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A Reynolds stress model for near-wall turbulence

Published online by Cambridge University Press:  26 April 2006

P. A. Durbin
P. A. Durbin
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA 94305-3030, USA
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Abstract

A tensorially consistent near-wall second-order closure model is formulated. Redistributive terms in the Reynolds stress equations are modelled by an elliptic relaxation equation in order to represent strongly non-homogeneous effects produced by the presence of walls; this replaces the quasi-homogeneous algebraic models that are usually employed, and avoids the need for ad hoc damping functions. A quasi-homogeneous model appears as the source term in the elliptic relaxation equation-here we use the simple Rotta return to isotropy and isotropization of production formulae. The formulation of the model equations enables appropriate boundary conditions to be satisfied.

The model is solved for channel flow and boundary layers with zero and adverse pressure gradients. Good predictions of Reynolds stress components, mean flow, skin friction and displacement thickness are obtained in various comparisons to experimental and direct numerical simulation data.

The model is also applied to a boundary layer flowing along a wall with a 90°, constant-radius, convex bend. Because the model is of a general, tensorially invariant form, special modifications for curvature effects are not needed; the equations are simply transformed to curvilinear coordinates. The model predicts many important features of this flow. These include: the abrupt drop of skin friction and Stanton number at the start of the curve, and their more gradual recovery after the bend; the suppression of turbulent intensity in the outer part of the boundary layer; a region of negative (counter-gradient) Reynolds shear stress; and recovery from curvature in the form of a Reynolds stress ‘bore’ propagating out from the surface. A shortcoming of the present model is that it overpredicts the rate of this recovery.

A heat flux model is developed. It is shown that curvature effects on heat transfer can also be accounted for automatically by a tensorially invariant formulation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 249 , April 1993 , pp. 465 - 498
Copyright
© 1993 Cambridge University Press

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References

Alving, A. E., Smits, A. J. & Watmuff, J. H. 1990 Turbulent boundary layer relaxation from convex curvature. J. Fluid Mech. 211, 529– 556.Google Scholar
Bandyopadhyay, P. R. & Ahmed, A. 1993 Turbulent boundary layers subject to multiple curvatures and pressure gradients. J. Fluid Mech. 246, 503– 527.Google Scholar
Bradshaw, P. 1973 The effect of streamline curvature on turbulent flow. AGARDograph 169.
Bradshaw, P., Mansour, N. N. & Piomelli, U. 1987 On local approximations of the pressure–strain term in turbulence models. Proc. Summer Program, Center for Turbulence Research, Stanford, University.
Bushnell, D. M. 1983 Turbulent drag reduction. AIAA Paper 83-0227.
Coles, D. E. & Hirst, E. A. (eds.) 1968 Computation of Turbulent Boundary Layers, vol. 2. AFOSR-IFP-Stanford Conference.
Dakos, T. & Gibson, M. M. 1987 On modelling the pressure terms of the scalar flux equations. Turbulent Shear Flows 5 (ed. F. Durst et al.), pp. 7– 18. Springer.
Durbin, P. A. 1991 Near-wall turbulence modeling without damping functions. Theoret. Comput. Fluid Dyn. 3, 1– 13.Google Scholar
Durbin, P. A. 1992 Application of a near-wall turbulence model to boundary layers and heat transfer. CTR Manuscript 132.Google Scholar
Durbin, P. A. & Speziale, C. G. 1991 Local anisotropy in strained turbulence at high Reynolds number. Trans. ASME I: J. Fluids Engng 113, 707– 709.Google Scholar
Gillis, J. C., Johnston, J. P., Kays, W. M. & Moffatt, R. J. 1980 Turbulent boundary layer on a convex curved surface. HMT-31. Department of Mechanical Engineering, Stanford University.
Hanjalic, K. & Launder, B. E. 1976 Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence. J. Fluid Mech. 74, 593– 610.Google Scholar
Hunt, J. C. R. & Graham, J. M. R. 1978 Free-stream turbulence near plane boundaries. J. Fluid Mech. 212, 497– 532.Google Scholar
Johnston, J. P. 1978 Internal Flows. In Turbulence (ed. P. Bradshaw). Topics in Applied Physics, vol. 12. Springer.
Kline, S. J. Morkovin, M. V. Sovran, G. & Cockrell, D. J. (eds.) 1968 Computation of Turbulent Boundary Layers, vol. 1. AFOSR-IFP-Stanford Conference.
Klebanoff, S. 1955 Characteristics of turbulence in a boundary layer with zero pressure gradient. NACA Rep. 1247.
Launder, B. E. 1978 Heat and Mass Transport. In Turbulence (ed. P. Bradshaw). Topics in Applied Physics, vol. 12. Springer.
Launder, B. E. 1989 Second-moment closure: present⃛ and future. Intl J. Heat Fluid Flow 10, 282– 300.Google Scholar
Lumley, J. L. & Newman, G. R. 1977 Return to isotropy of homogeneous turbulence. J. Fluid Mech. 82, 161– 178.Google Scholar
Mansour, N. N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation budgets in a turbulent channel flow. J. Fluid Mech. 194, 15– 44.Google Scholar
Mc Connell, A. J. 1957 Applications of Tensor Analysis. Dover.
Patel, V. C., Rodi, W. & Scheurer, G. 1985 Turbulence models for near-wall and low Reynolds number flows: a review. AIAA J. 23, 1308– 1319.Google Scholar
Reynolds, W. C., Kays, W. M. & Kline, S. J. 1958 Heat transfer in the turbulent incompressible boundary layer. NACA Memo. 12-1-58w.
Rodi, W. & Scheurer, G. 1986 Scrutinizing the k-ε model under adverse pressure gradient conditions. Trans. ASME I: J. Fluids Engng 108, 174– 180.Google Scholar
Samuel, A. E. & Joubert, P. N. 1974 A boundary layer developing in an increasingly adverse pressure gradient. J. Fluid Mech. 66, 481– 505.Google Scholar
Schubauer, G. B. & Spangenberg, W. G. 1960 Forced mixing in boundary layers. J. Fluid Mech. 8, 10– 32.Google Scholar
Simon, T. W., Moffatt, R. J., Johnston, J. P. & Kays, W. M. 1982 Turbulent boundary layer heat transfer experiments: curvature effects including introduction and recovery. NASA CR 3510.
So, R. M. C. & Mellor, G. L. 1973 Experiment on convex curvature effects in turbulent boundary layers. J. Fluid Mech. 60, 43– 62.Google Scholar
Spalart, P. R. 1988 Direct simulation of a turbulent boundary layer up to Rθ = 1410. J. Fluid Mech. 187, 61– 98.Google Scholar
Subramanian, C. S. & Antonia, R. A. 1981 Effect of Reynolds number on a slightly heated turbulent boundary layer. Intl J. Heat Mass Transfer 24, 1833– 1846.Google Scholar
Tavoularis, S. & Corrsin, S. 1981 Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient. Part 1. J. Fluid Mech. 104, 311– 347.Google Scholar
Tavoularis, S. & Karnik, U. 1989 Further experiments on the evolution of turbulent stresses and scales in uniformly sheared turbulence. J. Fluid Mech. 204, 457– 478.Google Scholar