scholarly journals Diffusive effects in local instabilities of a baroclinic axisymmetric vortex

2021 ◽  
Vol 928 ◽  
Author(s):  
Suraj Singh ◽  
Manikandan Mathur
Keyword(s):  
Local Stability ◽  
Base Flow ◽  
Mode Analysis ◽  
Neutral Stability ◽  
Curvature Effects ◽  
Axisymmetric Vortex ◽  
Out Of Plane ◽  
Double Diffusive ◽  
Symmetric Instability ◽  
Oscillatory Instabilities

We present a local stability analysis of an idealized model of the stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in the thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on the Schmidt number $Sc$ (ratio between momentum and mass diffusivities) is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including $Sc$ ), and two different instabilities are identified: (i) a monotonic instability (same as symmetric instability at $Sc = 1$ ), and (ii) an oscillatory instability (absent at $Sc = 1$ ). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from $Sc$ ) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison with $Sc = 1$ , monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as $Sc$ moves away from unity. Neutral stability boundaries on the plane of $Sc$ and a modified gradient Richardson number are then identified for both these instabilities. In the absence of curvature effects, our results are shown to be consistent with previous studies based on normal mode analysis, thus establishing that the local stability approach is well suited to capturing symmetric and double-diffusive instabilities. The paper concludes with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.

scholarly journals Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

2014 ◽  
Vol 758 ◽  
pp. 565-585 ◽  
Author(s):  
Manikandan Mathur ◽  
Sabine Ortiz ◽  
Thomas Dubos ◽  
Jean-Marc Chomaz
Keyword(s):  
Axial Velocity ◽  
Local Stability ◽  
Numerical Solutions ◽  
Axial Flow ◽  
Axial Direction ◽  
Threshold Value ◽  
Base Flow ◽  
Centrifugal Instability ◽  
Axisymmetric Vortex ◽  
Stuart Vortices

AbstractLinear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

Double-Diffusive Magneto Convection in a Compressible Couple-Stress Fluid Through Porous Medium

2011 ◽  
Vol 66 (5) ◽  
pp. 304-310 ◽  
Author(s):  
Pardeep Kumar ◽  
Hari Mohan
Keyword(s):  
Magnetic Field ◽  
Porous Medium ◽  
Couple Stress ◽  
Mode Analysis ◽  
Couple Stress Fluid ◽  
Onset Of Convection ◽  
Linearized Stability ◽  
Medium Permeability ◽  
Solute Gradient ◽  
Double Diffusive

The double-diffusive convection in a compressible couple-stress fluid layer heated and soluted from below through porous medium is considered in the presence of a uniform vertical magnetic field. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For stationary convection, the compressibility, stable solute gradient, magnetic field, and couple-stress postpone the onset of convection whereas medium permeability hastens the onset of convection. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. The stable solute gradient and magnetic field introduce oscillatory modes in the system, which were non-existent in their absence. A condition for the system to be stable is obtained by using the Rayleigh-Ritz inequality. The sufficient conditions for the non-existence of overstability are also obtained.

scholarly journals Influence of localised smooth steps on the instability of a boundary layer

2017 ◽  
Vol 817 ◽  
pp. 138-170 ◽  
Author(s):  
Hui Xu ◽  
Jean-Eloi W. Lombard ◽  
Spencer J. Sherwin
Keyword(s):  
Boundary Layer ◽  
Layer Thickness ◽  
Boundary Layer Thickness ◽  
Gaussian White Noise ◽  
Random Phase ◽  
Linear Analysis ◽  
Base Flow ◽  
Neutral Stability ◽  
Local Boundary ◽  
Type Transition

We consider a smooth, spanwise-uniform forward-facing step defined by a Gauss error function of height 4 %–30 % and four times the width of the local boundary layer thickness $\unicode[STIX]{x1D6FF}_{99}$. The boundary layer flow over a smooth forward-facing stepped plate is studied with particular emphasis on stabilisation and destabilisation of the two-dimensional Tollmien–Schlichting (TS) waves and subsequently on three-dimensional disturbances at transition. The interaction between TS waves at a range of frequencies and a base flow over a single or two forward-facing smooth steps is conducted by linear analysis. The results indicate that for a TS wave with a frequency ${\mathcal{F}}\in [140,160]$ (${\mathcal{F}}=\unicode[STIX]{x1D714}\unicode[STIX]{x1D708}/U_{\infty }^{2}\times 10^{6}$, where $\unicode[STIX]{x1D714}$ and $U_{\infty }$ denote the perturbation angle frequency and free-stream velocity magnitude, respectively, and $\unicode[STIX]{x1D708}$ denotes kinematic viscosity), the amplitude of the TS wave is attenuated in the unstable regime of the neutral stability curve corresponding to a flat plate boundary layer. Furthermore, it is observed that two smooth forward-facing steps lead to a more acute reduction of the amplitude of the TS wave. When the height of a step is increased to more than 20 % of the local boundary layer thickness for a fixed width parameter, the TS wave is amplified, and thereby a destabilisation effect is introduced. Therefore, the stabilisation or destabilisation effect of a smooth step is typically dependent on its shape parameters. To validate the results of the linear stability analysis, where a TS wave is damped by the forward-facing smooth steps direct numerical simulation (DNS) is performed. The results of the DNS correlate favourably with the linear analysis and show that for the investigated frequency of the TS wave, the K-type transition process is altered whereas the onset of the H-type transition is delayed. The results of the DNS suggest that for the perturbation with the non-dimensional frequency parameter ${\mathcal{F}}=150$ and in the absence of other external perturbations, two forward-facing smooth steps of height 5 % and 12 % of the boundary layer thickness delayed the H-type transition scenario and completely suppressed for the K-type transition. By considering Gaussian white noise with both fixed and random phase shifts, it is demonstrated by DNS that transition is postponed in time and space by two forward-facing smooth steps.

The Influence of Coupled Molecular Diffusion on Double-Diffusive Convection in a Porous Medium

1986 ◽  
Vol 108 (4) ◽  
pp. 872-876 ◽  
Author(s):  
N. Rudraiah ◽  
M. S. Malashetty
Keyword(s):  
Porous Medium ◽  
Molecular Diffusion ◽  
Normal Mode Analysis ◽  
Finite Amplitude ◽  
Mode Analysis ◽  
Double Diffusive Convection ◽  
Amplitude Analysis ◽  
Cross Diffusion ◽  
Diffusive Convection ◽  
Double Diffusive

The effect of coupled molecular diffusion on double-diffusive convection in a horizontal porous medium is studied using linear and nonlinear stability analyses. In the case of linear theory, normal mode analysis is employed incorporating two cross diffusion terms. It is found that salt fingers can form by taking cross-diffusion terms of appropriate sign and magnitude even when both concentrations are stably stratified. The conditions for the diffusive instability are compared with those for the formation of fingers. It is shown that these two types of instability will never occur together. The finite amplitude analysis is used to derive the condition for the maintenance of fingers. The stability boundaries are drawn for three different combinations of stratification and the effect of permeability is depicted.

A quasi-steady approach to the instability of time-dependent flows in pipes

2002 ◽  
Vol 465 ◽  
pp. 301-330 ◽  
Author(s):  
M. S. GHIDAOUI ◽  
A. A. KOLYSHKIN
Keyword(s):  
Experimental Data ◽  
Laminar Flow ◽  
Asymptotic Expansions ◽  
Unstable Mode ◽  
Landau Equation ◽  
Base Flow ◽  
Neutral Stability ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Weakly Nonlinear Analysis

Asymptotic solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe subject to rapid deceleration and/or acceleration are derived and their stability investigated using linear and weakly nonlinear analysis. In particular, base flow solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe are derived by the method of matched asymptotic expansions. The solutions are valid for short times and can be successfully applied to the case of an arbitrary (but unidirectional) axisymmetric initial velocity distribution. Excellent agreement between asymptotic and analytical solutions for the case of an instantaneous pipe blockage is found for small time intervals. Linear stability of the base flow solutions obtained from the asymptotic expansions to a three-dimensional perturbation is investigated and the results are used to re-interpret the experimental results of Das & Arakeri (1998). Comparison of the neutral stability curves computed with and without the planar channel assumption shows that this assumption is accurate when the ratio of the unsteady boundary layer thickness to radius (i.e. δ1/R) is small but becomes unacceptable when this ratio exceeds 0.3. Both the current analysis and the experiments show that the flow instability is non-axisymmetric for δ1/R = 0.55 and 0.85. In addition, when δ1/R = 0.18 and 0.39, the neutral stability curves for n = 0 and n = 1 are found to be close to one another at all times but the most unstable mode in these two cases is the axisymmetric mode. The accuracy of the quasi-steady assumption, employed both in this research and in that of Das & Arakeri (1998), is supported by the fact that the results obtained under this assumption show satisfactory agreement with the experimental features such as type of instability and spacing between vortices. In addition, the computations show that the ratio of the rate of growth of perturbations to the rate of change of the base flow is much larger than 1 for all cases considered, providing further support for the quasi-steady assumption. The neutral stability curves obtained from linear stability analysis suggest that a weakly nonlinear approach can be used in order to study further development of instability. Weakly nonlinear analysis shows that the amplitude of the most unstable mode is governed by the complex Ginzburg–Landau equation which reduces to the Landau equation if the amplitude is a function of time only. The coefficients of the Landau equation are calculated for two cases of the experimental data given by Das & Arakeri (1998). It is shown that the real part of the Landau constant is positive in both cases. Therefore, finite-amplitude equilibrium is possible. These results are in qualitative agreement with experimental data of Das & Arakeri (1998).

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.

Linear stability of two-layer Couette flows

2017 ◽  
Vol 826 ◽  
pp. 128-157 ◽  
Author(s):  
Alireza Mohammadi ◽  
Alexander J. Smits
Keyword(s):  
Interfacial Tension ◽  
Viscosity Ratio ◽  
Density Ratio ◽  
Initial Step ◽  
Base Flow ◽  
Neutral Stability ◽  
Low Viscosity ◽  
Tollmien Schlichting ◽  
And Performance ◽  
Interfacial Mode

The stability of two-layer Couette flow is investigated under variations in viscosity ratio, thickness ratio, interfacial tension and density ratio. The effects of the base flow on eigenvalue spectra are explained. A new type of interfacial mode is discovered at low viscosity ratio with properties that are different from Yih’s original interfacial mode (Yih, J. Fluid Mech., vol. 27, 1967, pp. 337–352). No unstable Tollmien–Schlichting waves were found over the range of parameters considered in this work. The results for thin films with different thicknesses can be collapsed onto a single curve if the Reynolds number and wavenumber are suitably defined based on the parameters of the thin layer. Interfacial tension always has a stabilizing effect, but the effects of density ratio cannot be so easily generalized. Neutral stability curves for water–alkane and water–air systems are presented as an initial step towards better understanding the effects of flow stability on the longevity and performance of liquid-infused surfaces and superhydrophobic surfaces.

Boussinesq global modes and stability sensitivity, with applications to stratified wakes

2017 ◽  
Vol 812 ◽  
pp. 1146-1188
Author(s):  
Kevin K. Chen ◽  
Geoffrey R. Spedding
Keyword(s):  
Bluff Body ◽  
Boussinesq Equations ◽  
Base Flow ◽  
Mode Analysis ◽  
Flow Density ◽  
Global Mode ◽  
Multiple Production ◽  
Small Density ◽  
Global Modes ◽  
The Stability

For the Boussinesq equations, we present a theory of linear stability sensitivity to base flow density and velocity modifications. Given a steady-state flow with small density variations, the sensitivity of the stability eigenvalues is computed from the direct and adjoint global modes of the linearised Boussinesq equations, similarly to Marquetet al.(J. Fluid Mech., vol. 615, 2008, pp. 221–252). Combinations of the density and velocity components of these modes reveal multiple production and transport mechanisms that contribute to the eigenvalue sensitivity. We present an application of the sensitivity theory to a stably linearly density-stratified flow around a thin plate at a$90^{\circ }$angle of attack, a Reynolds number of 30 and Froude numbers of 1, 8 and$\infty$. The global mode analysis reveals lightly damped undulations pervading through the entire domain, which are predicted by both inviscid uniform base flow theory and Orr–Sommerfeld theory. The sensitivity to base flow velocity modifications is primarily concentrated just downstream of the bluff body. On the other hand, the sensitivity to base flow density modifications is concentrated in regions both immediately upstream and immediately downstream of the plate. Both sensitivities have a greater upstream presence for lower Froude numbers.

scholarly journals Stability and nonlinear regimes of flow over a saturated porous medium

2013 ◽  
Vol 20 (4) ◽  
pp. 543-547 ◽  
Author(s):  
T. P. Lyubimova ◽  
D. T. Baydina ◽  
D. V. Lyubimov
Keyword(s):  
Porous Medium ◽  
Saturated Porous Medium ◽  
Base Flow ◽  
Nonlinear Flow ◽  
Wave Number ◽  
Neutral Stability ◽  
Steady Plane ◽  
Stability Of Water ◽  
Parameter Values ◽  
Nonlinear Regimes

Abstract. The paper deals with the investigation of stability and nonlinear regimes of flow over the saturated porous medium applied to the problem of stability of water flow over the bottom covered with vegetation. It is shown that the velocity profile of steady plane-parallel flow has two inflection points, which results in instability of this flow. The neutral stability curves, the dependencies of critical Reynolds number and the wave number of most dangerous perturbations on the ratio of porous layer thickness to the total thickness are obtained. The nonlinear flow regimes are investigated numerically by finite difference method. It is found that at supercritical parameter values waves travelling in the direction of the base flow take place.

scholarly journals Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics

2017 ◽  
Vol 473 (2205) ◽  
pp. 20170344 ◽  
Author(s):  
Oleg N. Kirillov
Keyword(s):  
Magnetic Field ◽  
Hopf Bifurcation ◽  
Smooth Transition ◽  
Neutral Stability ◽  
Ohmic Dissipation ◽  
Azimuthal Magnetic Field ◽  
Electrically Conducting ◽  
Whitney Umbrella ◽  
Finite Distance ◽  
Double Diffusive

We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. In the presence of the magnetic field, the hydrodynamically stable flow can demonstrate non-axisymmetric azimuthal magnetorotational instability (AMRI) both in the diffusionless case and in the double-diffusive case with viscous and ohmic dissipation. Performing stability analysis of amplitude transport equations of short-wavelength approximation, we find that the threshold of the diffusionless AMRI via the Hamilton–Hopf bifurcation is a singular limit of the thresholds of the viscous and resistive AMRI corresponding to the dissipative Hopf bifurcation and manifests itself as the Whitney umbrella singular point. A smooth transition between the two types of instabilities is possible only if the magnetic Prandtl number is equal to unity, Pm =1. At a fixed Pm ≠1, the threshold of the double-diffusive AMRI is displaced by finite distance in the parameter space with respect to the diffusionless case even in the zero dissipation limit. The complete neutral stability surface contains three Whitney umbrella singular points and two mutually orthogonal intervals of self-intersection. At these singularities, the double-diffusive system reduces to a marginally stable system which is either Hamiltonian or parity–time-symmetric.

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