skip to main content

Title: Stability of the anabatic Prandtl slope flow in a stably stratified medium
In the Prandtl model for anabatic slope flows, a uniform positive buoyancy flux at the surface drives an upslope flow against a stable background stratification. In the present study, we conduct linear stability analysis of the anabatic slope flow under this model and contrast it against the katabatic case as presented in Xiao & Senocak ( J. Fluid Mech. , vol. 865, 2019, R2). We show that the buoyancy component normal to the sloped surface is responsible for the emergence of stationary longitudinal rolls, whereas a generalised Kelvin–Helmholtz (KH) type of mechanism consisting of shear instability modulated by buoyancy results in a streamwise-travelling mode. In the anabatic case, for slope angles larger than $9^{\circ }$ to the horizontal, the travelling KH mode is dominant whereas, at lower inclination angles, the formation of the stationary vortex instability is favoured. The same dynamics holds qualitatively for the katabatic case, but the mode transition appears at slope angles of approximately $62^{\circ }$ . For a fixed slope angle and Prandtl number, we demonstrate through asymptotic analysis of linear growth rates that it is possible to devise a classification scheme that demarcates the stability of Prandtl slope flows into distinct regimes based on the dimensionless stratification perturbation number. We verify the existence of the instability modes with the help of direct numerical simulations, and observe close agreements between simulation results and predictions of linear analysis. For slope angle values in the vicinity of the junction point in the instability map, both longitudinal rolls and travelling waves coexist simultaneously and form complex flow structures.  more » « less
Award ID(s):
Author(s) / Creator(s):
Date Published:
Journal Name:
Journal of Fluid Mechanics
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Stationary longitudinal vortical rolls emerge in katabatic and anabatic Prandtl slope flows at shallow slopes as a result of an instability when the imposed surface buoyancy flux relative to the background stratification is sufficiently large. Here, we identify the self-pairing of these longitudinal rolls as a unique flow structure. The topology of the counter-rotating vortex pair bears a striking resemblance to speaker-wires and their interaction with each other is a precursor to further destabilization and breakdown of the flow field into smaller structures. On its own, a speaker-wire vortex retains its unique topology without any vortex reconnection or breakup. For a fixed slope angle $\alpha =3^{\circ }$ and at a constant Prandtl number, we analyse the saturated state of speaker-wire vortices and perform a bi-global linear stability analysis based on their stationary state. We establish the existence of both fundamental and subharmonic secondary instabilities depending on the circulation and transverse wavelength of the base state of speaker-wire vortices. The dominance of subharmonic modes relative to the fundamental mode helps to explain the relative stability of a single vortex pair compared to the vortex dynamics in the presence of two or an even number of pairs. These instability modes are essential for the bending and merging of multiple speaker-wire vortices, which break up and lead to more dynamically unstable states, eventually paving the way for transition towards turbulence. This process is demonstrated via three-dimensional flow simulations with which we are able to track the nonlinear temporal evolution of these instabilities. 
    more » « less
  2. We investigate the stability of katabatic slope flows over an infinitely wide and uniformly cooled planar surface subject to a downslope uniform ambient wind aloft. We adopt an extension of Prandtl’s original model for slope flows (Lykosov & Gutman, Izv. Acad. Sci. USSR Atmos. Ocean. Phys. , vol. 8 (8), 1972, pp. 799–809) to derive the base flow, which constitutes an interesting basic state in stability analysis because it cannot be reduced to a single universal form independent of external parameters. We apply a linear modal analysis to this basic state to demonstrate that for a fixed Prandtl number and slope angle, two independent dimensionless parameters are sufficient to describe the flow stability. One of these parameters is the stratification perturbation number that we have introduced in Xiao & Senocak ( J. Fluid Mech. , vol. 865, 2019, R2). The second parameter, which we will henceforth designate the wind forcing number, is hitherto uncharted and can be interpreted as the ratio of the kinetic energy of the ambient wind aloft to the damping due to viscosity and the stabilising effect of the background stratification. For a fixed Prandtl number, stationary transverse and travelling longitudinal modes of instabilities can emerge, depending on the value of the slope angle and the aforementioned dimensionless numbers. The influence of ambient wind forcing on the base flow’s stability is complicated, as the ambient wind can be both stabilising as well as destabilising for a certain range of the parameters. Our results constitute a strong counterevidence against the current practice of relying solely on the gradient Richardson number to describe the dynamic stability of stratified atmospheric slope flows. 
    more » « less
  3. Abstract

    Dynamically unstable katabatic Prandtl slope flows are studied via numerical simulations and spectral analysis. Results confirm the presence of aperiodic temporal and spatial oscillations in the flow fields due to the emergence and propagation of flow instabilities. Dampeden masseoscillations are observed to dominate the initial oscillatory stage of laminar katabatic slope flows. Stationary longitudinal rolls, which are dominant at shallow slopes, are observed to meander with increasing stratification perturbation parameter and the average distance between the rolls exhibits a strong dependence on slope inclination for slope angles less than . At much steeper slopes, traveling slope waves emerge and they are transported at the mean jet velocity. Both types of instability rolls coexist for certain combinations of dimensionless parameters, forming intricate structures that break into smaller eddies as the flow becomes more dynamically unstable. In the dynamically unstable nonturbulent regime,en masseoscillations are insignificant, but their normalised frequency can be used to discern the type of flow instability.

    more » « less
  4. The hydrodynamic stability of the radial Rayleigh-Benard-Poiseuille flow inside the collector of solar chimney power plants has not attracted much attention in the literature. Because the ground is heated, buoyancy driven instability is possible. In addition, viscosity driven instability may occur. Temporal and spatial simulations are employed for investigating the hydrodynamic stability of the radial flow. Towards this end a new compact finite difference Navier-Stokes code is being developed. Square channel ow simulations were performed for code validation purposes using both the new code and an existing finite-volume code. For certain Reynolds and Rayleigh number combinations transverse rolls or longitudinal rolls appear while for others the ow remains stable. A Fourier mode decomposition of the wall-normal velocity component at mid-channel height reveals linear (exponential) growth for the unstable modes and nonlinear interactions once the mode amplitudes exceed a certain threshold. The present square channel flow results are consistent with the neutral curves from the linear stability theory analysis by Gage and Reid. Simulations were also carried out for the collector of a 1:30 scale model of the Manzanares solar chimney power plant at the University of Arizona. The Rayleigh number is well above the critical Rayleigh number and steady longitudinal rolls appear shortly downstream of the collector inflow. 
    more » « less
  5. We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin–Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we refer to as an Alfvénic Dubrulle–Frisch instability, exists for any non-zero magnetic field strength as long as the magnetic Prandtl number ${{{Pm}}} < 1$ . We present a reduced model for this instability that reveals its excitation mechanism to be the negative eddy viscosity of periodic shear flows described by Dubrulle & Frisch ( Phys. Rev. A, vol. 43, 1991, pp. 5355–5364). Finally, we demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons. 
    more » « less