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Simplified theory of the near-wall turbulent layer of Newtonian and drag-reducing fluids

Published online by Cambridge University Press:  20 April 2006

M. A. Goldshtik ,
V. V. Zametalin  and
V. N. Shtern
M. A. Goldshtik
Affiliation:
Institute of Thermophysics, Siberian Branch of the U.S.S.R. Academy of Sciences, Novosibirsk-90, 630090, U.S.S.R.
V. V. Zametalin
Affiliation:
Institute of Thermophysics, Siberian Branch of the U.S.S.R. Academy of Sciences, Novosibirsk-90, 630090, U.S.S.R.
V. N. Shtern
Affiliation:
Institute of Thermophysics, Siberian Branch of the U.S.S.R. Academy of Sciences, Novosibirsk-90, 630090, U.S.S.R.
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Abstract

We propose a simplified theory of a viscous layer in near-wall turbulent flow that determines the mean-velocity profile and integral characteristics of velocity fluctuations. The theory is based on the concepts resulting from the experimental data implying a relatively simple almost-ordered structure of fluctuations in close proximity to the wall. On the basis of data on the greatest contribution to transfer processes made by the part of the spectrum associated with the main size of the observed structures, the turbulent fluctuations are simulated by a three-dimensional running wave whose parameters are found from the problem solution. Mathematically the problem reduces to the solution of linearized Navier-Stokes equations. The no-slip condition is satisfied on the wall, whereas on the outer boundary of a viscous layer the conditions of smooth conjunction with the asymptotic shape of velocity and fluctuation-energy profiles resulting from the dimensional analysis are satisfied. The formulation of the problem is completed by the requirement of maximum curvature of the mean-velocity profile on the outer boundary applied from stability considerations.

The solution of the problem does not require any quantitative empirical data, although the conditions of conjunction were formulated according to the well-known concepts obtained experimentally. As a result, the near-wall law for the averaged velocity has been calculated theoretically and is in good agreement with experiment, and the characteristic scales for fluctuations have also been determined. The developed theory is applied to turbulent-flow calculations in Maxwell and Oldroyd media. The elastic properties of fluids are shown to lead to near-wall region reconstruction and its associated drag reduction, as is the case in turbulent flows of dilute polymer solutions. This theory accounts for several features typical of the Toms effect, such as the threshold character of the effect and the decrease in the normal fluctuating velocity. The analysis of the near-wall Oldroyd fluid flow permits us to elucidate several new aspects of the drag-reduction effect. It has been established that the Toms effect does not always result in thickening of the viscous sublayer; on the contrary, the most intense drag reduction takes place without thickening in the viscous sublayer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 119 , June 1982 , pp. 423 - 441
Copyright
© 1982 Cambridge University Press

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References

Berman, N. S. 1977 Flow time scales and drag reduction. Phys. Fluids 20, 168–175.Google Scholar
Corcos, G. M. 1964 The structure of the turbulent pressure field in boundary layer flow. J. Fluid Mech. 18, 353–378.Google Scholar
Corino, E. K. & Brodkey, R. S. 1969 A visual investigation of the wall region in turbulent flows. J. Fluid Mech. 37, 1–30.Google Scholar
Davies, P. O. A. & Yule, A. J. 1975 Coherent structures in turbulence. J. Fluid Mech. 69, 513–537.Google Scholar
Einstein, N. A. & Li, H. 1956 The viscous sublayer along a smooth boundary. J. Engng Mech. Div. A.S.C.E. 82, 1–27.Google Scholar
Geshev, P. I. 1974 Turbulent exchange coefficients in viscous sub-layers. Prikl. Mekh. Teor. Fiz. 2, 61–66.Google Scholar
Goldshtik, M. A. 1968 Criterion of maximum stability of averaged turbulent flows. Dokl. Akad. Nauk S.S.S.R. 182, 1026–1028.Google Scholar
Goldshtik, M. A. & Shtern, V. N. 1977 Hydrodynamic Stability and Turbulence. Nauka.
Gorodtzov, V. A. & Belokon, V. S. 1973 Similarity laws for developed turbulence in diluted polymer solutions. J. Engng Phys. 25, 967–976.Google Scholar
Hatziavramidis, D. T. & Hanratty, T. J. 1979 The representation of the viscous wall region by a regular eddy pattern. J. Fluid Mech. 95, 655–679.Google Scholar
Hoyt, J. W. 1972 The effect of additives on fluid friction. Trans. A.S.M.E. D, J. Basic Engng 94, 258–285.Google Scholar
Kader, B. A. 1970 Turbulence in viscous sub-layer near a flat wall. In Turbulent Flows, pp. 69–73. Nauka.
Khabakhpasheva, E. M., Mikhailova, E. S., Perepelitza, B. V. & Efimenko, G. I. 1975 Experimental studies of the structure of near-wall turbulence. In Near- Wall Turbulent Flows, part 2, pp. 138–161. Institute of Thermophysics, Novosibirsk.
Khabakhpasheva, E. M. & Perepelitza, B. V. 1970 Peculiarities of near-wall turbulence in water flows with high-molecular additives. J. Engng Phys. 18, 1094–1097.Google Scholar
Kline, S. J., Reynolds, W. S., Shraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 26, 199–221.Google Scholar
Laufer, J. 1954 The structure of turbulence in fully developed pipe flow. NACA Rep. no. 1174.Google Scholar
Lawn, C. J. 1971 The determination of the rate of dissipation in turbulent pipe flow. J. Fluid Mech. 48, 477–508.Google Scholar
Metzner, A. B. & Park, M. G. 1964 Turbulent flow characteristics of viscoelastic fluids. J. Fluid Mech. 20, 291–303.Google Scholar
Mitchell, J. E. & Hanratty, T. J. 1966 A study of turbulence at a wall using an electrochemical wall shear-stress meter. J. Fluid Mech. 26, 199–221.Google Scholar
Morrison, W. R. B., Bullock, K. J. & Kronauer, R. E. 1971 Experimental evidence of waves in the sublayer. J. Fluid Mech. 47, 639–656.Google Scholar
Oldroyd, J. G. 1950 On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541.Google Scholar
Repik, E. U., Sosedko, Yu. P. & Tropinina, N. S. 1975 Studies of flow structure in the near-wall region of turbulent boundary layers. In Near-Wall Turbulent Flows, part II, pp. 138–161. Institute of Thermophysics, Novosibirsk.
Rudd, M. J. 1972 Velocity measurements made with a laser dopplermeter on the turbulent pipe flow of dilute polymer solutions. J. Fluid Mech. 51, 673–685.Google Scholar
Sadovskii, V. S., Sinitzyna, I. P. & Taganov, G. I. 1975 Numerical studies of a mathematical model for a viscous flow in turbulent boundary layer. In Near-Wall Turbulent Flow, part 1, pp. 94–116. Institute of Thermophysics, Novosibirsk.
Schubert, G. & Corcos, G. M. 1967 The dynamics of turbulence near a wall according to a linear model. J. Fluid Mech. 29, 113–135.Google Scholar
Sternberg, J. 1962 A theory for the viscous sublayer of a turbulent flow. J. Fluid Mech. 13, 241–271.Google Scholar
Toms, B. A. 1949 Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds number. In Proc. 1st Int. Cong. Rheol., vol. II, pp. 135–144. Amsterdam.
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.