Stability of thermoviscous Hele-Shaw flow

1996 ◽  
Vol 308 ◽  
pp. 111-128 ◽  
Author(s):  
S. J. S. Morris
Keyword(s):  
Variable Viscosity ◽  
Temporal Stability ◽  
Three Dimensional ◽  
Channel Length ◽  
Base Flow ◽  
Plane Flow ◽  
Entry Length ◽  
Specific Heats ◽  
The Stability ◽  
Long Channel

Viscous fingering can occur as a three-dimensional disturbance to plane flow of a hot thermoviscous liquid in a Hele-Shaw cell with cold isothermal walls. This work assumes the principle of exchange of stabilities, and uses a temporal stability analysis to find the critical viscosity ratio and finger spacing as functions of channel length, Lc. Viscous heating is taken as negligible, so the liquid cools with distance (x) downstream. Because the base flow is spatially developing, the disturbance equations are not fully separable. They admit, however, an exact solution for a liquid whose viscosity and specific heats are arbitrary functions of temperature. This solution describes the neutral disturbances in terms of the base flow and an amplitude, A(x). The stability of a given (computed) base flow is determined by solving an eigenvalue problem for A(x), and the critical finger spacing. The theory is illustrated by using it to map the instability for variable-viscosity flow with constant specific heat. Two fingering modes are predicted, one being a turning-point instability. The preferred mode depends on Lc. Finger spacing is comparable with the thermal entry length in a long channel, and is even larger in short channels. When applied to magmatic systems, the results suggest that fingering will occur on geological scales only if the system is about freeze.

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

scholarly journals Effects of the Control Parameters on the Stability of a Laminar Boundary Layer on a Porous Flat Plate

2021 ◽  
Vol 26 (4) ◽  
pp. 113-127
Author(s):  
T.F. Lihonou ◽  
A.V. Monwanou ◽  
C.H. Miwadinou ◽  
J.B. Chabi Orou
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Temporal Stability ◽  
Three Dimensional ◽  
Darcy Number ◽  
Basic Flow ◽  
Marginal Stability ◽  
Stability Diagrams ◽  
Plane Plate ◽  
The Stability

Abstract This work is devoted to the analysis of the linear temporal stability of a laminar dynamic boundary layer on a horizontal porous plane plate. The basic flow is assumed to be laminar and two-dimensional. The basic flow velocity profiles are obtained by numerically solving the Blasius equation using the Runge-Kutta method. The perturbations of these basic solutions are expressed in the form of three-dimensional Tollmien-Schlichting waves. The formulation of the stability problem leads to the Orr-Sommerfeld equation modified by the permeability parameter (Darcy number) and the small Reynolds number. This equation is given in a general form which can be applied to the Chebyshev domain and the boundary layer domain and solved numerically using the Chebyshev spectral collocation method. The marginal stability diagrams, the critical Reynolds numbers and the eigenvalue spectra are obtained for different values of the parameters which have modified the stability equation. Numerical solutions indicate the importance of the effect of these parameters on the flow stability characteristics.

Nonlinear regimes of spanwise modulated waves in plane Poiseuille flow

2017 ◽  
Vol 818 ◽  
pp. 492-527
Author(s):  
Homero G. Silva ◽  
M. A. F. Medeiros
Keyword(s):  
Reynolds Number ◽  
Data Interpretation ◽  
Dominant Mode ◽  
Base Flow ◽  
Nonlinear Phenomenon ◽  
Perfect Agreement ◽  
Local Conditions ◽  
Plane Flow ◽  
The Stability ◽  
Nonlinear Regimes

Wave modulation is an unavoidable ingredient of natural transition and wavepackets composed of a continuous range of frequencies and wavenumbers are considered as a good model for it. Conclusions regarding wavepacket nonlinear regimes are essentially based on comparison of the dominant mode in the modulated signal nonlinear bands with predictions of the most unstable mode in the corresponding unmodulated cases. The modulated signal bands are very broad and establishing the dominant mode is difficult. If the Reynolds number changes along the packet evolution, the bands also change to adapt to local conditions, which further hinders data interpretation and weakens the conclusions. In view of this, a study at a constant Reynolds number is proposed, the Poiseuille plane flow being chosen as the base flow. The flow choice also allowed an investigation of the phenomenon at different positions in the stability loop, an aspect that has never been addressed before. The work was numerical. Only spanwise modulation was considered and two different regimes were observed. Close to the first branch of the instability loop the packet splits into two patches. Oblique transition was the dominant nonlinear mechanism, which required spanwise interaction of packets. Elsewhere the dominant mechanism was fundamental instability (or$K$-type), which was fed by a previous nonlinear phenomenon and led to a subsequent one. The analysis involved comparison of growth rates, threshold amplitudes and amplitude scalings among other aspects, for modulated and corresponding unmodulated cases. Perfect agreement was found if appropriate variables were used, which enabled firm conclusions to be drawn about the phenomena investigated. This level of agreement was only possible because of the constant Reynolds number character of the flow, but the main conclusions are applicable to other wall shear flows.

Three-dimensional instability of the flow over a forward-facing step

2012 ◽  
Vol 695 ◽  
pp. 390-404 ◽  
Author(s):  
Daniel Lanzerstorfer ◽  
Hendrik C. Kuhlmann
Keyword(s):  
Stability Boundary ◽  
Temporal Stability ◽  
Three Dimensional ◽  
Plane Channel ◽  
Separated Flow ◽  
Critical Mode ◽  
Flow Deceleration ◽  
Forward Facing Step ◽  
Adjoint Analysis ◽  
The Stability

AbstractThe global, temporal stability of the two-dimensional, incompressible flow over a forward-facing step in a plane channel is investigated numerically. The geometry is varied systematically covering constriction ratios (step-to-inlet height) from 0.23 to 0.965. A three-dimensional linear stability analysis shows that the stability boundary is a smooth continuous function of the constriction ratio. If the critical Reynolds and wavenumbers are scaled appropriately, they approach a linear asymptotic behaviour for large step heights. The critical mode is found to be stationary and confined to the region of separated flow downstream of the step for all constriction ratios. An energy-transfer analysis reveals that the basic flow becomes unstable due to a combined effect involving lift-up and flow deceleration, leading to a critical mode exhibiting steady streaks. Moreover, the receptivity of the flow to initial as well as to structural perturbations is studied by means of an adjoint analysis.

scholarly journals Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities

2009 ◽  
Vol 622 ◽  
pp. 1-21 ◽  
Author(s):  
OLIVIER MARQUET ◽  
MATTEO LOMBARDI ◽  
JEAN-MARC CHOMAZ ◽  
DENIS SIPP ◽  
LAURENT JACQUIN
Keyword(s):  
High Speed ◽  
Passive Control ◽  
Three Dimensional ◽  
Adjoint Problem ◽  
Base Flow ◽  
Mode Analysis ◽  
Global Mode ◽  
Recirculation Bubble ◽  
Global Modes ◽  
The Stability

The stability of the recirculation bubble behind a smoothed backward-facing step is numerically computed. Destabilization occurs first through a stationary three-dimensional mode. Analysis of the direct global mode shows that the instability corresponds to a deformation of the recirculation bubble in which streamwise vortices induce low- and high-speed streaks as in the classical lift-up mechanism. Formulation of the adjoint problem and computation of the adjoint global mode show that both the lift-up mechanism associated with the transport of the base flow by the perturbation and the convective non-normality associated with the transport of the perturbation by the base flow explain the properties of the flow. The lift-up non-normality differentiates the direct and adjoint modes by their component: the direct is dominated by the streamwise component and the adjoint by the cross-stream component. The convective non-normality results in a different localization of the direct and adjoint global modes, respectively downstream and upstream. The implications of these properties for the control problem are considered. Passive control, to be most efficient, should modify the flow inside the recirculation bubble where direct and adjoint global modes overlap, whereas active control, by for example blowing and suction at the wall, should be placed just upstream of the separation point where the pressure of the adjoint global mode is maximum.

Transient growth in developing plane and Hagen Poiseuille flow

2005 ◽  
Vol 461 (2057) ◽  
pp. 1311-1333 ◽  
Author(s):  
Peter W Duck
Keyword(s):  
Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Alternative Mechanism ◽  
Fully Developed Flow ◽  
Circular Pipes ◽  
Flow Disturbances ◽  
High Reynolds ◽  
The Stability

The stability of developing entry flow in both two-dimensional channels and circular pipes is investigated for large Reynolds numbers. The basic flow is generated by uniform flow entering a channel/pipe, which then provokes the growth of boundary layers on the walls, until (far downstream) fully developed flow is attained; the length for this development is well known to be (Reynolds number)×the channel/pipe width/diameter. This enables the use of high-Reynolds-number theory, leading to boundary-layer-type equations which govern the flow; as such, there is no need to impose heuristic parallel-flow approximations. The resulting base flow is shown to be susceptible to significant, three-dimensional, transient (initially algebraic) growth in the streamwise direction, and, consequently, large amplifications to flow disturbances are possible (followed by ultimate decay far downstream). It is suggested that this initial amplification of disturbances is a possible and alternative mechanism for flow transition.

Optimal Disturbances in Three-Dimensional Natural Convection Flows

2012 ◽  
Author(s):  
Elia Merzari ◽  
Paul Fischer ◽  
W. David Pointer
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Natural Circulation ◽  
Three Dimensional ◽  
Boussinesq Approximation ◽  
Spectral Element ◽  
Base Flow ◽  
The Stability ◽  
Optimal Disturbances

Buoyancy-driven systems are subject to several types of flow instabilities. To evaluate the performance of such systems it is becoming increasingly crucial to be able to predict the stability of a given base flow configuration. Traditional Modal Linear stability Analysis requires the solution of very large eigenvalue systems for three-dimensional flows, which make this problem difficult to tackle. An alternative to modal Linear stability Analysis is the use of adjoint solvers [1] in combination with a power iteration [2]. Such methodology allows for the identification of an optimal disturbance or forcing and has been recently used to evaluate the stability of several isothermal flow systems [2]. In this paper we examine the extension of the methodology to non-isothermal flows driven by buoyancy. The contribution of buoyancy in the momentum equation is modeled through the Boussinesq approximation. The method is implemented in the spectral element code Nek5000. The test case is the flow is a two-dimensional cavity with differential heating and conductive walls and the natural circulation flow in a toroidal thermosiphon.

scholarly journals Temporal stability analysis of jets of lobed geometry

2018 ◽  
Vol 860 ◽  
pp. 5-39 ◽  
Author(s):  
Benshuai Lyu ◽  
Ann P. Dowling
Keyword(s):  
Stability Analysis ◽  
Temporal Stability ◽  
Vortex Sheet ◽  
Base Flow ◽  
Efficient Manner ◽  
New Approach ◽  
Instability Waves ◽  
Degenerate Eigenvalue ◽  
Penetration Ratio ◽  
The Stability

A two-dimensional temporal incompressible stability analysis is performed for lobed jets. The jet base flow is assumed to be parallel and of a vortex-sheet type. The eigenfunctions of this simplified stability problem are expanded using the eigenfunctions of a round jet. The original problem is then formulated as an innovative matrix eigenvalue problem, which can be solved in a very robust and efficient manner. The results show that the lobed geometry changes both the convection velocity and temporal growth rate of the instability waves. However, different modes are affected differently. In particular, mode 0 is not sensitive to the geometry changes, whereas modes of higher orders can be changed significantly. The changes become more pronounced as the number of lobes $N$ and the penetration ratio $\unicode[STIX]{x1D716}$ increase. Moreover, the lobed geometry can cause a previously degenerate eigenvalue ($\unicode[STIX]{x1D706}_{n}=\unicode[STIX]{x1D706}_{-n}$) to become non-degenerate ($\unicode[STIX]{x1D706}_{n}\neq \unicode[STIX]{x1D706}_{-n}$) and lead to opposite changes to the stability characteristics of the corresponding symmetric ($n$) and antisymmetric ($-n$) modes. It is also shown that each eigenmode changes its shape in response to the lobes of the vortex sheet, and the degeneracy of an eigenvalue occurs when the vortex sheet has more symmetric planes than the corresponding mode shape (including both symmetric and antisymmetric planes). The new approach developed in this paper can be used to study the stability characteristics of jets of other arbitrary geometries in a robust and efficient manner.

scholarly journals Linear stability and weakly nonlinear analysis of the flow past rotating spheres

2016 ◽  
Vol 807 ◽  
pp. 62-86 ◽  
Author(s):  
V. Citro ◽  
J. Tchoufag ◽  
D. Fabre ◽  
F. Giannetti ◽  
P. Luchini
Keyword(s):  
Three Dimensional ◽  
Base Flow ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Weakly Nonlinear ◽  
Rotating Sphere ◽  
Flow Past ◽  
The Stability ◽  
Planar Symmetry ◽  
Flow Past A Sphere

We study the flow past a sphere rotating in the transverse direction with respect to the incoming uniform flow, and particularly consider the stability features of the wake as a function of the Reynolds number $Re$ and the sphere dimensionless rotation rate $\unicode[STIX]{x1D6FA}$. Direct numerical simulations and three-dimensional global stability analyses are performed in the ranges $150\leqslant \mathit{Re}\leqslant 300$ and $0\leqslant \unicode[STIX]{x1D6FA}\leqslant 1.2$. We first describe the base flow, computed as the steady solution of the Navier–Stokes equation, with special attention to the structure of the recirculating region and to the lift force exerted on the sphere. The stability analysis of this base flow shows the existence of two different unstable modes, which occur in different regions of the $Re/\unicode[STIX]{x1D6FA}$ parameter plane. Mode I, which exists for weak rotations ($\unicode[STIX]{x1D6FA}<0.4$), is similar to the unsteady mode existing for a non-rotating sphere. Mode II, which exists for larger rotations ($\unicode[STIX]{x1D6FA}>0.7$), is characterized by a larger frequency. Both modes preserve the planar symmetry of the base flow. We detail the structure of these eigenmodes, as well as their structural sensitivity, using adjoint methods. Considering small rotations, we then compare the numerical results with those obtained using weakly nonlinear approaches. We show that the steady bifurcation occurring for $Re>212$ for a non-rotating sphere is changed into an imperfect bifurcation, unveiling the existence of two other base-flow solutions which are always unstable.

Sign in / Sign up

Export Citation Format

Share Document