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A revisit of the equilibrium assumption for predicting near-wall turbulence

  • Farid Karimpour (a1) and Subhas K. Venayagamoorthy (a1)

Abstract

In this study, we revisit the consequence of assuming equilibrium between the rates of production ( $P$ ) and dissipation  $({\it\epsilon})$ of the turbulent kinetic energy  $(k)$ in the highly anisotropic and inhomogeneous near-wall region. Analytical and dimensional arguments are made to determine the relevant scales inherent in the turbulent viscosity ( ${\it\nu}_{t}$ ) formulation of the standard $k{-}{\it\epsilon}$ model, which is one of the most widely used turbulence closure schemes. This turbulent viscosity formulation is developed by assuming equilibrium and use of the turbulent kinetic energy $(k)$ to infer the relevant velocity scale. We show that such turbulent viscosity formulations are not suitable for modelling near-wall turbulence. Furthermore, we use the turbulent viscosity $({\it\nu}_{t})$ formulation suggested by Durbin (Theor. Comput. Fluid Dyn., vol. 3, 1991, pp. 1–13) to highlight the appropriate scales that correctly capture the characteristic scales and behaviour of $P/{\it\epsilon}$ in the near-wall region. We also show that the anisotropic Reynolds stress ( $\overline{u^{\prime }v^{\prime }}$ ) is correlated with the wall-normal, isotropic Reynolds stress ( $\overline{v^{\prime 2}}$ ) as $-\overline{u^{\prime }v^{\prime }}=c_{{\it\mu}}^{\prime }(ST_{L})(\overline{v^{\prime 2}})$ , where $S$ is the mean shear rate, $T_{L}=k/{\it\epsilon}$ is the turbulence (decay) time scale and $c_{{\it\mu}}^{\prime }$ is a universal constant. ‘A priori’ tests are performed to assess the validity of the propositions using the direct numerical simulation (DNS) data of unstratified channel flow of Hoyas & Jiménez (Phys. Fluids, vol. 18, 2006, 011702). The comparisons with the data are excellent and confirm our findings.

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Corresponding author

Email address for correspondence: vskaran@colostate.edu

References

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del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135144.
Corrsin, S. 1958 Local isotropy in turbulent shear flow. NACA RM 58B11, 115.
Durbin, P. A. 1991 Near-wall turbulence closure modeling without damping functions. Theor. Comput. Fluid Dyn. 3, 113.
Durbin, P. A. & Pettersson Reif, B. A. 2011 Statistical Theory and Modeling for Turbulent Flows. John Wiley and Sons.
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to $\mathit{Re}_{{\it\tau}}=2003$ . Phys. Fluids 18, 011702.
Jiménez, J. 2013 How linear is wall-bounded turbulence? Phys. Fluids 25, 110814.
Jones, W. P. & Launder, B. E. 1973 The calculation of low-Reynolds-number phenomena with a two-equation model of turbulence. Intl J. Heat Mass Transfer 16, 11191130.
Karimpour, F. & Venayagamoorthy, S. K. 2013 Some insights for the prediction of near-wall turbulence. J. Fluid Mech. 723, 126139.
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Kolmogorov, A. N. 1942 Equations of turbulent motion of an incompressible fluid. Izv. Acad. Nauk, SSSR; Ser. Fiz. 6, 5658.
Lam, C. K. G. & Bremhorst, K. A. 1981 A modified form of the $k{-}{\it\epsilon}$ model for predicting wall turbulence. Trans. ASME I: J. Fluids Engng 103, 456460.
Launder, B. E. & Sharma, B. I. 1974 Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer 1, 131137.
Launder, B. E. & Spalding, D. B. 1972 Mathematical Models of Turbulence. Academic Press.
Lumley, J. L. 1978 Computational modeling of turbulent flows. Adv. Appl. Mech. 18, 123176.
Marusic, I. & Perry, A. E. 1995 A wall-wake model for the turbulence structure of boundary layers. Part 2. Further experimental support. J. Fluid Mech. 298, 389407.
Moser, R. D., Kim, J. & Mansour, N. N. 1999 Direct numerical simulation of turbulent channel flow up to $\mathit{Re}_{{\it\tau}}=590$ . Phys. Fluids 11, 943945.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Rodi, W. & Mansour, N. N. 1993 Low-Reynolds-number $k{-}{\it\epsilon}$ modelling with the aid of direct numerical simulation data. J. Fluid Mech. 250, 509529.
Schultz, M. P. & Flack, K. A. 2013 Reynolds-number scaling of turbulent channel flow. Phys. Fluids 25, 025104.
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A revisit of the equilibrium assumption for predicting near-wall turbulence

  • Farid Karimpour (a1) and Subhas K. Venayagamoorthy (a1)

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