Stratified flows with vertical layering of density: experimental and theoretical study of flow configurations and their stability

2011 ◽  
Vol 690 ◽  
pp. 571-606 ◽  
Author(s):  
Roberto Camassa ◽  
Richard M. McLaughlin ◽  
Matthew N. J. Moore ◽  
Kuai Yu
Keyword(s):  
Moving Boundary ◽  
Two Dimensions ◽  
Optical Tracking ◽  
Effective Length ◽  
Unstable Flow ◽  
Amplification Rate ◽  
Flow Configurations ◽  
The Stability ◽  
Horizontal Density Gradient ◽  
Stratified Viscous Fluid

AbstractA vertically moving boundary in a stratified fluid can create and maintain a horizontal density gradient, or vertical layering of density, through the mechanism of viscous entrainment. Experiments to study the evolution and stability of axisymmetric flows with vertically layered density are performed by towing a narrow fibre upwards through a stably stratified viscous fluid. The fibre forms a closed loop and thus its effective length is infinite. A layer of denser fluid is entrained and its thickness is measured by implementing tracking analysis of dyed fluid images. Thickness values of up to 70 times that of the fibre are routinely obtained. A lubrication model is developed for both a two-dimensional geometry and the axisymmetric geometry of the experiment, and shown to be in excellent agreement with dynamic experimental measurements once subtleties of the optical tracking are addressed. Linear stability analysis is performed on a family of exact shear solutions, using both asymptotic and numerical methods in both two dimensions and the axisymmetric geometry of the experiment. It is found analytically that the stability properties of the flow depend strongly on the size of the layer of heavy fluid surrounding the moving boundary, and that the flow is neutrally stable to perturbations in the large-wavelength limit. At the first correction of this limit, a critical layer size is identified that separates stable from unstable flow configurations. Surprisingly, in all of the experiments the size of the entrained layer exceeds the threshold for instability, yet no unstable behaviour is observed. This is a reflection of the small amplification rate of the instability, which leads to growth times much longer than the duration of the experiment. This observation illustrates that for finite times the hydrodynamic stability of a flow does not necessarily correspond to whether or not that flow can be realised from an initial-value problem. Similar instabilities that are neutral to leading order with respect to long waves can arise under the different physical mechanism of viscous stratification, as studied by Yih (J. Fluid Mech., vol. 27, 1967, pp. 337–352), and we draw a comparison to that scenario.

scholarly journals Levitation of non-magnetizable droplet inside ferrofluid

2018 ◽  
Vol 857 ◽  
pp. 398-448 ◽  
Author(s):  
Chamkor Singh ◽  
Arup K. Das ◽  
Prasanta K. Das
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Viscosity Ratio ◽  
Temporal Dynamics ◽  
Liquid Droplet ◽  
Droplet Surface ◽  
Two Dimensions ◽  
Projection Algorithm ◽  
The Stability ◽  
The Individual

The central theme of this work is that a stable levitation of a denser non-magnetizable liquid droplet, against gravity, inside a relatively lighter ferrofluid – a system barely considered in ferrohydrodynamics – is possible, and exhibits unique interfacial features; the stability of the levitation trajectory, however, is subject to an appropriate magnetic field modulation. We explore the shapes and the temporal dynamics of a plane non-magnetizable droplet levitating inside a ferrofluid against gravity due to a spatially complex, but systematically generated, magnetic field in two dimensions. The coupled set of Maxwell’s magnetostatic equations and the flow dynamic equations is integrated computationally, utilizing a conservative finite-volume-based second-order pressure projection algorithm combined with the front-tracking algorithm for the advection of the interface of the droplet. The dynamics of the droplet is studied under both the constant ferrofluid magnetic permeability assumption as well as for more realistic field-dependent permeability described by Langevin’s nonlinear magnetization model. Due to the non-homogeneous nature of the magnetic field, unique shapes of the droplet during its levitation, and at its steady state, are realized. The complete spatio-temporal response of the droplet is a function of the Laplace number $La$ , the magnetic Laplace number $La_{m}$ and the Galilei number $Ga$ ; through detailed simulations we separate out the individual roles played by these non-dimensional parameters. The effect of the viscosity ratio, the stability of the levitation path and the possibility of existence of multiple stable equilibrium states is investigated. We find, for certain conditions on the viscosity ratio, that there can be developments of cusps and singularities at the droplet surface; we also observe this phenomenon experimentally and compare with the simulations. Our simulations closely replicate the singular projection on the surface of the levitating droplet. Finally, we present a dynamical model for the vertical trajectory of the droplet. This model reveals a condition for the onset of levitation and the relation for the equilibrium levitation height. The linearization of the model around the steady state captures that the nature of the equilibrium point goes under a transition from being a spiral to a node depending upon the control parameters, which essentially means that the temporal route to the equilibrium can be either monotonic or undulating. The analytical model for the droplet trajectory is in close agreement with the detailed simulations.

scholarly journals Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation

2019 ◽  
Vol 24 (4) ◽  
pp. 101
Author(s):  
A. Karami ◽  
Saeid Abbasbandy ◽  
E. Shivanian
Keyword(s):  
Free Boundary ◽  
Least Squares ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Moving Least Squares ◽  
Heaviside Step Function ◽  
Mlpg Method ◽  
Mls Approximation

In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

scholarly journals Nonlinear higher-order spectral solution for a two-dimensional moving load on ice

2009 ◽  
Vol 621 ◽  
pp. 215-242 ◽  
Author(s):  
FÉLICIEN BONNEFOY ◽  
MICHAEL H. MEYLAN ◽  
PIERRE FERRANT
Keyword(s):  
Critical Speed ◽  
Moving Boundary ◽  
Moving Load ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Higher Order ◽  
Two Dimensions ◽  
Linear Solution ◽  
Nonlinear Solution ◽  
Bernoulli’S Equation

We calculate the nonlinear response of an infinite ice sheet to a moving load in the time domain in two dimensions, using a higher-order spectral method. The nonlinearity is due to the moving boundary, as well as the nonlinear term in Bernoulli's equation and the elastic plate equation. We compare the nonlinear solution with the linear solution and with the nonlinear solution found by Parau & Dias (J. Fluid Mech., vol. 460, 2002, pp. 281–305). We find good agreement with both solutions (with the correction of an error in the Parau & Dias 2002 results) in the appropriate regimes. We also derive a solitary wavelike expression for the linear solution – close to but below the critical speed at which the phase speed has a minimum. Our model is carefully validated and used to investigate nonlinear effects. We focus in detail on the solution at a critical speed at which the linear response is infinite, and we show that the nonlinear solution remains bounded. We also establish that the inclusion of nonlinearities leads to significant new behaviour, which is not observed in the linear solution.

scholarly journals Wind Accretion by Compact Objects: The “Flip-Flop” Instability

1992 ◽  
Vol 151 ◽  
pp. 185-194
Author(s):  
Mario Livio
Keyword(s):  
Random Walk ◽  
Accretion Rate ◽  
Compact Object ◽  
Two Dimensions ◽  
Compact Objects ◽  
Numerical Calculations ◽  
Three Dimensions ◽  
Flip Flop ◽  
Accretion Flows ◽  
The Stability

The problem of the stability of wind accretion onto compact objects is examined. Recent analytical and numerical calculations show that in two dimensions, Bondi-Hoyle accretion flows are unstable to a “flip-flop” instability. The instability can manifest itself as bursts in the accretion rate and as a random walk-type spin-up, spin-down behaviour of the accreting compact object. The nature of the flow in three dimensions needs further clarification. Possible observational implications are reviewed.

LIQUID FOAM DRAINAGE: AN OVERVIEW

2008 ◽  
Vol 22 (15) ◽  
pp. 2333-2354 ◽  
Author(s):  
QICHENG SUN ◽  
LIANGHUI TAN ◽  
GUANGQIAN WANG
Keyword(s):  
Surfactant Solution ◽  
Two Dimensions ◽  
Limiting Factors ◽  
Liquid Foams ◽  
Measuring Techniques ◽  
Substantial Progress ◽  
The Stability ◽  
Liquid Foam ◽  
Concentrated Dispersions ◽  
Foam Drainage

Liquid foams are concentrated dispersions of gas bubbles in a small amount of surfactant solution, which are perpetually out of equilibrium systems. The process of liquid draining through networks of Plateau borders in a fresh foam is so-called foam drainage, as a result of both gravitational and capillary forces, which has great effect on the stability of foams. From the view of foam physics and dynamics, this paper briefly introduces foam structure and major lifetime limiting factors of foam. The substantial progress on the theory of drainage, measuring techniques for liquid fractions, drainage in both one dimension and two dimensions, and drainage in microgravity circumstances are overviewed throughout. Remaining tasks are discussed and a multiscale methodology for foam drainage is proposed for future investigations.

The Stability of a Radial Wall Jet

1977 ◽  
Vol 28 (4) ◽  
pp. 247-258 ◽  
Author(s):  
Yutaka Tsuji ◽  
Yoshinobu Morikawa ◽  
Masaaki Sakou
Keyword(s):  
Linear Stability ◽  
Water Flow ◽  
Wall Jet ◽  
Two Dimensional ◽  
Radial Wall ◽  
Linear Region ◽  
Stability Calculation ◽  
Amplification Rate ◽  
Vortex Patterns ◽  
The Stability

SummaryMeasured stability characteristics in a radial wall jet were compared with calculated results for a two-dimensional wall jet. It was found that the stability of the radial wall jet is similar in many respects to that of the two-dimensional wall jet. An exception is that the local amplification rate of the disturbance velocity is much higher than in the two-dimensional case. It was also found that quarter-harmonics appear in the non-linear region, as well as half-harmonics, and that their amplitude distributions show profiles similar to that of the fundamental component. Further, vortex patterns were visualised in water flow, and results corresponding to measurements in air flow and to the linear stability calculation were obtained.

Numerical Analysis of Unstable Flow in Last-Stage Blades of Steam Turbines

2006 ◽  
Author(s):  
J. C. Garci´a ◽  
J. Kubiak ◽  
F. Sierra ◽  
G. Gonza´lez ◽  
G. Urquiza
Keyword(s):  
Steady State ◽  
Pressure Field ◽  
Stokes Equations ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Steam Turbines ◽  
Unstable Flow ◽  
Geometry Model ◽  
Last Stage ◽  
Exit Edge

As well known steam turbines are strongly affected because of vibrations. Unstable vibrations can appear together with steady-state vibrations. We present the results of numerical computations about unstable flow and its interaction on blades of steam turbines, which can lead to unstable modes of vibration. Unstable phenomena appear as a result of interaction of blades with the stream of steam flow where the pressure field provides the force. The analysis centers particularly in the last stage or L-0 of a 110 MW turbine. Navier-Stokes equations are resolved in two dimensions using a commercial program called Fluent based on finite-volume method. A 2-D geometry model was built in order to represent the dimensional aspects of the diaphragm as well as the rotor located in the last stage of the turbine. Periodic boundary conditions were applied to both sides of the blade with the purpose of simplifying the computation avoiding resolve for the whole wheel. The computations were conducted in both modes, steady state and time dependent. The results show the distribution of pressure fields as a function of the distance to the exit edge of the diaphragm blades. Also, the pressure and velocity fields are shown through contours along the flow channel between the diaphragm blades. The paper includes the time-dependence behavior of pressure field. A Fourier analysis is used to determine the characteristic frequencies of the system, based on numerical results.

The stability and transition of an axisymmetric wake

1966 ◽  
Vol 26 (2) ◽  
pp. 237-253 ◽  
Author(s):  
Hiroshi Sato ◽  
Osami Okada
Keyword(s):  
Stability Equation ◽  
Velocity Fluctuations ◽  
Closed Loops ◽  
Artificial Disturbance ◽  
Linearized Stability ◽  
Amplification Rate ◽  
Theoretical Investigations ◽  
High Reynolds ◽  
The Stability ◽  
Good Agreement

Experimental and theoretical investigations were made of the instability and transition of the wake behind an axisymmetric slender body at high Reynolds number. The sound from a loudspeaker was used as an artificial disturbance. The velocity fluctuations induced by the sound are selectively amplified depending on the frequency. The wavelength, the phase velocity and the amplification rate of the velocity fluctuations were measured. A linearized stability equation of the disturbance superposed on the axisymmetric wake was solved numerically for both neutral and amplified disturbances. Theoretical results on eigenvalues and eigenfunctions of the stability equation show good agreement with experimental results.Measurements on phase distributions in streamwise and azimuthal directions indicate that the line of same phase forms a helix and not a series of discrete closed loops.

Inversion of inductive electromagnetic data in highly conductive terrains

Geophysics ◽  
2005 ◽  
Vol 70 (1) ◽  
pp. G16-G28 ◽  
Author(s):  
G. Schultz ◽  
C. Ruppel
Keyword(s):  
A Priori ◽  
Two Dimensions ◽  
Forward Model ◽  
Practical Applications ◽  
Shallow Subsurface ◽  
Subsurface Structures ◽  
Conductivity Structure ◽  
Actual Field ◽  
Priori Information ◽  
The Stability

Despite the increasing use of controlled-source frequency-domain EM data to characterize shallow subsurface structures, relatively few inversion algorithms have been widely applied to data from real-world settings, particularly in high-conductivity terrains. In this study, we develop robust and convergent regularized, least-squares inversion algorithms based on both linear and nonlinear formulations of mutual dipole induction for the forward problem. A modified version of the discrepancy principle based on a priori information is implemented to select optimal smoothing parameters that simultaneously guarantee the stability and best-fit criteria. To investigate the problems of resolution and equivalence, we consider typical layered-earth models in one and two dimensions using both synthetic and observed data. Synthetic examples show that inversions based on the nonlinear forward model more accurately resolve subsurface structure, and that inversions based on the linear forward model tend to drastically underpredict high conductivities at depth. Inversions of actual field data from well-characterized sites (e.g., National Geotechnical Experimentation Site; sand-dominated coastal aquifer in the Georgia Bight) are used to test the applicability of the model to terrains with different characteristic conductivity structure. A comparison of our inversion results with existing cone-penetrometer and downhole-conductivity data from these field sites demonstrates the ability of the inversions to constrain conductivity variations in practical applications.

Sign in / Sign up

Export Citation Format

Share Document