Skip to main content Accessibility help
×
Home

On the scaling of shear-driven entrainment: a DNS study

  • Harm J. J. Jonker (a1), Maarten van Reeuwijk (a2), Peter P. Sullivan (a3) and Edward G. Patton (a3)

Abstract

The deepening of a shear-driven turbulent layer penetrating into a stably stratified quiescent layer is studied using direct numerical simulation (DNS). The simulation design mimics the classical laboratory experiments by Kato & Phillips (J. Fluid Mech., vol. 37, 1969, pp. 643–655) in that it starts with linear stratification and applies a constant shear stress at the lower boundary, but avoids sidewall and rotation effects inherent in the original experiment. It is found that the layers universally deepen as a function of the square root of time, independent of the initial stratification and the Reynolds number of the simulations, provided that the Reynolds number is large enough. Consistent with this finding, the dimensionless entrainment velocity varies with the bulk Richardson number as $R{i}^{- 1/ 2} $ . In addition, it is observed that all cases evolve in a self-similar fashion. A self-similarity analysis of the conservation equations shows that only a square root growth law is consistent with self-similar behaviour.

Copyright

Corresponding author

Email address for correspondence: h.j.j.jonker@tudelft.nl

References

Hide All
Beare, R. J., Macvean, M. K., Holtslag, A. A. M., Cuxart, J., Esau, I., Golaz, J.-C., Jimenez, M. A., Khairoutdinov, M., Kosovic, B., Lewellen, D., Lund, T. S., Lundquist, J. K., McCabe, A., Moene, A. F., Noh, Y., Raasch, S. & Sullivan, P. 2006 An intercomparison of large-eddy simulations of the stable boundary layer. Boundary-Layer Meteorol. 118 (2), 247272.
Conzemius, R. J. & Fedorovich, E. 2006 Dynamics of sheared convective boundary layer entrainment. Part II: evaluation of bulk model predictions of entrainment flux. J. Atmos. Sci. 63, 11791199.
Deardorff, J. W. & Willis, G. E. 1982 Dependence of mixed-layer entrainment on shear stress and velocity jump. J. Fluid Mech. 115, 123149.
Deardorff, J. W. & Yoon, S.-C. 1984 On the use of an annulus to study mixed-layer entrainment. J. Fluid Mech. 142, 97120.
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455493.
Galperin, B., Kantha, L. H., Hassid, S. & Rosati, A. 1988 A quasi-equilibrium turbulent energy model for geophysical flows. J. Atmos. Sci. 45, 5562.
Jones, I. S. F. & Mulhearn, P. J. 1983 The influence of external turbulence on sheared interfaces. Geophys. Astrophys. Fluid Dyn. 24, 4962.
Kantha, L. H., Phillips, O. M. & Azad, R. S. 1977 On turbulent entrainment at a stable density interface. J. Fluid Mech. 79, 753768.
Kato, H. & Phillips, O. M. 1969 On the penetration of a turbulent layer into stratified fluid. J. Fluid Mech. 37 (04), 643655.
Kundu, P. K. K. 1981 Self-similarity in stress-driven entrainment experiments. J. Geophys. Res. 86 (C3), 19791988.
Large, W. G., McWilliams, J. C. & Doney, S. C. 1994 Oceanic vertical mixing: a review and a model with a non-local boundary layer parameterization. Rev. Geophys. 32 (4), 363403.
Ligniéres, F., Califano, F. & Mangeney, A. 1998 Stress-driven mixed layer in a stably stratified fluid. Geophys. Astrophys. Fluid Dyn. 88, 81113.
Mellor, G. L. & Durbin, P. A. 1975 The structure and dynamics of the ocean surface mixed layer. J. Phys. Oceanogr. 5, 718728.
Nieuwstadt, F. T. M. 1984 The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci. 41 (14), 22022216.
Pollard, R. T., Rhines, P. B. & Thompson, R. O. R. Y. 1973 The deepening of the wind-mixed layer. Geophys. Fluid. Dyn. 3, 381404.
Price, J. F. 1979 On the scaling of stress-driven entrainment experiments. J. Fluid Mech. 90, 509529.
Price, J. F. 1981 Upper ocean response to a hurricane. J. Phys. Oceanogr. 11 (4), 153175.
van Reeuwijk, M. 2007 Direct simulation and regularization modelling of turbulent thermal convection. PhD thesis, Delft University of Technology.
van Reeuwijk, M., Jonker, H. J. J. & Hanjalić, K. 2008 Wind and boundary layers in Rayleigh–Bénard convection. I. Analysis and modelling. Phys. Rev. E 77, 036311.
Scranton, D. R. & Lindberg, W. R. 1983 An experimental study of entraining, stressdriven, stratified flow in an annulus. Phys. Fluids 26, 11981205.
Stull, 1998 An Introduction to Boundary Layer Meteorology. Kluwer.
Sullivan, P. P., Moeng, C.-H., Stevens, B., Lenschow, D. H. & Major, S. D. 1998 Structure of the entrainment zone capping the convective atmospheric boundary layer. J. Atmos. Sci. 55, 30423064.
Tennekes, H. 1973 A model for the dynamics of the inversion above a convective boundary layer. J. Atmos. Sci. 30 (4), 558566.
Thompson, R. O. R. Y. 1979 A re-examination of the entrainment process in some laboratory flows. Dyn. Atmos. Oceans 4, 4555.
Thorpe, S. A. 2005 The Turbulent Ocean. Cambridge University Press.
Troen, I. & Mahrt, L. 1986 A simple model of the atmospheric boundary layer; sensitivity to surface evapouration. Boundary-Layer Meteorol. 37, 129148.
Verstappen, R. W. C. P. & Veldman, A. E. P. 2003 Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187 (1), 343368.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

On the scaling of shear-driven entrainment: a DNS study

  • Harm J. J. Jonker (a1), Maarten van Reeuwijk (a2), Peter P. Sullivan (a3) and Edward G. Patton (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed