No Access Submitted: 06 April 2017 Accepted: 05 May 2017 Published Online: 17 May 2017
Physics of Fluids 29, 051702 (2017); https://doi.org/10.1063/1.4983724
more...View Affiliations
View Contributors
• L. Djenidi
• L. Danaila
• R. A. Antonia
• S. Tang
We develop an analytical expression for the velocity derivative flatness factor, F, in decaying homogenous and isotropic turbulence (HIT) starting with the transport equation of the third-order moment of the velocity increment and assuming self-preservation. This expression, fully consistent with the Navier-Stokes equations, relates F to the product between the second-order pressure derivative ($∂2p/∂x2$) and second-order moment of the longitudinal velocity derivative ($(∂u/∂x)2$), highlighting the role the pressure plays in the scaling of the fourth-order moment of the longitudinal velocity derivative. It is also shown that F has an upper bound which follows the integral of $k*4Ep*(k*)$ where Ep and k are the pressure spectrum and the wavenumber, respectively (the symbol * represents the Kolmogorov normalization). Direct numerical simulations of forced HIT suggest that this integral converges toward a constant as the Reynolds number increases.
The financial support of the Australian Research Council is acknowledged. L.D. thanks the Institut National des Sciences Appliquées (INSA, Rouen, France) for its financial support for this work during his stay at CORIA (University of Rouen Normandie). The Labex EMC3 is thanked for its financial support.
1. 1. A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds number,” Dokl. Akad. Nauk SSSR 30(4), 301–305 (1941) Google Scholar
[A. N. Kolmogorov, [Proc. R. Soc. London A 434, 9–13 (1991)]. Google ScholarCrossref
2. 2. L. Landau and E. Lifshitz, Fluid Mechanics (Pergamon Press, London, 1963). Google Scholar
3. 3. A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. Fluid Mech. 13, 82–85 (1962). https://doi.org/10.1017/s0022112062000518, Google ScholarCrossref, ISI
4. 4. R. Antonia, S. L. Tang, L. Djenidi, and L. Danaila, “Boundedness of the velocity derivative skewness in various turbulent flows,” J. Fluid Mech. 781, 727–744 (2015). https://doi.org/10.1017/jfm.2015.539, Google ScholarCrossref, ISI
5. 5. S. L. Tang, R. Antonia, L. Djenidi, H. Abe, T. Zhou, L. Danaila, and Y. Zhou, “Transport equation for the meant turbulent energy dissipation rate on the centreline of a fully developed channel flow,” J. Fluid Mech. 777, 151–177 (2015). https://doi.org/10.1017/jfm.2015.342, Google ScholarCrossref
6. 6. S. L. Tang, R. Antonia, L. Djenidi, and Y. Zhou, “Transport equation for the meant turbulent energy dissipation rate in the far-wake of a circular cylinder,” J. Fluid Mech. 784, 109–129 (2015). https://doi.org/10.1017/jfm.2015.597, Google ScholarCrossref
7. 7. R. J. Hill, “Equations relating structure functions of all orders,” J. Fluid Mech. 434, 379–388 (2001). https://doi.org/10.1017/s0022112001003949, Google ScholarCrossref, ISI
8. 8. L. Danaila, F. Anselmet, T. Zhou, and R. A. Antonia, “A generalization of Yaglom’s equation which accounts for the large-scale forcing in heated decaying turbulence,” J. Fluid Mech. 391, 359–372 (1999). https://doi.org/10.1017/s0022112099005418, Google ScholarCrossref
9. 9. R. J. Hill and O. N. Boratav, “Next-order structure-function equations,” Phys. Fluids 13, 276–283 (2001). https://doi.org/10.1063/1.1327294, Google ScholarScitation, ISI
10. 10. T. Gotoh and T. Nakano, “Role of pressure in turbulence,” J. Stat. Phys. 113, 855–874 (2003). https://doi.org/10.1023/a:1027316804161, Google ScholarCrossref
11. 11. B. R. Pearson and R. A. Antonia, “Rλ behaviour of 2nd & 4th-order moments of velocity increments & derivatives,” in Proceedings of the First Symposium on Turbulence and Shear Flow Phenomena (TSFP-1), Santa Barbara, California, 12–15 September 1999, edited by N. Kasagi (Begell House, 1999), pp. 91–96, ISBN 1-56700-135-1. Google ScholarCrossref
12. 12. E. D. Siggia, “Invariants for the one-point vorticity and strain correlation functions,” Phys. Fluids 24, 1934–1936 (1981). https://doi.org/10.1063/1.863289, Google ScholarScitation, ISI
13. 13. R. M. Kerr, “Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence,” J. Fluid Mech. 153, 31–58 (1985). https://doi.org/10.1017/s0022112085001136, Google ScholarCrossref
14. 14. T. Ishihara, Y. Kaneda, M. Yokokawa, K. Itakura, and A. Uno, “Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics,” J. Fluid Mech. 592, 335–366 (2007). https://doi.org/10.1017/s0022112007008531, Google ScholarCrossref, ISI
15. 15. T. Zhou and R. A. Antonia, “Reynolds number dependence of the small-scale structure of grid turbulence,” J. Fluid Mech. 406, 81–107 (2000). https://doi.org/10.1017/s0022112099007296, Google ScholarCrossref
16. 16. A. Tsinober, E. Kit, and T. Dracos, “Experimental investigation of the field of velocity gradients in turbulent flows,” J. Fluid Mech. 242, 169–192(1992). https://doi.org/10.1017/s0022112092002325, Google ScholarCrossref
17. 17. T. Ishihara, Y. Kaneda, M. Yokokawa, K. Itakura, and A. Uno, “Spectra of energy dissipation, enstrophy and pressure by high-resolution direct numerical simulations of turbulence in a periodic box,” J. Phys. Soc. Jpn. 72, 983–986 (2003). https://doi.org/10.1143/jpsj.72.983, Google ScholarCrossref
18. 18. V. Yakhot, “Pressure–velocity correlations and scaling exponents in turbulence,” J. Fluid Mech. 495, 135–143 (2003). https://doi.org/10.1017/s0022112003006281, Google ScholarCrossref