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What Equations Describe Turbulence Adequately?

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The Essence of Turbulence as a Physical Phenomenon

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Abstract

As for today the standpoint of continuum mechanics reflected by the Navier–Stokes equations as a coarse graining over the molecular effects is considered as adequate and Perhaps the biggest fallacy about turbulence is that it can be reliably described (statistically) by a system of equations which is far easier to solve than the full time-dependent three-dimensional Navier–Stokes equations (Bradshaw, Exp Fluids 16:203–216, 1994).

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Notes

  1. 1.

    But see Goldstein (1972) concerning the success of NSE equations for the laminar flows of viscous fluids, but even in this case, it is, in fact, surprising that the assumption of linearity in the relation between \(\tau _{ij}\) and \(s_{ij}\) as usually employed in continuum theory,...works as well, and over as large a range, as it does. Unless we are prepared simply to accept this gratefully, without further curiosity, it seems clear that a deeper explanation must be sought.

    Also, Ladyzhenskaya (1975) and McComb (1990), Friedlander and Pavlović (2004) on alternatives to NSE, and Tsinober (1993, 2009) and references therein. In any case it is safe to keep in mind that no equations are Nature.

    Note aslo the statement by Ladyzhenskaya (1969): ... it is hardly possible to explain the transition from laminar to turbulent flows within the framework of the classical Navier-Stokes theory.

    Finally, since Leray (1934) one was not sure about the (theoretical, but not observational) possibility that turbulence is a manifestation of breakdown of the Navier–Stokes equations. Today it is clear that these doubts are definitely not correct.

  2. 2.

    However, a far less-trivial issue is ergodicity, i.e. if the flow is statistically stationary, the common practice is to use one long enough realization, i.e. it may suffice to have such a realization at least for those who believe that statistics is enough. The basis of this is the ergodicity hypothesis, see Chap. 6.

  3. 3.

    To quote Constantin (2016): An asymptotic description of the Navier-Stokes equation, based on a given “nearby” smooth Euler flow is difficult because the connection between the imposed Euler flow and the Navier-Stokes equation is illusory near the boundary.

    This is absolutely correct but attempting to imagine both flows (NSE and Euler) in some “similar conditions” one is tempted to doubt that this is the only “location” where these flows are not “nearby”. They seem to be drastically far from being such in most turbulent flows and at most of locations and times.

  4. 4.

    Massive evidence supports the view that most real fluids are Newtonian i.e, obeying the NSE with the viscous term \(\nu \nabla ^{2}u_{i}\), resulting from the product of the deviatoric stress and the strain tensors \(\tau _{ij}s_{ij}\) with linear relation between them \(\tau _{ij}=2\rho \nu s_{ij}\). It is for this reason the rate dissipation rate \(\epsilon \) in newtonian fluids is proportional to \(s^{2}\), \(\epsilon =2\nu s_{ij}s_{ij}\).

    In the hyperviscous case the viscous term \((-1)^{h+1}v_{h}\nabla ^{2h}u_{i}\) does not seem to correspond to any realistic relation between \(\tau _{ij}\) and \(s_{ij}\) even in rheology.

  5. 5.

    Saffman wrote in (1968): A property of turbulent motion is that the boundary conditions do not suffice to determine the detailed flow field but only average or mean properties. For example, pipe flow or the flow behind a grid in a wind tunnel at large Reynolds number is such that it is impossible to determine from the equations of motion the detailed flow at any instant. The true aim of turbulence theory is to predict the mean properties and their dependence on the boundary conditions.

    The latter view is still very popular in the community. Such an aim may be interpreted as giving up important aspects of understanding the physics of basic processes of turbulent flows. Indeed, there exist a multitude of various “theories” predicting at least some of the mean properties of some flows, but none seem to claim penetration into the physics of turbulence.

    It may be said that most of the theoretical work on the dynamics of turbulence has been devoted (and still is devoted) to ways of overcoming the difficulties associated with the closure problem, Monin and Yaglom (1971, p. 9).

    These difficulties have notbeen overcome and it does not seem that that this will happen in the near future if at all. There are several reasons for this. One of the hardest is the nonlocality property which is discussed in the Chap. 7.

    We mention that formally there exist two closed formulations which in reality are suspect to be just a formal restatement of the Navier Stokes equations, at least, as concerns the results obtained to date. One is due to Keller and Friedmann (1925) infinite chain of equations for the moments and the equivalent to this chain is an equation in term’s of functional integrals by Hopf (1952).

  6. 6.

    As a fictitious “mechanism” of delivery of energy from large to the small scales to be dissipated in the latter there - hence the frequently used term “Richardson Kolmogorov cascade”, for more on this and related issues, see below Sect. 7.3.

  7. 7.

    From time to time there appear claims to decompositions with weakly interacting elements/objects. This appears to be true only if some parameter is small as in RDT like theories, but not for genuinely nonlinear/strong turbulent flows whith no hope that small parameter does exist.

  8. 8.

    In fact it is also due to Euler, see Lamb (1932). A detailed account on the ‘misnomer’ by which the ‘Lagrangian’ equations are ascribed to Lagrange is found in Truesdell (1954), pp. 30–31.

  9. 9.

    Victor Youdovich used to say that one of the best solutions of NSE is experiment, 1971, private communication.

  10. 10.

    There are “smaller” problems such as uniqueness as shown in a recent example by Buckmaster and Vicol (2017) that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy .

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  1. School of Mechanical Engineering, Tel Aviv University, Tel Aviv, Israel

    Arkady Tsinober

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Tsinober, A. (2019). What Equations Describe Turbulence Adequately?. In: The Essence of Turbulence as a Physical Phenomenon. Springer, Cham. https://doi.org/10.1007/978-3-319-99531-1_3

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