scholarly journals On the non-parallel instability of the rotating-sphere boundary layer

2017 ◽  
Vol 818 ◽  
pp. 288-318 ◽  
Author(s):  
Antonio Segalini ◽  
Stephen J. Garrett

We present a new solution for the steady boundary-layer flow over the rotating sphere that also accounts for the eruption of the boundary layer at the equator and other higher-order viscous effects. Non-parallel corrections to the local type I and type II convective instability modes of this flow are also computed as a function of spin rate. Our instability results are associated with the previously observed spiral vortices and remarkable agreement between our predictions of the number of vortices and experimental observations is found. Vortices travelling at 70 %–80 % of the local surface speed are found to be the most amplified for sufficient spin rates, also consistent with prior experimental observations.

Author(s):  
Dip Mukherjee ◽  
Bikash Sahoo

The Bödewadt boundary-layer flow is induced by the rotation of a viscous fluid rotating with a constant angular velocity over a stationary disk. In this paper, the Bödewadt boundary-layer flow has been studied in the presence of the Coriolis force to observe the effect of radial stretch of the lower disk on the flow. For the first time in the literature, a numerical investigation of the effects of both stretching mechanism and the Coriolis force on the flow behaviour and on the convective instability characteristics of the above flow has been carried out. In this paper, the Kármán similarity transformations have been considered in order to convert the system of PDEs representing the momentum equations of the flow into a system of highly non-linear coupled ODEs and solved numerically to obtain the velocity profiles of the Bödewadt flow. Then, a convective instability analysis has been performed by using the Chebyshev collocation method in order to obtain the neutral curves. From the neutral curves it is observed that radial stretch has a globally stabilising effect on both the inviscid Type-I and the viscous Type-II instability modes. This underlying physical phenomena has been verified by performing an energy analysis of the flow. The results obtained excellently supports the previous works and will be prominently treated as a benchmark for our future studies.


Author(s):  
A. Samad ◽  
S. J. Garrett

The continuous development of spinning projectiles and other industrial applications has led to the need to understand the laminar boundary-layer flow and subsequent onset of transition over the general family of rotating spheroids. We begin by finding the laminar boundary-layer flow over a general spheroid. In particular, we distinguish between prolate and oblate spheroids and use an appropriate spheroidal coordinate system in each case. The laminar-flow equations are established for each family of spheroid rotating in otherwise still fluid. An eccentricity parameter e is used to distinguish particular bodies within the oblate or prolate families. In each case, setting e = 0 reduces the equations to those already established by Howarth [2] and Banks [4] for the rotating sphere. In this preliminary study the laminar-flow equations at each latitude are solved by extending the original series solutions due to Howarth and Banks for the rotating sphere. The laminar flows obtained are consistent with established results for the rotating sphere as e tends to zero, and tend to the von Ka´rma´n [5] solution for the rotating disk as the latitude is reduced close to the nose. Analyses of the convective instability are performed on the rotating prolate family. These extend the linear analyses previously published by Malik, Lingwood and Garrett & Peake [6–10] on related geometries. An investigation into the relative importance of type I (crossflow) and type II (streamline curvature) modes is also presented. At low latitudes increasing eccentricity has negligible effects on the stability characteristics of the flow. However as the latitude increases, eccentricity is seen to lower the upper (type I) branch of the neutral curve, reducing the region of instability.


1962 ◽  
Vol 13 (4) ◽  
pp. 519-528 ◽  
Author(s):  
D. L. Turcotte ◽  
J. M. Lyons

It is the purpose of this paper to solve a boundary-value problem posed by induction electromagnetic pumps and generators. Solutions are obtained by an expansion technique and a momentum method for the laminar, incompressible flow problem. For large values of the interaction parameter (μ2σH20λ/ρμe viscous effects are shown to be restricted to periodic boundary layers. In regions of high-field strength a local Hartmann solution is valid. Where the applied field is weak an inertial boundary layer is present which thickens in the upstream direction.A logical explanation of this phenomenon is given. The condition that a boundary-layer type flow exist is obtained and is shown to be in general satisfied. The results show that inviscid theory may be used to calculate the overall performance of electromagnetic pumps and generators while the boundary-layer theory developed here may be used to obtain the wall shear stress.


1985 ◽  
Vol 107 (1) ◽  
pp. 97-104 ◽  
Author(s):  
M. A. I. El-Shaarawi ◽  
M. F. El-Refaie ◽  
S. A. El-Bedeawi

A finite-difference scheme is developed for solving the boundary layer equations governing the laminar flow about a rotating sphere which is subjected to a uniform stream in the direction of the axis of rotation. Numerical results are presented for the meridional and azimuthal velocities and for the wall-shear-stress components. Also, the angle at which the meridional velocity gradient normal to the wall vanishes is given at values of the parameter Ta/Re2 ranged from zero (the stationary sphere case) to 10000. As compared with the momentum integral technique of Schlichting [8], the present scheme succeeded in obtaining solutions for very considerably larger values of the parameter Ta/Re2.


2006 ◽  
Vol 11 (3) ◽  
pp. 227-245 ◽  
Author(s):  
O. Anwar Bég ◽  
H. S. Takhar ◽  
G. Nath ◽  
A. J. Chamkha

The convective heat transfer on a rotating sphere in the presence of magnetic field, buoyancy forces and impulsive motion is examined theoretically and numerically in this paper. We apply a boundary layer model comprising the balance equations for x and y direction translational momentum and heat transfer, and solve these coupled non-linear partial differential equations using Blottner’s finite-difference method [1]. The numerical solutions are benchmarked with the earlier study by Lee [2] on laminar boundary layer flow over rotating bodies in forced flow and found to be in excellent agreement. The effects of magnetic field, buoyancy parameter, Prandtl number and thermal conductivity parameter on translational velocities and temperature and other variables (shear stress etc) are presented graphically and discussed at length. The problem finds applications in chemical engineering technologies, aerodynamics and planetary astrophysics.


2018 ◽  
Author(s):  
Jutta Vüllers ◽  
Georg J. Mayr ◽  
Ulrich Corsmeier ◽  
Christoph Kottmeier

Abstract. This paper investigates frequently occurring foehn in the Dead Sea valley. For the first time, sophisticated, high- resolution measurements were performed to investigate the horizontal and vertical flow field. In up to 72 % of the days in summer, foehn was observed at the eastern slope of the Judean Mountains around sunset. Furthermore, the results also revealed that in approximately 10 % of the cases the foehn detached from the slope and only effected elevated layers of the valley atmosphere. Lidar measurements showed that there are two main types of foehn. Type I has a duration of approximately 2–3 h and mean maximum velocities of around 5 m s−1 and does not propagate far into the valley, whereas type II affects the whole valley, as it propagates across the valley to the eastern side. Type II reaches mean maximum wind velocities of 11 m s−1 and has a duration of about 4–5 h. A case study of a type II foehn shows that foehn is initiated by the horizontal temperature gradient across the mountain range. In the investigated case this was caused by an amplified heating and delayed cooling of the valley boundary layer in the afternoon, compared to the upstream boundary layer over the mountain ridge. The foehn was further intensified by the advection of cool maritime air masses upstream over the coastal plains leading to a transition of subcritical to supercritical flow conditions downstream and the formation of a hydraulic jump and rotor beneath. These foehn events are of particular importance for the local climatic conditions, as they modify the temperature and humidity fields in the valley and, furthermore, they are important because they enhance evaporation from the Dead Sea and influence the aerosol distribution in the valley.


Sign in / Sign up

Export Citation Format

Share Document