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Large Reynolds Number Behavior, Symmetries, Universality

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The Essence of Turbulence as a Physical Phenomenon

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Abstract

The large Reynolds number behavior is of special importance from several points of fundamental nature which include such issues as restoring (or not) in somesensethe symmetries of Navier–Stokes equations (e.g. locally in Kolmogorov, Dokl Akad Nauk SSSR 30:299–303, (1941a)), universality, the role of viscosity/dissipation and the concept of inertial range, the role of the nature of forcing/excitation, inflow, initial and boundary conditions. In view of the arguments and experimental results on nonlocality and the direct and bidirectional coupling between large and small scales in the previous section a natural question arises what is the impact of nonlocality on all the above and whether there are enough reasons and evidence for a discussion and reexamination of the above issues, generally, and in relation to the nonlocal properties of turbulent flows among others, especially. Navier–Stokes equations at sufficiently large Reynolds number have the property of intrinsic mechanisms of becoming complex without any external aid such as strain and vorticity self-amplification. There is no guarantee that the outcome is the same from, e.g., natural “self-randomization” and with random forcing, on one hand, and different kinds of forcing, boundary and initial conditions, on the other hand. Moreover, there is serious evidence that the outcome may be and indeed is different.

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Notes

  1. 1.

    This latter is due to Kolmogorov (1941a). We emphasize that the correction he proposed in 1962 concerns the impact of the large scales on the scaling of structure functions in CDIR.

    A typical statement is found in Frisch (1995, p. 81): A consequence of Kolmogorov’s formula (6.5) (i.e the 4/5 law - see the (8.2) above) is the existence of a range of \( \ell \)‘s over which \( \left\langle \epsilon \right\rangle \)(\( \ell \) ) is independent of \( \ell \).

  2. 2.

    Noteworthy is the pure kinematic relation involving the third order structure function, Hosokawa (2007), Kholmyansky and Tsinober (2008), Germano (2012) \(\left\langle (\Delta u)^{3}\right\rangle =3\left\langle u_{sum}^{2}\Delta u\right\rangle \), with \(2u_{sum}=\)\(u(x+r)+u(x)\), which together with the 4/5 law results in a relation equivalent to the 4/5 law \( \left\langle u_{sum}^{2}\Delta u\right\rangle =\left\langle \epsilon \right\rangle r/30\). Thus the 4/5 law provides a clear indication of absence of statistical independence between the quantities residing at large and small scales.

    Of interest is the issue concerning the (an)isotropy of the CDIR and in particular the impact of the SDE’s. The information is scarse. On a qualitative level the tendency to anisotropyas manifested by the relations of the second order velocity structure functions is somewhat weaker as compared to the 4/5 law, Gulitskii et al. (2007a, b, c), Iyer et al. (2017).

  3. 3.

    This is precisely the question by Kraichnan (1974): How a theoretical attack on the inertial-range (IR) problem should proceed is far from clear, This question seems to be irrelevant as there is no such an object (IR) in existence.

  4. 4.

    There is a general difficulty in the experimental context associated with the direct large/small-scale coupling, see Chap. 7. As the Reynolds number increases the characteristic scale between (!) the regions of strong small-scale activity (‘intermittency’) is increasing, whereas the characteristic scale of these regions themselves is decreasing. On the other hand when looking at higher- order structure functions \(S^{p}\sim r^{\zeta p} \) (assuming some scaling) one is forced to use very long time records to achieve statistical convergence. This means that very large scales (which are anisotropic) are involved in the process. In other words, as the Reynolds number and the order of the structure function increase so (most likely) does the ‘deviation’ from \(\zeta _{p}\text { }\tilde{}p/3\). This means that experimentally (!) it may be not realistic to get a reasonable and reliable estimate of the exponents \(\zeta _{p}\) if such exist. This becomes less realistic as the order of the structure function and the Reynolds number are increasing. All the above provided that the experimental errors are small. Here again the larger the order of the structure function the larger is the influence of the errors.

  5. 5.

    The differences in the behavior of systems with different dissipation at finite Reynolds numbers points to a possibility that the hypothetical limiting solution will depend on the kind of dissipation we have at finite Reynolds number. For example, assume that for a hyperviscous case the mean dissipation \(\epsilon \) \(\rightarrow const\) (or just nonvanishing) as some viscosity \(\nu _{h}\) goes to zero, then velocity derivatives (both vorticity and strain) grow on the average as \(\nu ^{-1/2h}\), which compared to the Newtonian case \(h=2\) is pretty slow, if say, \(h=8\) as used in many simulations.

  6. 6.

    Frisch (1995) presents this in the form of his hypothesis H1 (p. 74), but omits to mention that it is due to Kolmogorov: there is no presentation of the hypothesis of local isotropy in his book. A similar omitting is pretty ubiquitous, among the latest example being Falkovich (2009).

  7. 7.

    That far that one of the principal incentives for writing this book was a desire to summarize the development of the idea of a universal local structure in any turbulent flow for sufficiently large Reynolds number, [Monin and Yaglom 1971, p. 21]. This is 1600 pages total.

  8. 8.

    For other negative statements about universality see, for example, Saffman (1978) and Hunt and Carruthers (1990).

  9. 9.

    Hopf (1948) conjectured that the underlying attractor is finite dimensional due to presence of viscosity.

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

    Arkady Tsinober

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Tsinober, A. (2019). Large Reynolds Number Behavior, Symmetries, Universality. In: The Essence of Turbulence as a Physical Phenomenon. Springer, Cham. https://doi.org/10.1007/978-3-319-99531-1_8

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