Settling-driven instability in two-component stably stratified Hele-Shaw flows

2018 ◽  
Vol 843 ◽  
Author(s):  
Rafael M. Oliveira ◽  
Eckart Meiburg
Keyword(s):  
Stability Problem ◽  
Settling Velocity ◽  
Critical Value ◽  
Neutral Stability ◽  
Instability Mode ◽  
Two Component ◽  
Linear Stability Problem ◽  
Constant Gradient ◽  
Double Diffusive ◽  
Hele Shaw Cell

We investigate the onset of instability in a stably stratified two-component fluid in a vertical Hele-Shaw cell when the unstably stratified scalar has a settling velocity. This linear stability problem is analysed on the basis of Darcy’s law, for constant-gradient base states. The settling velocity is found to trigger a novel instability mode characterized by pairs of inclined waves. For unequal diffusivities, this new settling-driven mode competes with the traditional double-diffusive mode. Below a critical value of the settling velocity, the double-diffusive elevator mode dominates, while, above this threshold, the inclined waves associated with the settling-driven instability exhibit faster growth. The analysis yields neutral stability curves and allows for the discussion of various asymptotic limits.

scholarly journals A settling-driven instability in two-component, stably stratified fluids

2017 ◽  
Vol 816 ◽  
pp. 243-267 ◽  
Author(s):  
A. Alsinan ◽  
E. Meiburg ◽  
P. Garaud
Keyword(s):  
Linear Stability ◽  
Settling Velocity ◽  
Scalar Fields ◽  
State Density ◽  
Stratified Fluids ◽  
Instability Mechanism ◽  
Two Component ◽  
Constant Gradient ◽  
Stability Results ◽  
Double Diffusive

We analyse the linear stability of stably stratified fluids whose density depends on two scalar fields where one of the scalar fields is unstably stratified and involves a settling velocity. Such conditions may be found, for example, in flows involving the transport of sediment in addition to heat or salt. A linear stability analysis for constant-gradient base states demonstrates that the settling velocity generates a phase shift between the perturbation fields of the two scalars, which gives rise to a novel, settling-driven instability mode. This instability mechanism favours the growth of waves that are inclined with respect to the horizontal. It is active for all density and diffusivity ratios, including for cases in which the two scalars diffuse at identical rates. If the scalars have unequal diffusivities, it competes with the elevator mode waves of the classical double-diffusive instability. We present detailed linear stability results as a function of the governing dimensionless parameters, including for lateral gradients of the base state density fields that result in predominantly horizontal intrusion instabilities. Highly resolved direct numerical simulation results serve to illustrate the nonlinear competition of the various instabilities for such flows in different parameter regimes.

On the stability of the asymptotic suction boundary-layer profile

1965 ◽  
Vol 23 (4) ◽  
pp. 715-735 ◽  
Author(s):  
T. H. Hughes ◽  
W. H. Reid
Keyword(s):  
Exact Solution ◽  
Stability Problem ◽  
Adjoint Problem ◽  
Analytical Continuation ◽  
Neutral Stability ◽  
The Stability ◽  
Asymptotic Suction ◽  
Linear Stability Problem ◽  
Original Treatment ◽  
General Method

This paper presents a discussion of some aspects of the linear stability problem for the asymptotic suction profile. An exact solution of the inviscid equation is first obtained in terms of the usual hypergeometric function and its analytical continuation. This exact solution provides both a corrected version of an earlier treatment by Freeman and an independent check on the more general method suggested for solving the inviscid equation numerically. Various approximations to the characteristic equation, and hence to the curve of neutral stability, are then considered. In particular, it is found that, in a consistent asymptotic treatment of the related adjoint problem, at least one viscous correction to the singular inviscid solution must be considered. Based on the present results for the adjoint problem, it is suggested that Tollmien's original treatment of the viscous corrections must be slightly modified.

scholarly journals Thermohaline layering in dynamically and diffusively stable shear flows

2016 ◽  
Vol 805 ◽  
pp. 147-170 ◽  
Author(s):  
Timour Radko
Keyword(s):  
Shear Flows ◽  
World Ocean ◽  
Critical Value ◽  
Dissipative Systems ◽  
Widespread Occurrence ◽  
Salty Water ◽  
Two Component ◽  
Stratified Ocean ◽  
Double Diffusive ◽  
Horizontal Layers

In this study we examine two-component shear flows that are stable with respect to Kelvin–Helmholtz and to double-diffusive instabilities individually. Our focus is on diffusively stratified ocean regions, where relatively warm and salty water masses are located below cool fresh ones. It is shown that such systems may be destabilized by the interplay between shear and thermohaline effects, caused by unequal molecular diffusivities of density components. Linear stability analysis suggests that parallel two-component flows can be unstable for Richardson numbers exceeding the critical value for non-dissipative systems $(Ri=1/4)$ by up to four orders of magnitude. Direct numerical simulations indicate that these instabilities transform the initially linear density stratification into a series of well-defined horizontal layers. It is hypothesized that the combined thermohaline–shear instabilities could be ultimately responsible for the widespread occurrence of thermohaline staircases in diffusively stable regions of the World Ocean.

Finite amplitude cellular convection

1958 ◽  
Vol 4 (3) ◽  
pp. 225-260 ◽  
Author(s):  
W. V. R. Malkus ◽  
G. Veronis
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Heat Transport ◽  
Stability Problem ◽  
Dominant Role ◽  
Linear Equations ◽  
Finite Amplitude ◽  
Set Up ◽  
Rayleigh Numbers ◽  
Linear Stability Problem

When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.

Sediment-laden fresh water above salt water: linear stability analysis

2011 ◽  
Vol 691 ◽  
pp. 279-314 ◽  
Author(s):  
P. Burns ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
Fresh Water ◽  
Settling Velocity ◽  
Interface Thickness ◽  
Particle Settling ◽  
Salinity Field ◽  
Unstable Layer ◽  
Stability Results ◽  
Double Diffusive

AbstractWhen a layer of particle-laden fresh water is placed above clear, saline water, both Rayleigh–Taylor and double diffusive fingering instabilities may arise. For quasi-steady base profiles, we obtain linear stability results for such situations by means of a rational spectral approximation method with adaptively chosen grid points, which is able to resolve multiple steep gradients in the base state density profile. In the absence of salinity and for a step-like concentration profile, the dominant parameter is the ratio of the particle settling velocity to the viscous velocity scale. As long as this ratio is small, particle settling has a negligible influence on the instability growth. However, when the particles settle more rapidly than the instability grows, the growth rate decreases inversely proportional to the settling velocity. This damping effect is a result of the smearing of the vorticity field, which in turn is caused by the deposition of vorticity onto the fluid elements passing through the interface between clear and particle-laden fluid. In the presence of a stably stratified salinity field, this picture changes dramatically. An important new parameter is the ratio of the particle settling velocity to the diffusive spreading velocity of the salinity, or alternatively the ratio of the unstable layer thickness to the diffusive interface thickness of the salinity profile. As long as this quantity does not exceed unity, the instability of the system and the most amplified wavenumber are primarily determined by double diffusive effects. In contrast to situations without salinity, particle settling can have a destabilizing effect and significantly increase the growth rate. Scaling laws obtained from the linear stability results are seen to be largely consistent with earlier experimental observations and theoretical arguments put forward by other authors. For unstable layer thicknesses much larger than the salinity interface thickness, the particle and salinity interfaces become increasingly decoupled, and the dominant instability mode becomes Rayleigh–Taylor-like, centred at the lower boundary of the particle-laden flow region.

The Onset of Double-Diffusive Convection in a Two-Layer System With a Viscoelastic Fluid-Saturated Porous Medium Under High-Frequency Vibration

2020 ◽  
Vol 143 (1) ◽  
Author(s):  
Moli Zhao ◽  
Huan Zhao ◽  
Shaowei Wang ◽  
Chen Yin
Keyword(s):  
Viscoelastic Fluid ◽  
High Frequency ◽  
Physical Parameters ◽  
Neutral Stability ◽  
Double Diffusive Convection ◽  
Layer System ◽  
High Frequency Vibration ◽  
Diffusive Convection ◽  
Wave Numbers ◽  
Double Diffusive

Abstract The effect of high frequency vibration in the gravity field on the double-diffusive convection in a two-layer system with a viscoelastic fluid-saturated porous layer is studied. The averaging method is employed to split the unknown functions into a periodic rapidly varying part and a slower mean part. Then, the governing equation of perturbations is numerically solved by the Chebyshev tau method and QZ decomposition method. The influence of physical parameters on the stability of system is investigated. It is shown that the neutral stability curves are bimodal under high frequency vibration. The parameter of the high frequency vibration mainly stabilizes the pure fluid layer for greater wave numbers and has a weak impact on the whole system for smaller wave numbers.

Stability of columnar convection in a porous medium

2013 ◽  
Vol 737 ◽  
pp. 205-231 ◽  
Author(s):  
Duncan R. Hewitt ◽  
Jerome A. Neufeld ◽  
John R. Lister
Keyword(s):  
Porous Medium ◽  
Linear Stability ◽  
Stability Problem ◽  
Columnar Structure ◽  
Marginal Stability ◽  
Vertically Propagating ◽  
The Stability ◽  
Rayleigh Benard ◽  
Linear Stability Problem ◽  
Dimensional Domain

AbstractConvection in a porous medium at high Rayleigh number $\mathit{Ra}$ exhibits a striking quasisteady columnar structure with a well-defined and $\mathit{Ra}$-dependent horizontal scale. The mechanism that controls this scale is not currently understood. Motivated by this problem, the stability of a density-driven ‘heat-exchanger’ flow in a porous medium is investigated. The dimensionless flow comprises interleaving columns of horizontal wavenumber $k$ and amplitude $\widehat{A}$ that are driven by a steady balance between vertical advection of a background linear density stratification and horizontal diffusion between the columns. Stability is governed by the parameter $A= \widehat{A}\mathit{Ra}/ k$. A Floquet analysis of the linear-stability problem in an unbounded two-dimensional domain shows that the flow is always unstable, and that the marginal-stability curve is independent of $A$. The growth rate of the most unstable mode scales with ${A}^{4/ 9} $ for $A\gg 1$, and the corresponding perturbation takes the form of vertically propagating pulses on the background columns. The physical mechanism behind the instability is investigated by an asymptotic analysis of the linear-stability problem. Direct numerical simulations show that nonlinear evolution of the instability ultimately results in a reduction of the horizontal wavenumber of the background flow. The results of the stability analysis are applied to the columnar flow in a porous Rayleigh–Bénard (Rayleigh–Darcy) cell at high $\mathit{Ra}$, and a balance of the time scales for growth and propagation suggests that the flow is unstable for horizontal wavenumbers $k$ greater than $k\sim {\mathit{Ra}}^{5/ 14} $ as $\mathit{Ra}\rightarrow \infty $. This stability criterion is consistent with hitherto unexplained numerical measurements of $k$ in a Rayleigh–Darcy cell.

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