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Turbulent plane Poiseuille flow

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Abstract

The understanding of fully developed turbulence remains a major unsolved problem of statistical physics. A challenge there is how to use what we understand of this problem to build-up a closure method, that is to express the time-averaged turbulent stress tensor as a function of the time-averaged velocity field u(x). We have shown that the closure problem is strongly restricted due to constraints on the time-averaged quantities, and to scaling laws derived from the idea that dissipation in fully developed turbulence is by singular events resulting from an evolution described by the Euler equations. It implies that the turbulent stress is a non-local function in space of the time-averaged velocity u(x), involving an integral kernel, an extension of classical Boussinesq theory of turbulent viscosity. We treat one of the simplest possible physical situation, the turbulent Poiseuille flow between two parallel plates. In this case, the integral kernel takes a simple form leading to full analysis of the time-averaged turbulent flow. In the limit of a very large Reynolds number, one has to match a viscous boundary layer near the walls bounding the flow and an outer solution in the bulk of the flow, a non-trivial asymptotic analysis because of logarithms. Besides the boundary layers close to the walls, there is another “inner” boundary layer near the center plane of the flow. Our expression for the turbulent stress tensor yields ultimately the complete structure of the boundary layer, including in locations where viscosity becomes important.

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Notes

  1. This is the classical drag law of Newton, quadratic with respect to the velocity. Notice that Newton’s drag law was also derived by Maxwell (without the absolute value) who did not quote Newton.

  2. In Sect. (35) of [4], Landau wrote ” the dissipation occurs mainly in the region of rotational turbulent flows and hardly at all outside this region.”

  3. In the cases of circular pipes and mixing layer set up considered in our previous papers [1, 2], we have chosen to take a full 3D propagator, although the velocity fields depend on two variables only. Following the present argument, in the 2D case the kernel would be of the form \(\log \sqrt{x^{2} +z^{2}}\).

  4. Prandtl boundary conditions include the conditions on the surface itself, taking into account the possible roughness, and the Prandtl boundary layers equations. Let us notice that the classical Prandtl equations were shown to be still valid for smooth (no slip boundary conditions) curved surfaces when the viscosity goes to zero [6].

  5. In the more general case of non-smooth surfaces, the RST components \(\sigma _{ij}\) including the velocity fluctuations normal to the solid surface vanish, but not the longitudinal ones.

  6. This is an idea going back to the second half of the eighteenth century and due to the French engineer Chézy [8] with \(g/2\rho \) replaced by the slope of the bottom of a river times the acceleration of gravity and h a length depending on the depth and width of the flowing river.

  7. See for example Eq. (17.6) p. 518 in Ref. [9], 8th revised and enlarged edition. Note that \(u_{\tau }\) in this Eq. (17.6) corresponds to our \(u_{*}\).

  8. There has been attempts to compare accurately the log law of the wall with experimental data. This meets several challenges. First, the fitting of the experimental data and the theory requires to know accurately the velocity parameter \(u_*\), a parameter which can be inferred only indirectly from measurements of the local stress and/or from fit of the velocity data. Secondly, as we show, this log-law of the wall is only valid in a range of distances to the wall that is intermediate between the thickness of the viscous sublayer and the far wall dependence of the velocity profile that depends in a non-trivial way of the Reynolds number (and so of the average flow speed). All this, added to the difficulty of making a clear-cut difference between a logarithm and a power law with a small exponent, makes it hard to pinpoint the range of values of the distance to the wall where the log law applies, see Ref. [11].

  9. Finally explained by Prandtl.

  10. The flow is potential upstream because the Kelvin theorem states that circulation is convected along flow lines.

  11. Along a half-plate at zero incidence, observations made a hundred years ago display a great increase of the boundary layer thickness at the transition from laminar (close to x=0 where the thickness behaves as \(h(x)=5 (\nu x/u)^{1/2} \)) to turbulent behavior, occurring at the critical value \(ux/\nu \approx 5 \cdot 10^{5}\).

  12. See section 36 of [4] where it is stated that the turbulent domain formed behind a plate displays an angular dependence, because there are no constant at our disposal having the dimension of a length. Therefore, the average velocity in the turbulent boundary layer is expected to depend on the ratio z/x, as studied in our recent mixing layer paper [2].

References

  1. C. Josserand, M. Le Berre, Y. Pomeau, Scaling laws in turbulence. Chaos 30(7), 073137 (2020). https://doi.org/10.1063/1.5144147

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Y. Pomeau, M. Le Berre, Turbulence in a wedge: the case of the mixing layer. Phys. Rev. Fluids 6, 074603 (2021)

    Article  ADS  Google Scholar 

  3. Y. Pomeau, M. Le Berre, Leray turbulence: what can we learn from acceleration compared to velocity. arXiv:1909.01724 (2019)

  4. L. Landau, E.M. Lifshitz, Course of Theoretical Physics, Fluid Mechanics (Pergamon, Oxford, 1987)

    Google Scholar 

  5. F.G. Schmitt, About Boussinesq turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity. C. R. Mecanique 335, 617–627 (2007)

    Article  ADS  Google Scholar 

  6. C.G. Liu, Y.G. Wang, Derivation of the Prandtl boundary layer equations for the incompressible Navier–Stokes equations in a curved domain. Appl. Math. Lett. 34, 81–85 (2014)

    Article  MathSciNet  Google Scholar 

  7. G.I. Taylor, Flow in pipes and between parallel planes. Proc. R. Soc A 159, 496–506 (1937)

    ADS  MATH  Google Scholar 

  8. G. Mouret, A. Chézy, Histoire d’une formule d’hydraulique, Annales des ponts et chaussées, Paris, vol. LXI, n. 11, mars-avril 1921, pp. 165 –269

  9. H. Schlichting, K. Gersten, Boundary Layer Theory (Springer, Berlin, 1999)

    MATH  Google Scholar 

  10. A.J. Hutchinson, N. Hale, K. Born, D.P. Mason. Prandtl’s extended mixing length model applied to the two-dimensional turbulent classical far wake. Proc. R. Soc. A 477(2246), 20200875 (2021)

  11. M.H. Buschmann, M. Gad-el-Hak, Turbulent boundary layers: reality and myth. In: G. Lube, G. Rapin (Eds.), International Conference on Boundary and Interior Layers BAIL 2006, University of Gottingen (2006)

  12. L.F. Moody, Friction factors for pipe flow, Transactions of the ASME pp. 671–684, Proceedings of the Semi-Annual meeting of the Hydraulic division of ASME, Pittsburg, June 19–22 (1944). In this interesting publication Moody refers to our \(f\) as Darcy coefficient for pipe flows and to Chézy for its application to free surface flows. The theory behind the friction factor of more or less rough pipes is not discussed by Moody. It is referred to works by von Karman and by the Prandtl school (ref [9])

  13. U. Schumann, Realizability of Reynolds-stress turbulence models. Phys. Fluids 20, 721 (1977)

    Article  ADS  Google Scholar 

  14. M.J. Walsh, A.M. Lindemann, Optimization and application of riblets for turbulent drag reduction. AIAA paper 84-0347 (1984)

  15. R. García-Mayoral, J. Jimenez, Drag reduction by riblets. Philos. Trans. R. Soc. A. 369, 1412–1427 (2011). https://doi.org/10.1098/rsta.2010.0359

    Article  ADS  Google Scholar 

  16. A. Legeay, O. Cadot, On the recirculating flow of three-dimensional asymmetric bluff bodies. Exp. Fluids 61, 249 (2020)

    Article  Google Scholar 

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Acknowledgements

We thank Christophe Josserand for his interest in this work and for fruitful and stimulating discussions. Yves Pomeau thanks the Simons Foundation for support of this work through Targeted Grant in MPS-663054, “Revisiting the Turbulence Problem Using Statistical Mechanics.”

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  1. Laboratoire d’Hydrodynamique, Ladhyx, (CNRS UMR 7646), Ecole Polytechnique, 91128, Palaiseau, France

    Yves Pomeau & Martine Le Berre

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  1. Yves Pomeau
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  2. Martine Le Berre
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Correspondence to Martine Le Berre.

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Pomeau, Y., Le Berre, M. Turbulent plane Poiseuille flow. Eur. Phys. J. Plus 136, 1114 (2021). https://doi.org/10.1140/epjp/s13360-021-02118-z

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