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Mixing and Diffusion in Regular/Fractal Grid Turbulence

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Fractal Flow Design: How to Design Bespoke Turbulence and Why

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 568))

Abstract

Scalar mixing in turbulent flows is widely observed in nature as well as in industrial plants. In this chapter, we deal with three topics related to mixing and diffusion in grid-generated turbulence. The first topic (shown in Sect. 1) is experimental research on an axisymmetric CO\(_2\) jet issuing into free-stream turbulent flows generated by a square-mesh biplane round-rod grid (referred to as a regular grid in Sect. 1) and a square fractal grid. The CO\(_2\) jet issues from a small pipe located in the decaying region of these grid turbulences. A composite probe consisting of two concentration-sensitive I-type hot-wire sensors is used. For both flows, the mesh Reynolds number in the free stream is 6,000, and the jet Reynolds number based on the relative velocity between the free stream and the exit velocity of the jet is 5,000. The Taylor Reynolds numbers are about 100 and 35 for the square fractal grid and the regular grid, respectively. The results show that the half-widths of the mean velocity and concentration of the jets increase more rapidly, and the root mean square velocity and concentration in the axial direction decay more slowly for stronger free-stream turbulence. The second topic (shown in Sect. 2) is the development of a mixing layer of a high-Schmidt-number passive scalar in turbulent flows generated by a square-mesh biplane square-bar grid (referred to as a regular grid in Sect. 2) and a square fractal grid with the same mesh Reynolds number of 2500. A uniform passive scalar (Rhodamine B) is supplied only from the lower stream; therefore, scalar mixing layers with an initial step profile develop downstream of the grids. Particle image velocimetry and planar laser-induced fluorescence are used to investigate the velocity and concentration statistics. It is reconfirmed that the square fractal grid produces a higher turbulence intensity than the regular grid. The eddy diffusivity of the mass in the square fractal grid turbulence is approximately 4.2 times larger than that in the regular grid turbulence. The third topic (shown in Sect. 3) is direct numerical simulation of the mixing layer developed in grid turbulence. The simulations include the heat transfer in turbulent flows generated by four types of grid: (a) a square-mesh biplane square-bar grid (referred to as a regular grid in Sect. 3), (b) a square-mesh single-plane square-bar grid, (c) a composite grid consisting of parallel square-bars and (d) a square fractal grid. Two fluids at different temperatures are provided separately in the upper and lower streams upstream of the grids, generating a thermal mixing layer behind the grid. For grid (a), simulations with two different Prandtl numbers of 0.71 and 7.1, corresponding to air and water flows, respectively, are performed. The results show that the typical grid turbulence and thermal mixing layer can be simulated downstream of the grids, and a larger vertical turbulent heat flux is observed when the Prandtl number is large. Next the mixing layers in regular and square fractal grid turbulences are compared. In particular, the effects of the ratio of the largest to the smallest bar for the fractal grid, \(t_r\), are investigated. The results show that turbulent mixing is enhanced to a greater extent in fractal grid turbulence than in regular grid turbulence, especially at large \(t_r\). In Sect. 4, the conclusion and future prospects are presented.

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References

  • Brown, G. L., & Rebollo, M. R. (1972). A small, fast-response probe to measure composition of a binary gas mixture. AIAA Journal, 10(5), 649–652.

    Article  Google Scholar 

  • Chassaing, P. (1977). Heat transfer from cylindrical anemometer probes in CO\(_2\)-air mixtures. Physics of Fluids, 20(8), 1260–1262.

    Article  Google Scholar 

  • Chassaing, P. (1979). Mélange Turbulent de Gaz Inertes Dans un Jet Tubelibre. Ph.D. thesis, Institut Natl. Polytech. Toulouse.

    Google Scholar 

  • Chate, H., Villermaux, E., & Chomaz, J.-M. (1999). Mixing: Chaos and turbulence. New York: Kluwer Academic/Plenum Publishers.

    Book  Google Scholar 

  • Comte-Bellot, G., & Corrsin, S. (1966). The use of a contraction to improve the isotropy of grid-generated turbulence. Journal of Fluid Mechanics, 25(4), 657–682.

    Article  Google Scholar 

  • Culbertson, C. T., Jacobson, S. C., & Ramsey, J. M. (2002). Diffusion coefficient measurements in microfluidic devices. Talanta, 56(2), 365–373.

    Article  Google Scholar 

  • Davidson, P. A. (2004). Turbulence, an introduction for scientists and engineers. Oxford: Oxford University Press.

    MATH  Google Scholar 

  • Dimotakis, P. E. (2005). Turbulent mixing. Annual Review of Fluid Mechanics, 37(1), 329–356.

    Article  MathSciNet  MATH  Google Scholar 

  • Dimotakis, P. E., & Catrakis, H. J. (1999). Turbulence, fractals, and mixing. In H. Chate, et al. (Eds.), Mixing: Chaos and turbulence (pp. 59–143). New York: Kluwer Academic/Plenum Publishers.

    Chapter  Google Scholar 

  • Fadlun, E. A., Verzicco, R., Orlandi, P., & Mohd-Yusof, J. (2000). Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. Journal of Computational Physics, 161, 35–60.

    Article  MathSciNet  MATH  Google Scholar 

  • George, W. K., & Wang, H. (2009). The exponential decay of homogeneous turbulence. Physics of Fluids, 21, 025108.

    Article  MATH  Google Scholar 

  • Gomes-Fernandes, R., Ganapathisubramani, B., & Vassilicos, J. C. (2012). Particle image velocimetry study of fractal-generated turbulence. Journal of Fluid Mechanics, 711, 306–336.

    Article  MATH  Google Scholar 

  • Gomes-Fernandes, R., Ganapathisubramani, B., & Vassilicos, J. C. (2014). Evolution of the velocity-gradient tensor in a spatially developing turbulent flow. Journal of Fluid Mechanics, 756, 252–292.

    Article  Google Scholar 

  • Hart, D. P. (2000). PIV error correction. Experiments in Fluids, 29(1), 13–22.

    Article  Google Scholar 

  • Hinze, J. O. (1975). Turbulence (2nd ed.). New York: McGraw-Hill Inc.

    Google Scholar 

  • Huq, P., & Britter, R. E. (1995). Mixing due to grid-generated turbulence of a two-layer scalar profile. Journal of Fluid Mechanics, 285, 17–40.

    Article  Google Scholar 

  • Hurst, D., & Vassilicos, J. C. (2007). Scalings and decay of fractal-generated turbulence. Physics of Fluids, 19, 035103.

    Article  MATH  Google Scholar 

  • Ikeno, T., & Kajishima, T. (2007). Finite-difference immersed boundary method consistent with wall conditions for incompressible turbulent flow simulations. Journal of Computational Physics, 226, 1485–1508.

    Article  MathSciNet  MATH  Google Scholar 

  • Keane, R., & Adrian, R. (1992). Theory of cross-correlation analysis of PIV images. Applied Scientific Research, 49, 191–215.

    Google Scholar 

  • Komori, S., & Nagata, K. (1996). Effects of molecular diffusivities on counter-gradient scalar and momentum transfer in strongly stable stratification. Journal of Fluid Mechanics, 326, 205–237.

    Article  Google Scholar 

  • Kong, H., Choi, H., & Lee, J. S. (2001). Dissimilarity between the velocity and temperature fields in a perturbed turbulent thermal boundary layer. Physics of Fluids, 13(5), 1466–1479.

    Article  MATH  Google Scholar 

  • Krogstad, P.-Å., & Davidson, P. A. (2010). Is grid turbulence Saffman turbulence? Journal of Fluid Mechanics, 642, 373–394.

    Article  MathSciNet  MATH  Google Scholar 

  • Laizet, S., & Vassilicos, J. C. (2011a). DNS of fractal-generated turbulence. Flow, Turbulence and Combustion, 87(4), 673–705.

    Google Scholar 

  • Laizet, S., & Vassilicos, J. C. (2011b). Fractal grid turbulence and acoustic predictions. In Seventh International Symposium on Turbulence and Shear Flow Phenomena (TSFP-7), P30P (6 pages). Ottawa, Canada: Ottawa Convention Centre.

    Google Scholar 

  • Laizet, S., & Vassilicos, J. C. (2012). Fractal space-scale unfolding mechanism for energy-efficient turbulent mixing. Physical Review E, 86(4), 046302.

    Article  Google Scholar 

  • Laizet, S., & Vassilicos, J. C. (2015). Stirring and scalar transfer by grid-generated turbulence in the presence of a mean scalar gradient. Journal of Fluid Mechanics, 764, 52–75.

    Article  Google Scholar 

  • Laizet, S., Lamballais, E., & Vassilicos, J. C. (2010). A numerical strategy to combine high-order schemes, complex geometry and parallel computing for high resolution DNS of fractal generated turbulence. Computers & Fluids, 39, 471–484.

    Article  MATH  Google Scholar 

  • Laizet, S., Vassilicos, J. C., & Cambon, C. (2013). Interscale energy transfer in decaying turbulence and vorticity-strain-rate dynamics in grid-generated turbulence. Fluid Dynamics Research, 45(6), 061408.

    Article  MathSciNet  MATH  Google Scholar 

  • LaRue, J. C., & Libby, P. A. (1981). Thermal mixing layer downstream of half-heated turbulence grid. Physics of Fluids, 24(4), 597–603.

    Article  Google Scholar 

  • Lele, S. K. (1992). Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103, 16–42.

    Article  MathSciNet  MATH  Google Scholar 

  • Libby, P. A. (1975). Diffusion of heat downstream of a turbulence grid. Acta Astronautica, 2, 867–878.

    Article  Google Scholar 

  • Mazellier, N., & Vassilicos, J. C. (2010). Turbulence without Richardson-Kolmogorov cascade. Physics of Fluids, 22(7), 075101.

    Article  MATH  Google Scholar 

  • Mohamed, M. S., & LaRue, J. C. (1990). The decay power law in grid-generated turbulence. Journal of Fluid Mechanics, 219, 195–214.

    Article  Google Scholar 

  • Morinishi, Y., Lund, T. S., Vasilyev, O. V., & Moin, P. (1998). Fully conservative higher order finite difference schemes for incompressible flow. Journal of Computational Physics, 143, 90–124.

    Article  MathSciNet  MATH  Google Scholar 

  • Nagata, K., & Komori, S. (2001). The difference in turbulent diffusion between active and passive scalars in stable thermal stratification. Journal of Fluid Mechanics, 430, 361–380.

    Article  MATH  Google Scholar 

  • Nagata, K., Suzuki, H., Sakai, Y., Hayase, T., & Kubo, T. (2008a). Direct numerical simulation of turbulence characteristics generated by fractal grids. International Review of Physics, 2(6), 400–409.

    Google Scholar 

  • Nagata, K., Suzuki, H., Sakai, Y., Hayase, T., & Kubo, T. (2008b). Direct numerical simulation of turbulent mixing in grid-generated turbulence. Physica Scripta, T132, 014054.

    Article  Google Scholar 

  • Nagata, K., Suzuki, H., Sakai, Y., & Hayase, T. (2010). In Angermann, L. (Ed.), Numerical simulations-examples and applications in computational fluid dynamics. Chap. 3 (pp. 39–62). INTECH open access publishers.

    Google Scholar 

  • Nagata, K., Sakai, Y., Inaba, T., Suzuki, H., Terashima, O., & Suzuki, H. (2013). Turbulence structure and turbulence kinetic energy transport in multiscale/fractal-generated turbulence. Physics of Fluids, 25(6), 065102.

    Article  Google Scholar 

  • Nakamura, I., Sakai, Y., & Miyata, M. (1987). Diffusion of matter by a non-buoyant plume in grid-generated turbulence. Journal of Fluid Mechanics, 178, 379–403.

    Article  Google Scholar 

  • Okamoto, K., Nishino, S., Saga, T., & Kobayashi, T. (2000). Standard images for particle-image velocimetry. Measurement Science and Technology, 11, 685–691.

    Article  Google Scholar 

  • Panchapakesan, N. R., & Lumley, J. L. (1993). Turbulence measurements in axisymmetric jet of air and helium. Part 2. helium jet. Journal of Fluid Mechanics, 246, 225–247.

    Article  Google Scholar 

  • Sakai, Y. (2007). Characteristics of turbulent diffusion and reacting fields. Journal of JSME, 110(1059), 18–22. (in Japanese).

    Google Scholar 

  • Sakai, Y., Okada, Y., Kobayashi, N., Nakamura, I. & Miwa, M. (1999). Measurements of velocity-scalar joint statistics in a high Schmidt number axisymmetric turbulent jet. Proceedings of 3rd ASME/JSME Joint Fluids Engineering Conference, FEDSM-7765 (8 pages).

    Google Scholar 

  • Sakai, Y., Watanabe, T., Kamohara, S., Kushida, T., & Nakamura, I. (2001). Simultaneous measurements of concentration and velocity in a CO\(_2\) jet issuing into a grid turbulence by two-sensor hot-wire probe. International Journal of Heat and Fluid Flow, 22(3), 227–236.

    Article  Google Scholar 

  • Seoud, R. E., & Vassilicos, J. C. (2007). Dissipation and decay of fractal-generated turbulence. Physics of Fluids, 19(10), 105108.

    Article  MATH  Google Scholar 

  • Sreenivasan, K. R. (1991). Fractals and multifractals in fluid turbulence. Annual Review of Fluid Mechanics, 23, 539–600.

    Article  MathSciNet  Google Scholar 

  • Sreenivasan, K. R., Tavoularis, S., Henry, R., & Corrsin, S. (1980). Temperature fluctuations and scales in grid-generated turbulence. Journal of Fluid Mechanics, 100(3), 597–621.

    Article  Google Scholar 

  • Sugii, Y., Nishino, S., & Okubo, T. (2000). A highly accurate iterative PIV technique using a gradient method. Measurement Science and Technology, 11, 1666–1673.

    Article  Google Scholar 

  • Suzuki, H., Nagata, K., Sakai, Y., & Hayase, T. (2010a). Direct numerical simulation of regular and fractal-grid turbulence using the immersed boundary method and fully conservative higher-order finite-difference schemes. International Review of Physics, 4(2), 83–90.

    Google Scholar 

  • Suzuki, H., Nagata, K., Sakai, Y., & Hayase, T. (2010b). Direct numerical simulation of turbulent mixing in regular and fractal grid turbulence. Physica Scripta, 142, 014065.

    Article  Google Scholar 

  • Suzuki, H., Nagata, K., Sakai, Y., & Ukai, R. (2010c). High-Schmidt-number scalar transfer in regular and fractal grid turbulence. Physica Scripta, 142, 014069.

    Article  Google Scholar 

  • Suzuki, H., Nagata, K., & Sakai, Y. (2012a). Quantitative visualization of high-Schmidt-number turbulent mixing in grid turbulence by means of PLIF. Journal of Visualization, 15(2), 109–117.

    Google Scholar 

  • Suzuki, H., Nagata, K., Sakai, Y., & Hasegawa, Y. (2012b). Improved PIV analysis using an interpolation technique (analysis on improvement effects of the Fourier interpolation using quasi particle-images). Transactions of the JSME, Series B, 78(790), 1248–1259. (in Japanese).

    Google Scholar 

  • Suzuki, H., Nagata, K., Sakai, Y., Hayase, T., Hasegawa, Y., & Ushijima, T. (2013a). Direct numerical simulation of fractal-generated turbulence. Fluid Dynamics Research, 45, 061409.

    Article  MathSciNet  MATH  Google Scholar 

  • Suzuki, H., Nagata, K., Sakai, Y., & Hasegawa, Y. (2013b). Fractal analysis of turbulent mixing in fractal-generated turbulence by PLIF. Physica Scripta, 155, 014062.

    Article  Google Scholar 

  • Ulbrecht, J. J., & Patterson, G. K. (Eds.). (1985). Mixing of Liquids by Mechanical Agitation. New York: Gordon and Breach Science Publishers.

    Google Scholar 

  • Valente, P. C., & Vassilicos, J. C. (2011). The decay of turbulence generated by a class of multiscale grids. Journal of Fluid Mechanics, 687, 300–340.

    Article  MATH  Google Scholar 

  • Valente, P. C., & Vassilicos, J. C. (2012). Universal dissipation scaling for non-equilibrium turbulence. Physical Review Letters, 108, 214503.

    Article  Google Scholar 

  • Vassilicos, J. C. (2015). Dissipation in turbulent flows. Annual Review of Fluid Mechanics, 47, 95–114.

    Article  Google Scholar 

  • Veeravalli, S., & Warhaft, Z. (1989). The Shearless turbulence mixing layer. Journal of Fluid Mechanics, 207, 191–229.

    Article  Google Scholar 

  • Warhaft, Z. (1984). The interference of thermal fields from line sources in grid turbulence. Journal of Fluid Mechanic, 144, 363–387.

    Article  Google Scholar 

  • Warhaft, Z. (2000). Passive scalars in turbulent flows. Annual Review of Fluid Mechanics, 32, 203–240.

    Article  MathSciNet  MATH  Google Scholar 

  • Westerweel, J., Dabiri, D., & Gharib, M. (1997). The effect of a discrete window offset on the accuracy of cross-correlation analysis of digital PIV recordings. Experiments in Fluids, 23(1), 20–28.

    Article  Google Scholar 

  • Way, J., & Libby, P. A. (1971). Application of hot-wire anemometry and digital techniques to measurements in a turbulent helium jet. AIAA Journal, 9(8), 1567–1573.

    Article  Google Scholar 

  • Yoon, K., & Warhaft, Z. (1991). Turbulent mixing and transport in a thermally stratified interfacial layer in decaying grid turbulence. Physics of Fluids A, 3(5), 1143–1155.

    Article  Google Scholar 

  • Zhou, Y., Nagata, K., Sakai, Y., Suzuki, H., Ito, Y., Terashima, O., & Hayase, H. (2014). Relevance of turbulence behind the single square grid to turbulence generated by regular-and multiscale-grids. Physics of Fluids, 26, 075105.

    Google Scholar 

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Acknowledgments

The authors acknowledge Professor J. Christos Vassilicos (Imperial College London, UK) and Dr. Sylvain Laizet (Imperial College London, UK) for providing many valuable comments and suggestions for preparing this article. They also acknowledge Professor Toshiyuki Hayase (Tohoku University, Japan), Dr. Takashi Kubo (Meijo University, Japan) and Dr. Osamu Terashima (Honda R&D Co. Ltd., Japan) for their help with this research. They thank Mr. Takuya Yamaguchi (Nagoya University, Japan), Mr. Syuhei Ichino (Nagoya University) and Mr. Kenji Horiuchi (Nagoya University) for their support of the experiments. A part of this work was carried out under the Collaborative Research Project of the Institute of Fluid Science, Tohoku University, Japan, and the Research Cooperative Program between the Japan Society for the Promotion of Science and The Royal Society. A part of this study was supported by Grants-in-Aid (Nos. 20008010, 21656051, 22360076, 22360077, 25289030, 25289031, and 18686015) from the Japanese Ministry of Education, Culture, Sports, Science and Technology.

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Authors and Affiliations

  1. Department of Mechanical Science and Engineering, Nagoya University, Nagoya, Japan

    Yasuhiko Sakai & Yasumasa Ito

  2. Department of Aerospace Engineering, Nagoya University, Nagoya, Japan

    Koji Nagata

  3. Department of Mechanical Engineering, Yamaguchi University, Yamaguchi, Japan

    Hiroki Suzuki

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  1. Yasuhiko Sakai
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  2. Koji Nagata
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  4. Yasumasa Ito
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Correspondence to Yasuhiko Sakai .

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Editors and Affiliations

  1. Dept. of Mech. Science and Engineering, Nagoya University, Nagoya, Japan

    Yasuhiko Sakai

  2. Department of Aeronautics, Imperial College London, London, United Kingdom

    Christos Vassilicos

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Sakai, Y., Nagata, K., Suzuki, H., Ito, Y. (2016). Mixing and Diffusion in Regular/Fractal Grid Turbulence. In: Sakai, Y., Vassilicos, C. (eds) Fractal Flow Design: How to Design Bespoke Turbulence and Why. CISM International Centre for Mechanical Sciences, vol 568. Springer, Cham. https://doi.org/10.1007/978-3-319-33310-6_2

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