Siegfried Herzog. Shortly thereafter Lumley moved to Cornell, leaving “my” experiment and Herzog behind to finish it. (Siggy eventually wrote his dissertation at Cornell about 10 years later.) As per normal the experiments had proven more difficult than we had hoped, mostly because of probe and data storage issues. Herzog did complete them, but by this time the DNS efforts of Moin and Kim had caught up, in part because of our interaction with them. Our combined efforts proved important in laying the groundwork for the dynamic systems work of Aubrey et al. [1] later.

At Buffalo starting in 1974 I set out to do the same type of experiments that I had started at Penn State, but in the axisymmetric jet mixing layer. With a modest grant from NSF and some support from AFOSR in a collaborative program with Roger Arndt at Minnesota and Hassan Nagib at IIT, my students and I began the series of experiments which continue until today. Given the acceptance and wide-spread use of POD-based techniques today, it is hard in hindsight to imagine the hostility we faced at every step of the way. While I had gotten used to the negative proposal reviews, I really was quite surprised at how ready opponents were to make their disdain obvious in public. It was never clear, at least early on, whether the opposition was to Lumley’s idea, or to Lumley himself with me as surrogate for their attacks.

The turning point for me came at the 1976 APS meeting in Eugene, OR where in an invited talk, Hans Liepman of Cal Tech went out of his way to trash Lumley’s ideas. His specific comment after a brief tirade: “I’ve never seen any structures just sitting there.” There was really no opportunity to question him in the plenary session, but seeing him with a small group at the coffee break which followed I tried to engage him about his comment (quite gently—since I was only 31 at the time). I suggested he was being unfair and clearly lacked understanding of what Lumley had actually done. And I tried to explain briefly why the time-dependence was really all there. Hans was not in a mood to listen and pretty much exploded in my face. By this time a rather larger group of 20 or so had gathered around us, probably smelling my blood. At the most intense moment I felt an arm pushing me aside and Bill Reynolds of Stanford stepped between us. He tapped Hans on the stomach with this program and pointed over his shoulder with his thumb at me and said with a big smile: “The kids’s right, Hans!”. Liepman said not a word, turned on his heel and left.² Needless to say, Bill Reynolds was my hero after that. And while I got my reputation as “controversial,” not many took cheap shots in public after that at either me or Lumley or his decomposition.

The real breakthrough in our collective thinking about Lumley’s decomposition actually came with the work of Mark Glauser and Stewart Lieb, both Ph.D. students of mine in the early 1980s. My colleague Andres Soom at Buffalo had several MS students design for us a special computer controlled rig to make the measurement program possible—a real novelty in the late 1970s. It was constructed by Scott Woodward, who for many years afterward was an important part of my life and lab. Mark copied an idea from Hassan Nagib of making rakes of probes instead of individual ones. So we had finally both space and time information simultaneously. Figures 2.1 and 2.2 show the results of this decomposition. It was clear to us from the moment we saw the first reconstructed velocity traces that the Lumley integral had produced almost exactly the most dynamic events. And it had done so with only three eigenfunctions.

We showed these very plots to John Lumley during a break in a meeting at Cornell in the summer of 1983 and presented them about the same time [17, 27]. Looking at the plots together was truly the moment I think that John first realized the meaning and potential for what he had done. And it is also this moment I think that his interest in POD was reborn, but recast this time in the context of dynamic systems. Our paths also diverged—no longer was his decomposition about coherent structures or large eddies. He, with the collaboration of Phil Holmes and students Aubrey and Podvin and Bergooz, went for how to model these dynamic events. A whole new field of dynamic systems approach to turbulence was born—about which much has been written (c.f. [1, 18, 20]).

My group, by contrast, continued our quest to find the eigenfunctions and learn how the flow itself put them together. And that quest continues until this day. Not as much as been written, but enough to merit some discussion of its theoretical underpinnings. That is what this article was about—understanding what Lumley’s integral truly implies about (and demands of) the underlying flows, whether experimental, computational, or theoretical.

Appendix 2: The Problem with the “Snapshot POD”

The so-called Snatpshot POD was introduced by Sirovich in the early 1980s and has been used extensively since for a variety of purposes. It has been extremely popular for the dynamics system attempts to understand and control turbulence, especially since the pioneering study of Aubrey et al. [1] and the book by Holmes et al. [20]. It is easily implemented if one has many ‘snapshots’ of data like those commonly produced by PIV. But it is a mistake to confuse it with what has come to be known as the ‘classical POD’, and it has only a superficial connection to the Lumley decomposition discussed above.

The primary problem with it for real turbulent flows can be demonstrated quite easily. Understanding why there is a problem is a bit more subtle. To simplify things, consider a field of only the spatial variable, x, and time, t. The snapshot POD basically replaces the instantaneous velocity with the following expansion:

$$\displaystyle{ u(x,t) =\sum _{ n=1}^{N}a_{ n}(t)\ \phi ^{n}(x) }$$

(2.29)

Like the classical POD, the eigenfunctions, ϕ ⁿ(x), are still orthogonal and the random coefficients, a _n (t), are uncorrelated at different times. The two-point two-time correlation can therefore readily be computed as:

$$\displaystyle{ \langle u(x,t)u(x',t')\rangle =\sum _{ n=1}^{N}\langle a_{ n}^{{\ast}}(t)a_{ n}(t')\rangle \ \phi ^{n{\ast}}(x)\phi ^{n}(x') }$$

(2.30)

The connection to the classical POD (first noted by Sirovich in the early 1980s) comes about by replacing the classical POD integral with its finite difference approximation over space. In most applications this number is quite low so the matrices involved are quite manageable. But when using all the data available from DNS or PIV, these spatial arrays can be very large—typically 10⁶ or more. It might be argued, why not just take a smaller number of points—say a subset of those available. Unfortunately this causes serious aliasing, exactly like what happens if a time series is sampled too slowly. Now this can be overcome by spatially filtering the instantaneous data, but this can also be quite computationally intensive.

So enter the snapshot POD. By assuming the flow to be stationary, time averaging can be recognized to also be a summation, not over space but over snapshots. And by comparison the number of snapshots can be quite manageable, thousands instead of millions. Now comes Sirovich’s clever trick: interchange the order of summation so the time ‘average’ is outside the double summation and solve that eigenvalue problem instead. Presto, the classical POD and snapshot appear to have produced exactly the same result! So where is the problem?

Implicit in the derivation of the snapshot POD is the assumption of statistical stationarity, hence 〈a _n (t)a _n (t′)〉 = F ⁿ(t′ − t) only. So letting τ = t′ − t, we can rewrite Eq. (2.30) as:

$$\displaystyle{ \langle u(x,t)u(x',t')\rangle =\sum _{ n=1}^{N}F^{n}(\tau )\ R^{n}(x,x') }$$

(2.31)

where R ⁿ(x, x′) = ϕ ^n∗(x)ϕ ⁿ(x′), There is nothing in principle wrong with this except for the fact that I know of no turbulence which behaves this way. It would be a very rare flow indeed were the turbulence scales uncoupled from the temporal evolution of the flow.

This problem with the snapshot was first pointed out by me and Mark Glauser in 1986 in an APS/DFD presentation, but ignored and even disputed (see note by Aubrey et al. [1]). The failure of near-wall models and other dynamic system models to advance beyond relatively simple or separated flows, I believe can still be largely attributed to this underlying deficiency.

The underlying rationale for the snapshot POD lies in the clever interchanging of order of summation and the numerical approximations to both the Lumley integral and the finite sum arithmetic used to estimate an averaged value. The basic problem lies in the fact that Lumley’s optimization applied to a field which is inhomogeneous in space but stationary in time implies that the time-modes are Fourier modes in frequency. And this means all of the spatial eigenfunctions are functions of both space and frequency, not separate functions of space and time. This was one of the most important points of the body of this paper. Similarly, if the field has some directions which are homogeneous and/or periodic, then the eigenfunctions in these directions are Fourier modes, and so in the other directions they depend on frequency and mode number as well as space. Moreover, stationarity implies complex coefficients so that the same eigenfunctions can be used with different phases between them. The snapshot POD cannot reflect this since it is missing information, and simply mixes them.

Now my negativity about the snapshot POD should not be interpreted to mean that I think it cannot be useful. It can be very useful—just not in the sense of Lumley’s projection. Sometimes the periodic spatial pieces can be sorted as noted in the work of my students and co-workers and those of Glauser, Tinney, and co-workers. But in general, the time–frequency problem does not appear to be tractable, and this complicates attempts to use it for understanding dynamics. On the other hand, the snapshot POD can be a VERY useful way to sort and reduce data sets. One example previously mentioned was suggested to me by J. Freund who stored snapshot POD coefficients to provide a starting field for large scale CFD computations of a compressible mixing layer well into the run. Another example is from the work of Wänström et al. [41, 42] who used the snapshot POD results to filter and reconstruct the cross-correlation before applying the classical POD to the snapshot results.