Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 1. Linear stability analysis

2013 ◽  
Vol 721 ◽  
pp. 268-294 ◽  
Author(s):  
L. Talon ◽  
N. Goyal ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stokes Equations ◽  
Variable Density ◽  
Local Concentration ◽  
Miscible Displacements ◽  
Instability Modes ◽  
Finger Tip

AbstractA computational investigation of variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells is presented. As a first step, two-dimensional base states are obtained by means of simulations of the Stokes equations, which are nonlinear due to the dependence of the viscosity on the local concentration. Here, the vertical position of the displacement front is seen to reach a quasisteady equilibrium value, reflecting a balance between viscous and gravitational forces. These base states allow for two instability modes: first, there is the familiar tip instability driven by the unfavourable viscosity contrast of the displacement, which is modulated by the presence of density variations in the gravitational field; second, a gravitational instability occurs at the unstably stratified horizontal interface along the side of the finger. Both of these instability modes are investigated by means of a linear stability analysis. The gravitational mode along the side of the finger is characterized by a wavelength of about one half to one full gap width. It becomes more unstable as the gravity parameter increases, even though the interface is shifted closer to the wall. The growth rate is largest far behind the finger tip, where the interface is both thicker, and located closer to the wall, than near the finger tip. The competing influences of interface thickness and wall proximity are clarified by means of a parametric stability analysis. The tip instability mode represents a gravity-modulated version of the neutrally buoyant mode. The analysis shows that in the presence of density stratification its growth rate increases, while the dominant wavelength decreases. This overall destabilizing effect of gravity is due to the additional terms appearing in the stability equations, which outweigh the stabilizing effects of gravity onto the base state.

Variable-density miscible displacements in a vertical Hele-Shaw cell: linear stability

2007 ◽  
Vol 584 ◽  
pp. 357-372 ◽  
Author(s):  
N. GOYAL ◽  
H. PICHLER ◽  
E. MEIBURG
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stokes Equations ◽  
Capillary Tube ◽  
Computational Study ◽  
Stability Limit ◽  
Dominant Wavelength ◽  
Miscible Displacements

A computational study based on the Stokes equations is conducted to investigate the effects of gravitational forces on miscible displacements in vertical Hele-Shaw cells. Nonlinear simulations provide the quasi-steady displacement fronts in the gap of the cell, whose stability to spanwise perturbations is subsequently examined by means of a linear stability analysis. The two-dimensional simulations indicate a marked thickening (thinning) and slowing down (speeding up) of the displacement front for flows stabilized (destabilized) by gravity. For the range investigated, the tip velocity is found to vary linearly with the gravity parameter. Strongly stable density stratifications lead to the emergence of flow patterns with spreading fronts, and to the emergence of a secondary needle-shaped finger, similar to earlier observations for capillary tube flows. In order to investigate the transition between viscously driven and purely gravitational instabilities, a comparison is presented between displacement flows and gravity-driven flows without net displacements.The linear stability analysis shows that both the growth rate and the dominant wavenumber depend only weakly on the Péclet number. The growth rate varies strongly with the gravity parameter, so that even a moderately stable density stratification can stabilize the displacement. Both the growth rate and the dominant wavelength increase with the viscosity ratio. For unstable density stratifications, the dominant wavelength is nearly independent of the gravity parameter, while it increases strongly for stable density stratifications. Finally, the kinematic wave theory of Lajeunesse et al. (J. Fluid Mech. vol. 398, 1999, p. 299) is seen to capture the stability limit quite accurately, while the Darcy analysis misses important aspects of the instability.

scholarly journals Random distributions of initial porosity trigger regular necking patterns at high strain rates

2018 ◽  
Vol 474 (2211) ◽  
pp. 20170575 ◽  
Author(s):  
K. E. N’souglo ◽  
A. Srivastava ◽  
S. Osovski ◽  
J. A. Rodríguez-Martínez
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
High Strain ◽  
Initial Porosity ◽  
Strain Rates ◽  
High Strain Rates ◽  
Ductile Materials ◽  
Instability Modes ◽  
Localization Of Plastic Deformation

At high strain rates, the fragmentation of expanding structures of ductile materials, in general, starts by the localization of plastic deformation in multiple necks. Two distinct mechanisms have been proposed to explain multiple necking and fragmentation process in ductile materials. One view is that the necking pattern is related to the distribution of material properties and defects. The second view is that it is due to the activation of specific instability modes of the structure. Following this, we investigate the emergence of necking patterns in porous ductile bars subjected to dynamic stretching at strain rates varying from 10 3   s −1 to 0.5×10 5   s −1 using finite-element calculations and linear stability analysis. In the calculations, the initial porosity (representative of the material defects) varies randomly along the bar. The computations revealed that, while the random distribution of initial porosity triggers the necking pattern, it barely affects the average neck spacing, especially, at higher strain rates. The average neck spacings obtained from the calculations are in close agreement with the predictions of the linear stability analysis. Our results also reveal that the necking pattern does not begin when the Considère condition is reached but is significantly delayed due to the stabilizing effect of inertia.

Linear Stability Analysis and Application of a New Solution Method of the Elastodynamic Equations Suitable for a Unified Fluid-Structure-Interaction Approach

2008 ◽  
Vol 130 (3) ◽  
Author(s):  
C. G. Giannopapa ◽  
G. Papadakis
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Solution Method ◽  
Stokes Equations ◽  
Fluid Structure Interaction ◽  
Analytic Solutions ◽  
Computational Domain ◽  
Fluid Structure ◽  
Structure Interaction

In the conventional approach for fluid-structure-interaction problems, the fluid and solid components are treated separately and information is exchanged across their interface. According to the conventional terminology, the current numerical methods can be grouped in two major categories: partitioned methods and monolithic methods. Both methods use separate sets of equations for fluid and solid that have different unknown variables. A unified solution method has been presented in the previous work of Giannopapa and Papadakis (2004, “A New Formulation for Solids Suitable for a Unified Solution Method for Fluid-Structure Interaction Problems,” ASME PVP 2004, San Diego, CA, July, PVP Vol. 491–1, pp. 111–117), which is different from these methods. The new approach treats both fluid and solid as a single continuum; thus, the whole computational domain is treated as one entity discretized on a single grid. Its behavior is described by a single set of equations, which are solved fully implicitly. In this paper, the elastodynamic equations are reformulated so that they contain the same unknowns as the Navier–Stokes equations, namely, velocities and pressure. Two time marching and one spatial discretization scheme, widely used for fluid equations, are applied for the solution of the reformulated equations for solids. Using linear stability analysis, the accuracy and dissipation characteristics of the resulting difference equations are examined. The aforementioned schemes are applied to a transient structural problem (beam bending) and the results compare favorably with available analytic solutions and are consistent with the conclusions of the stability analysis. A parametric investigation using different meshes, time steps, and beam dimensions is also presented. For all cases examined, the numerical solution was stable and robust and therefore is suitable for the next stage of application to full fluid-structure-interaction problems.

Pulsatile flow in stenotic geometries: flow behaviour and stability

2009 ◽  
Vol 622 ◽  
pp. 291-320 ◽  
Author(s):  
M. D. GRIFFITH ◽  
T. LEWEKE ◽  
M. C. THOMPSON ◽  
K. HOURIGAN
Keyword(s):  
Stability Analysis ◽  
Shear Layer ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Base Flow ◽  
Arterial Stenosis ◽  
Working Fluid ◽  
Absolute Instability ◽  
Straight Tube ◽  
Instability Modes

Pulsatile inlet flow through a circular tube with an axisymmetric blockage of varying size is studied both numerically and experimentally. The geometry consists of a long, straight tube and a blockage, semicircular in cross-section, serving as a simplified model of an arterial stenosis. The stenosis is characterized by a single parameter, the aim being to highlight fundamental behaviours of constricted pulsatile flows. The Reynolds number is varied between 50 and 700 and the stenosis degree by area between 0.20 and 0.90. Numerically, a spectral element code is used to obtain the axisymmetric base flow fields, while experimentally, results are obtained for a similar set of geometries, using water as the working fluid. For low Reynolds numbers, the flow is characterized by a vortex ring which forms directly downstream of the stenosis, for which the strength and downstream propagation velocity vary with the stenosis degree. Linear stability analysis is performed on the simulated axisymmetric base flows, revealing a range of absolute instability modes. Comparisons are drawn between the numerical linear stability analysis and the observed instability in the experimental flows. The observed flows are less stable than the numerical analysis predicts, with convective shear layer instability present in the experimental flows. Evidence is found of Kelvin–Helmholtz-type shear layer roll-ups; nonetheless, the possibility of the numerically predicted absolute instability modes acting in the experimental flow is left open.

On the Surface Stability of a Spherical Void Embedded in a Stressed Matrix

2005 ◽  
Vol 74 (1) ◽  
pp. 8-12 ◽  
Author(s):  
Jérôme Colin
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Applied Stress ◽  
Spherical Harmonics ◽  
Linear Stability Analysis ◽  
Spherical Cavity ◽  
Spherical Void ◽  
Surface Stability ◽  
Infinitesimal Perturbation

The linear stability analysis of the shape of a spherical cavity embedded in an infinite-size matrix under stress has been performed when infinitesimal perturbation from sphericity of the rod is assumed to appear by surface diffusion. Developing the perturbation on a basis of complete spherical harmonics, the growth rate of each harmonic Ylm(θ,φ) has been determined and the conditions for the development of the different fluctuations have been discussed as a function of the applied stress and the order l of the perturbation.

scholarly journals Deep-water sediment wave formation: linear stability analysis of coupled flow/bed interaction

2011 ◽  
Vol 680 ◽  
pp. 435-458 ◽  
Author(s):  
L. LESSHAFFT ◽  
B. HALL ◽  
E. MEIBURG ◽  
B. KNELLER
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Internal Wave ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stokes Equations ◽  
Wave Mode ◽  
Bottom Current ◽  
Field Observations ◽  
Sediment Bed

A linear stability analysis is carried out for the interaction of an erodible sediment bed with a sediment-laden, stratified flow above the bed, such as a turbidity or bottom current. The fluid motion is described by the full, two-dimensional Navier–Stokes equations in the Boussinesq approximation, while erosion is modelled as a diffusive flux of particles from the bed into the fluid. The stability analysis shows the existence of both Tollmien–Schlichting and internal wave modes in the stratified boundary layer. For the internal wave mode, the stratified boundary layer acts as a wave duct, whose height can be determined analytically from the Brunt–Väisälä frequency criterion. Consistent with this criterion, distinct unstable perturbation wavenumber regimes exist for the internal wave mode, which are associated with different numbers of pressure extrema in the wall-normal direction. For representative turbidity current parameters, the analysis predicts unstable wavelengths that are consistent with field observations. As a key condition for instability to occur, the base flow velocity boundary layer needs to be thinner than the corresponding concentration boundary layer. For most of the unstable wavenumber ranges, the phase relations between the sediment bed deformation and the associated wall shear stress and concentration perturbations are such that the sediment waves migrate in the upstream direction, which again is consistent with field observations.

Linear stability analysis of two-way coupled particle-laden density current

2017 ◽  
Vol 95 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Pouriya Amini ◽  
Ehsan Khavasi ◽  
Navid Asadizanjani
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Base Flow ◽  
Interfacial Instability ◽  
Linear Stability Theory ◽  
Flow Density ◽  
Density Current ◽  
Density Currents

Stability of two-way coupled particle-laden density current is studied with the aim of linear stability analysis. Interfacial instability can be found in density currents, which effects entrainment and the rate of effective mixing. In this paper, we investigate the density current interfacial instability using linear stability theory, considering the particles attendance. The ultimate goal is to extract the governing equation for current with particles and study the effect of different parameters on stability of such currents. Base flow has velocity and density profiles of tangent hyperbolic type. Main current and particles are studied in two separate phases. It is found that current will be more stable as M0 (M0 = S∗N∗/ρ∗ where ρ∗ is the non-dimensional flow density, S∗ is the Stokes’ drag coefficient, and N∗ is the particles’ number density) grows, this is a result of number of particles and their radius, and also viscosity effects. The current is more stable as the growth rate increases. As the Richardson number in M0 rises, the growth rate value decreases. As the slope of the river bed increases, the current is less stable.

scholarly journals The Growth and Saturation of Submesoscale Instabilities in the Presence of a Barotropic Jet

2018 ◽  
Vol 48 (11) ◽  
pp. 2779-2797 ◽  
Author(s):  
Megan A. Stamper ◽  
John R. Taylor ◽  
Baylor Fox-Kemper
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Initial Conditions ◽  
Boussinesq Equations ◽  
Small Region ◽  
Computational Domain ◽  
Mean Flow ◽  
The Mean

AbstractMotivated by recent observations of submesoscales in the Southern Ocean, we use nonlinear numerical simulations and a linear stability analysis to examine the influence of a barotropic jet on submesoscale instabilities at an isolated front. Simulations of the nonhydrostatic Boussinesq equations with a strong barotropic jet (approximately matching the observed conditions) show that submesoscale disturbances and strong vertical velocities are confined to a small region near the initial frontal location. In contrast, without a barotropic jet, submesoscale eddies propagate to the edges of the computational domain and smear the mean frontal structure. Several intermediate jet strengths are also considered. A linear stability analysis reveals that the barotropic jet has a modest influence on the growth rate of linear disturbances to the initial conditions, with at most a ~20% reduction in the growth rate of the most unstable mode. On the other hand, a basic state formed by averaging the flow at the end of the simulation with a strong barotropic jet is linearly stable, suggesting that nonlinear processes modify the mean flow and stabilize the front.

A linear stability analysis for miscible displacements

1986 ◽  
Vol 1 (2) ◽  
pp. 179-199 ◽  
Author(s):  
Shih-Hsien Chang ◽  
John C. Slattery
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Miscible Displacements
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