Skip to main content Accessibility help
×
Home

Stability of the Prandtl model for katabatic slope flows

  • Cheng-Nian Xiao (a1) and Inanc Senocak (a1)

Abstract

We investigate the stability of the Prandtl model for katabatic slope flows using both linear stability theory and direct numerical simulations. Starting from Prandtl’s analytical solution for uniformly cooled laminar slope flows, we use linear stability theory to identify the onset of instability and features of the most unstable modes. Our results show that the Prandtl model for parallel katabatic slope flows is prone to transverse and longitudinal modes of instability. The transverse mode of instability manifests itself as stationary vortical flow structures aligned in the along-slope direction, whereas the longitudinal mode of instability emerges as waves propagating in the base-flow direction. Beyond the stability limits, these two modes of instability coexist and form a complex flow structure crisscrossing the plane of flow. The emergence of a particular form of these instabilities depends strongly on three dimensionless parameters, which are the slope angle, the Prandtl number and a newly introduced stratification perturbation parameter, which is proportional to the relative importance of the disturbance to the background stratification due to the imposed surface buoyancy flux. We demonstrate that when this parameter is sufficiently large, then the stabilising effect of the background stratification can be overcome. For shallow slopes, the transverse mode of instability emerges despite meeting the Miles–Howard stability criterion of $Ri>0.25$ . At steep slope angles, slope flow can remain linearly stable despite attaining Richardson numbers as low as $3\times 10^{-3}$ .

Copyright

Corresponding author

Email address for correspondence: senocak@pitt.edu

References

Hide All
Candelier, J., Le Dizès, S. & Millet, C. 2011 Shear instability in a stratified fluid when shear and stratification are not aligned. J. Fluid Mech. 685, 191201.10.1017/jfm.2011.306
Chen, J., Bai, Y. & Le Dizès, S. 2016 Instability of a boundary layer flow on a vertical wall in a stably stratified fluid. J. Fluid Mech. 795, 262277.10.1017/jfm.2016.202
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.10.1017/CBO9780511616938
Facchini, G., Favier, B., Le Gal, P., Wang, M. & Le Bars, M. 2018 The linear instability of the stratified plane Couette flow. J. Fluid Mech. 853, 205234.10.1017/jfm.2018.556
Fedorovich, E. & Shapiro, A. 2009 Structure of numerically simulated katabatic and anabatic flows along steep slopes. Acta Geophys. 57 (4), 9811010.10.2478/s11600-009-0027-4
Görtler, H. 1959 Über eine Analogie zwischen den Instabilitäten laminarer Grenzschichtströmungen an konkaven Wänden und an erwärmten Wänden. Ing.-Arch. 28 (1), 7178.10.1007/BF00536101
Jacobsen, D. A. & Senocak, I. 2013 Multi-level parallelism for incompressible flow computations on GPU clusters. Parallel Comput. 39 (1), 120.10.1016/j.parco.2012.10.002
Kaminski, A. K., Caulfield, C. P. & Taylor, J. R. 2017 Nonlinear evolution of linear optimal perturbations of strongly stratified shear layers. J. Fluid Mech. 825, 213244.10.1017/jfm.2017.396
Kolář, V. 2007 Vortex identification: new requirements and limitations. Intl J. Heat Fluid Flow 28 (4), 638652.10.1016/j.ijheatfluidflow.2007.03.004
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10 (4), 496508.10.1017/S0022112061000305
Miller, J. R. & Gage, K. S. 1972 Prandtl number dependence of the stability of a stratified free shear layer. Phys. Fluids 15 (5), 723725.10.1063/1.1693973
Prandtl, L. 1942 Führer durch die Strömungslehre. Vieweg und Sohn.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.10.1007/978-1-4613-0185-1
Shapiro, A. & Fedorovich, E. 2004 Unsteady convectively driven flow along a vertical plate immersed in a stably stratified fluid. J. Fluid Mech. 498, 333352.10.1017/S0022112003006803
Shapiro, A. & Fedorovich, E. 2014 A boundary-layer scaling for turbulent katabatic flow. Boundary-Layer Meteorol. 153 (1), 117.10.1007/s10546-014-9933-3
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223 (605–615), 289343.10.1098/rsta.1923.0008
Umphrey, C., DeLeon, R. & Senocak, I. 2017 Direct numerical simulation of turbulent katabatic slope flows with an immersed boundary method. Boundary-Layer Meteorol. 164 (3), 367382.10.1007/s10546-017-0252-3
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

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