Fractional Turbulence Models

Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 196)

Abstract

In this article we propose to generalize Reynolds shear stresses in local zero-equation turbulence models to nonlocal and fractional forms. In the well-accepted general method, starting with a Kraichnanian convolution-integral as Reynolds shear stress, different weighting functions are possible candidates to serve this purpose; e.g. the Liouville weighting function leads to the left-handed Riemann–Liouville fractional derivative and the Heaviside distribution to a mean velocity difference, respectively a difference quotient. Therefore, this weighting function transforms the first gradient (the one in the eddy diffusivity) of Prandtl’s 1925 mixing-length model to an eddy diffusivity with a mean velocity difference and, thereby, directly leads to the (modified) Prandtl shear-layer model of 1942. Prandtl’s intuitive development—which is in agreement with fractional calculus—does not serve as a proof of correctness, but is a welcome coincidence. By further following Prandtl’s intuition and applying the Heaviside distribution also to the remaining driving gradient, yields the Difference-Quotient Turbulence Model (DQTM), which was discovered by other means and had been published by Egolf in 1991. As a result, it becomes clear that the DQTM is a natural nonlocal extension of Prandtl’s models and contains a special case of a simple fractional derivative, namely a difference quotient, which stands for the highest possible nonlocality and minimum calculation time to solve a turbulent flow problem.

References

1. 1.
R. Dedekind, M. Weber (ed.), Bernard Riemann’s gesammelte mathematische Werke und wissenschaftlicher Nachlass, Chap. XIX: Versuch einer allgemeinen Auffassung der Integration und Differentiation. Cambridge University Press, Cambridge (2013)Google Scholar
2. 2.
R.K. Raina, C.L. Koul, On Weyl fractional calculus. Proc. Am. Math. Soc. 73(2), 188–192 (1979)
3. 3.
B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman and Co., New York, 1977)Google Scholar
4. 4.
C. Beck, F. Schlögl, Thermodynamics of Chaotic Systems. Cambridge Nonlinear Science Series (Cambridge University Press, New York, 1993)Google Scholar
5. 5.
B.J. West, Fractional Calculus—View of Complexity: Tomorrow’s Science (CRC Press, Taylor & Francis Group, Boca Raton, 2016)Google Scholar
6. 6.
P.W. Egolf, K. Hutter, From linear and local to nonlinear and nonlocal zero-equation turbulence models, in Proceedings of IMA Conference for Turbulence, Waves and Mixing (Kings College, Cambridge, 2016), pp. 71–74, 6–8 July 2016Google Scholar
7. 7.
O. Reynolds, On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil. Trans. R. Soc. Lond. A 186, 123–164 (1895)Google Scholar
8. 8.
K. Hutter, K. Jöhnk, Continuum Methods of Physical Modeling (Springer, Berlin, 2004)Google Scholar
9. 9.
J.O. Hinze, Turbulence, 2nd edn (McGraw-Hill, New York, 1975)Google Scholar
10. 10.
R.H. Kraichnan, The Closure Problem of Turbulent Theory. Research Report of Office of Naval Research No. HSN-3, pp. 1-48 (1961)Google Scholar
11. 11.
L. Prandtl, Bericht über Untersuchungen zur ausgebildeten Turbulenz. ZAMM 5(2), 136–139 (1925)
12. 12.
L. Prandtl, Bemerkungen zur Theorie der freien Turbulenz. ZAMM 22(5), 241–243 (1942)
13. 13.
R. Herrmann, Fractional Calculus (World Scientific, Singapore, 2011)Google Scholar
14. 14.
I.M. Gelfand, G.E. Shilov, Generalized Functions, vol. 1 (American Mathematical Society, Providence, 1964)Google Scholar
15. 15.
P.W. Egolf, D.A. Weiss, Difference-quotient turbulence model: the axi-symmetric isothermal jet. Phys. Rev. E 58(1), 459–470 (1998)Google Scholar
16. 16.
P.W. Egolf, Lévy statistics and beta model: a new solution of “wall” turbulence with a critical phenomenon. Int. J. Refrig. 32, 1815–1836 (2009)
17. 17.
K. Hutter, Y. Wang, Fluid and Thermodynamics, volume 2: Advanced Fluid Mechanics and Thermodynamic Fundamentals, Chap. 16 (Springer, Berlin, 2016)Google Scholar
18. 18.
P.W. Egolf, K. Hutter, in Turbulent Shear Flow Described by the Algebraic Difference-Quotient Turbulence Model, ed, by J. Peinke, G. Kampers, M. Oberlack, M. Waclawczyk, A. Talamelli. Progress in Turbulence VI, Springer Proceedings in Physics (Springer, Heidelberg, 2016), pp. 105–109Google Scholar 