Averaging operators in turbulence

Although Gregory Falkovich and Katepalli Sreenivasan review important lessons from hydrodynamic turbulence (PHYSICS TODAY, April 2006, page 43), we think the field has left us a legacy of Reynolds averaging whose worth needs to be reevaluated. The foremost reason why turbulent flows "confound any simple attempts to understand them" is that, as the authors point out, "questions about turbulent flows can be posed and answered only in terms of statistical averages" [emphasis ours]. Falkovich and Sreenivasan represent this averaging with angle brackets, ⟨ . . . ⟩, on page 44 but gloss over the fundamental importance of averaging operators in turbulence; they say only that angle brackets denote "a suitable average."

Experimentalists have inherited Reynolds averaging for obtaining estimates of ⟨ . . . ⟩, but such averaging is appropriate only when the turbulence is in steady state. The atmosphere, for example, is a turbulent fluid that is rarely in steady state.

Early work by Sreenivasan and coworkers¹ and by others^2,3 revealed that Reynolds averaging of turbulence time series leads to lagged autocorrelation functions whose net area under the curve is zero. That is, they imply zero integral scale. Our recent work⁴ has built on that result to conclude that block averaging, the recommended modern version of Reynolds averaging⁵ formulated to analyze turbulence time series recorded over long periods, generates turbulence statistics whose time evolution is incompatible with the Navier–Stokes equation. A comparable result emerges for the conservation equation for passive scalars described on page 47 of the PHYSICS TODAY article. The authors say those "who study turbulence believe that all its important properties are contained" in those equations. Although we concur with that statement, the newly found incompatibility⁴ is unacceptable.

Reynolds averages evidently have subtle features that conflict with fundamental physical laws. These features are a consequence of using an averaging method appropriate for data that are stationary and independent to analyze data that are stationary and correlated. Therefore, the links "between turbulence, critical phenomena, and other problems of condensed matter physics and field theory" that Falkovich and Sreenivasan anticipate from future research may remain hidden until more robust methods for assessing the time-specific as well as time-invariant average properties of turbulence are formulated. Standard Reynolds averaging and its modern refinements, unfortunately, are not reliable for deducing the statistical properties of turbulence.

References

1. 1. K. R. Sreenivasan, A. J. Chambers, R. A. Antonia, Boundary-Layer Meteorol. 14, 341 (1978) [INSPEC].
2. 2. G. I. Taylor, Proc. London Math. Soc. 20, 196 (1921).
3. 3. G. Comte-Bellot, S. Corrsin, J. Fluid Mech. 48, 273 (1971) .
4. 4. G. Treviño, E. L. Andreas, Boundary-Layer Meteorol. 120, 497 (2006) .
5. 5. X. Lee, W. Massman, B. Law, eds., Handbook of Micrometeorology: A Guide for Surface Flux Measurement and Analysis, Kluwer, Boston (2004).

Falkovich and Sreenivasan reply: Our review was devoted to fundamental physical properties of turbulence. These properties manifest themselves most clearly in instances that are statistically steady and homogeneous. We interpret the letter writers' concern to mean that one has to be careful, in general circumstances, about the choice of the averaging procedure. Indeed, one needs to exercise care in defining averages for nonstationary processes or those with insufficient data. However, that fact does not invalidate the Navier–Stokes equations or the advection–diffusion equation.

One possible explanation for the zero values of the inferred integral scale is the inadvertent filtering out of the very lowest frequencies from a measured turbulent signal. This was an attribute of much of the instrumentation used some 30 years earlier, before the digital revolution became commonplace.