Experimental survey of linear and nonlinear inertial waves and wave instabilities in a spherical shell

2016 ◽  
Vol 789 ◽  
pp. 589-616 ◽  
Author(s):  
Michael Hoff ◽  
U. Harlander ◽  
C. Egbers
Keyword(s):  
Spherical Shell ◽  
Experimental Studies ◽  
Vertical Shear ◽  
Meridional Plane ◽  
Inertial Waves ◽  
Supercritical Regime ◽  
Laser Plane ◽  
Linear And Nonlinear ◽  
Wave Instabilities ◽  
Time Periodic

We experimentally study linear and nonlinear inertial waves in a spherical shell with a radius ratio of ${\it\eta}=1/3$. The shell rotates with a mean angular velocity ${\it\Omega}_{0}$ around its vertical axis. This rotation is overlaid by a time-periodic libration of the inner sphere in the range $0<{\it\omega}_{lib}<2{\it\Omega}_{0}$ to excite inertial waves with a defined frequency. In the first part, we investigate linear inertial waves. The influence of the libration amplitude and the libration frequency on the waves and further the efficiency of the forcing to excite linear inertial waves will be discussed. For this, qualitative data from Kalliroscope visualisation in a meridional laser plane, as well as quantitative particle image velocimetry (PIV) data in a horizontal plane, have been analysed. A simple two-dimensional ray-tracing model is applied for the meridional plane to interpret the visualisations with respect to energy focusing and wave attractors. For sufficiently high/low libration amplitudes/frequencies, the Stewartson layer, a vertical shear layer tangential to the inner sphere’s equator, becomes unstable. This so-called ‘supercritical’ regime, where centrifugal and shear instabilities occur, allows for nonlinear wave coupling. PIV analyses in the horizontal laser plane in the corotating frame show low-frequency structures that correspond to Rossby-wave instabilities of the Stewartson layer. Some of these are travelling retrograde and are trapped near the Stewartson layer, others are travelling prograde filling the whole gap outside the Stewartson layer. Since libration can be viewed as a time-periodic variation of differential rotation, we assume that these two different structures are related to either the retrograde $(Ro_{d}<0)$ or the prograde $(Ro_{d}>0)$ phase of the libration cycle. The experimental results confirm theoretical, numerical as well as other experimental studies on Stewartson-layer instabilities.

Study on the influence of linear and nonlinear distribution of crosswind on the motion of aircraft wake vortex

2021 ◽  
pp. 095441002098743
Author(s):  
Dong Li ◽  
Ziming Xu ◽  
Ke Zhang ◽  
Zeyu Zhang ◽  
Jinxin Zhou ◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Dissipation Rate ◽  
Vortex Pair ◽  
Vertical Shear ◽  
Wake Vortex ◽  
Analysis Method ◽  
Velocity Magnitude ◽  
Aircraft Wake ◽  
Linear And Nonlinear ◽  
Nonlinear Distribution

Environmental crosswind can greatly affect the development of aircraft wake vortex pair. Previous numerical simulations and experiments have shown that the nonlinear vertical shear of the crosswind velocity can affect the dissipation rate of the aircraft wake vortex, causing each vortex of the vortex pair descent with different velocity magnitude, which will lead to the asymmetrical settlement and tilt of the wake vortex pair. Through numerical simulations, this article finds that uniform crosswind convection and linear vertical shear crosswind convection can also have an effect on the strength of the vortex. This effect is inversely proportional to the cube of the vortex spacing, so it is more intense on small separation vortex pair. In addition, the superposition of crosswind and vortex-induced velocities will lead to the asymmetrical pressure distribution around the vortex pair, which will also cause the tilt of the vortex pair. Furthermore, a new analysis method for wake vortex is proposed, which can be used to predict the vortex trajectory.

scholarly journals The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Jifeng Chu ◽  
Ting Xia
Keyword(s):  
Periodic Function ◽  
Lyapunov Stability ◽  
Stability Criterion ◽  
Normal Forms ◽  
Sufficient Condition ◽  
Periodic Functions ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Time Periodic ◽  
Damped Oscillator

Leta(t),b(t)be continuousT-periodic functions with∫0Tb(t)dt=0. We establish one stability criterion for the linear damped oscillatorx′′+b(t)x′+a(t)x=0. Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillatorx′′+b(t)x′+a(t)x+c(t)x2n-1+e(t,x)=0, wheren≥2,c(t)is a continuousT-periodic function,e(t,x)is continuousT-periodic intand dominated by the powerx2nin a neighborhood ofx=0.

Inertial waves in a rotating spherical shell

1997 ◽  
Vol 341 ◽  
pp. 77-99 ◽  
Author(s):  
M. RIEUTORD ◽  
L. VALDETTARO
Keyword(s):  
Spherical Shell ◽  
Cross Section ◽  
Shear Layers ◽  
Inertial Waves ◽  
Ekman Number ◽  
Distribution Of Eigenvalues ◽  
Simple Rules ◽  
Meridional Cross Section ◽  
Zero Viscosity ◽  
The Web

The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.

Vertical shear alters chemical front speed in thin-layer flows

2019 ◽  
Vol 874 ◽  
pp. 235-262 ◽  
Author(s):  
Thomas D. Nevins ◽  
Douglas H. Kelley
Keyword(s):  
Thin Layer ◽  
Experimental Studies ◽  
Reaction Diffusion ◽  
Quantitative Agreement ◽  
Vertical Shear ◽  
Front Speed ◽  
Reaction Fronts ◽  
Diffusion Dynamics ◽  
Reactive Mixing ◽  
Front Profile

The mixing of a reactive scalar by a fluid flow can have a significant impact on reaction dynamics and the growth of reacted regions. However, experimental studies of the fluid mechanics of reactive mixing present significant challenges and puzzling results. The observed speed at which reacted regions expand can be separated into a contribution from the underlying flow and a contribution from reaction–diffusion dynamics, which we call the chemical front speed. In prior work (Nevins & Kelley, Chaos, vol. 28 (4), 2018, 043122), we were surprised to observe that the chemical front speed increased where the underlying flow in a thin layer was faster. In this paper, we show that the increase is physical and is caused by smearing of reaction fronts by vertical shear. We show that the increase occurs not only in thin-layer flows with a free surface, but also in Hele-Shaw systems. We draw these conclusions from a series of simulations in which reaction fronts are located according to depth-averaged concentration, as is common in laboratory experiments. Where the front profile is deformed by shear, the apparent front speed changes as well. We compare the simulations to new experimental results and find close quantitative agreement. We also show that changes to the apparent front speed are reduced approximately 80 % by adding a lubrication layer.

scholarly journals Artificial Bee Colony Algorithm with Time-Varying Strategy

2015 ◽  
Vol 2015 ◽  
pp. 1-17 ◽  
Author(s):  
Quande Qin ◽  
Shi Cheng ◽  
Qingyu Zhang ◽  
Li Li ◽  
Yuhui Shi
Keyword(s):  
Swarm Intelligence ◽  
Artificial Bee Colony Algorithm ◽  
Artificial Bee Colony ◽  
Experimental Studies ◽  
Population Based ◽  
Time Varying ◽  
Abc Algorithm ◽  
Bee Colony ◽  
Linear And Nonlinear ◽  
Proper Balance

Artificial bee colony (ABC) is one of the newest additions to the class of swarm intelligence. ABC algorithm has been shown to be competitive with some other population-based algorithms. However, there is still an insufficiency that ABC is good at exploration but poor at exploitation. To make a proper balance between these two conflictive factors, this paper proposed a novel ABC variant with a time-varying strategy where the ratio between the number of employed bees and the number of onlooker bees varies with time. The linear and nonlinear time-varying strategies can be incorporated into the basic ABC algorithm, yielding ABC-LTVS and ABC-NTVS algorithms, respectively. The effects of the added parameters in the two new ABC algorithms are also studied through solving some representative benchmark functions. The proposed ABC algorithm is a simple and easy modification to the structure of the basic ABC algorithm. Moreover, the proposed approach is general and can be incorporated in other ABC variants. A set of 21 benchmark functions in 30 and 50 dimensions are utilized in the experimental studies. The experimental results show the effectiveness of the proposed time-varying strategy.

Equatorial wave attractors and inertial oscillations

2007 ◽  
Vol 570 ◽  
pp. 47-67 ◽  
Author(s):  
LEO R. M. MAAS ◽  
UWE HARLANDER
Keyword(s):  
Spherical Shell ◽  
Gravity Waves ◽  
Equatorial Region ◽  
Meridional Plane ◽  
Kinetic Energy Density ◽  
Inertial Oscillations ◽  
Homogeneous Fluid ◽  
Plane Model ◽  
Equatorial Wave ◽  
Inertia Gravity Waves

Three different approximations to the axisymmetric small-disturbance dynamics of a uniformly rotating thin spherical shell are studied for the equatorial region assuming time-harmonic motion. The first is the standard β-plane model. The second is Stern's (Tellus, vol. 15, 1963, p. 246) homogeneous, equatorial β-plane model of inertial waves (that includes all Coriolis terms). The third is a version of Stern's equation extended to include uniform stratification. It is recalled that the boundary value problem (BVP) that governs the streamfunction of zonally symmetric waves in the meridional plane becomes separable only for special geometries. These separable BVPs allow us to make a connection between the streamfunction field and the underlying geometry of characteristics of the governing equation. In these cases characteristics are each seen to trace a purely periodic path. For most geometries, however, the BVP is non-separable and characteristics and therefore wave energy converge towards a limit cycle, referred to as an equatorial wave attractor. For Stern's model we compute exact solutions for wave attractor regimes. These solutions show that wave attractors correspond to singularities in the velocity field, indicating an infinite magnification of kinetic energy density along the attractor. The instability that arises occurs without the necessity of any ambient shear flow and is referred to as geometric instability.For application to ocean and atmosphere, Stern's model is extended to include uniform stratification. Owing to the stratification, characteristics are trapped near the equator by turning surfaces. Characteristics approach either equatorial wave attractors, or point attractors situated at the intersections of turning surfaces and the bottom. At these locations, trapped inertia–gravity waves are perceived as near-inertial oscillations. It is shown that trapping of inertia–gravity waves occurs for any monochromatic frequency within the allowed range, while equatorial wave attractors exist in a denumerable, infinite set of finite-sized continuous frequency intervals. It is also shown that the separable Stern equation, obtained as an approximate equation for waves in a homogeneous fluid confined to the equatorial part of a spherical shell, gives an exact description for buoyancy waves in uniformly but radially stratified fluids in such shells.

Sign in / Sign up

Export Citation Format

Share Document