On the inertial instability of the Kolmogorov flow in a rotating stratified fluid

2021 ◽  
Author(s):  
Michael Kurgansky
Keyword(s):  
Critical Reynolds Number ◽  
Stationary Regime ◽  
Main Flow ◽  
Density Stratification ◽  
Necessary Condition ◽  
Uniform Density ◽  
Kolmogorov Flow ◽  
Inertial Instability ◽  
Prandtl Numbers ◽  
Rotating Stratified Fluid

<p>The linear and non-linear inertial stability of the Kolmogorov flow in a rotating viscous fluid of uniform density is investigated. A necessary condition for instability is the violation of the criterion of non-viscous inertial stability, and the sufficient condition of instability is formulated in terms of the Reynolds criterion. The existence of stable secondary stationary regimes in the problem is shown, developing in a context of loss of stability of the main flow and having the shape of rolls (cloud streets in the atmosphere) oriented along it. Stable density stratification is taken into account when the direction of gravity coincides with the direction of rotation of the fluid. In this case, the necessary condition for the inertial instability of the main flow remains the same, but the critical Reynolds number for the instability depends on two additional dimensionless parameters that appear in the problem: the stratification parameter and the Prandtl number. The case of Prandtl numbers less than or equal to unity has been studied in greater detail, when there is a secondary stationary regime, which can be unstable - in contrast to the case of a fluid that is uniform in density - and density stratification is a destabilizing factor.</p>

Stability of thermocapillary flows in non-cylindrical liquid bridges

2002 ◽  
Vol 458 ◽  
pp. 35-73 ◽  
Author(s):  
CH. NIENHÜSER ◽  
H. C. KUHLMANN
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Critical Reynolds Number ◽  
Volume Fraction ◽  
Thermocapillary Flow ◽  
Surface Shape ◽  
Liquid Bridges ◽  
Gravity Level ◽  
Neutral Curves ◽  
Prandtl Numbers

The thermocapillary flow in liquid bridges is investigated numerically. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. When gravity acts parallel to the axis of the liquid bridge the shape is axisymmetric. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, Prandtl number, and aspect ratio is explained by analysing the energy budgets of the neutral modes. For small Prandtl numbers (Pr = 0.02) the critical Reynolds number exhibits a smooth minimum near volume fractions which approximately correspond to the volume of a cylindrical bridge. When the Prandtl number is large (Pr = 4) the intersection of two neutral curves results in a sharp peak of the critical Reynolds number. Since the instabilities for low and high Prandtl numbers are markedly different, the influence of gravity leads to a distinctly different behaviour. While the hydrostatic shape of the bridge is the most important effect of gravity on the critical point for low-Prandtl-number flows, buoyancy is the dominating factor for the stability of the flow in a gravity field when the Prandtl number is high.

scholarly journals Large-scale instability of a generalized turbulent Kolmogorov flow

2009 ◽  
Vol 16 (4) ◽  
pp. 569-577 ◽  
Author(s):  
B. Legras ◽  
B. Villone
Keyword(s):  
Turbulent Flows ◽  
Mirror Symmetry ◽  
Large Scale ◽  
Necessary Condition ◽  
Scale Separation ◽  
Kolmogorov Flow ◽  
Nonlinear Contribution ◽  
Basic Scale ◽  
Negative Viscosity ◽  
Molecular Viscosity

Abstract. We present an analytical study of the large scale instability of a generalized turbulent Kolmogorov flow, i.e. a periodic shear flow where the molecular viscosity has been substituted by an eddy viscosity parameterized with the Clark-Smagorinsky model and where the external forcing is adapted to maintain the flow against this dissipation. We employ multiscaling technique assuming a scale separation between the basic scale of such a generalized turbulent Kolmogorov flow and the largest scales of the flow. The main result is that an amplitude equation for the large-scale secondary flow is obtained which exhibits, like for the standard Kolmogorov flow, an instability of the negative viscosity type. We find that the presence of mirror symmetry in the basic flow is a necessary condition and that further propagative and nonlinear contribution are produced otherwise. The result is encouraging for the generic existence of large-scale instabilities of the negative viscosity type in fully turbulent flows.

scholarly journals The Nutations of Neutron Stars and Core-Crust Coupling

1987 ◽  
Vol 125 ◽  
pp. 454-454
Author(s):  
C. R. Gwinn
Keyword(s):  
Simple Model ◽  
Neutron Stars ◽  
Ideal Fluid ◽  
Density Stratification ◽  
Uniform Density ◽  
Rotating Fluid ◽  
The Core ◽  
Inertial Coupling ◽  
The Earth

Neutron stars, like the earth, are rotating fluid-filled ellipsoids. Poincaré (Bull. Astron. 27, 321, 1910), Hough (Phil. Trans. R. Soc. A186, 469, 1895) and others have discussed the nutations of such objects through a simple model, which treats the crust as rigid and the core as an ideal fluid of uniform density and vorticity. The core and crust are coupled by inertial coupling: the forces which constrain the fluid to its cavity within the crust can produce a net torque, since the cavity is ellipsoidal. Additional torques, and the effects of the elasticity in the crust and density stratification in the core, may be accomodated in such models as well (Sasao et al., Proc. IAU Symposium 78, p. 165, 1980, and references therein).

Stability of Nonparallel Developing Flow in a Pipe to Nonaxisymmetric Disturbances

1983 ◽  
Vol 50 (1) ◽  
pp. 210-214 ◽  
Author(s):  
V. K. Garg
Keyword(s):  
Experimental Data ◽  
Critical Reynolds Number ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Main Flow ◽  
Wave Number ◽  
Developing Flow ◽  
Azimuthal Wave Number ◽  
Flow Velocity Profile ◽  
Axisymmetric Disturbances

Linear spatial stability of the nonparallel developing flow in a rigid circular pipe has been studied at several axial locations for nonaxisymmetric disturbances. The main flow velocity profile is obtained by Hornbeck’s finite-difference method assuming uniform flow at entry to the pipe. The method of multiple scales is used to account for all the nonparallel effects. It is found that the nonparallel developing flow is most unstable to nonaxisymmetric disturbances with azimuthal wave number n equal to unity. Axisymmetric disturbances are, however, more unstable than nonaxisymmetric disturbances with n ≥ 2 except in the near-entry region. The results show that the parallel flow theory overpredicts the critical Reynolds number by as much as 136.5 percent in the near entry region for the n = 1 disturbance. The present results compare well with the available experimental data.

On the alignment and axisymmetrization of a vertically tilted geostrophic vortex

1995 ◽  
Vol 289 ◽  
pp. 29-50 ◽  
Author(s):  
F. Viera
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Density Stratification ◽  
Uniform Density ◽  
General Tendency ◽  
Cylindrical Shape ◽  
Vertical Alignment ◽  
Horizontal Scale ◽  
Geostrophic Vortex

We investigate the evolution of a vertically tilted geostrophic vortex of cylindrical shape and circular horizontal cross-section using the recently developed method of boundary surface dynamics. The vortex consists of a finite volume of constant potential vorticity immersed in a spatially unbounded fluid of uniform density stratification. The fully nonlinear three-dimensional problem is then reduced to the calculation of the Lagrangian evolution of the boundary surface of the vortex region, thus decreasing the dimensionality by one. In the numerical simulations presented here, the vortex shows a general tendency to attain vertical alignment and a horizontal axisymmetrical shape by wobbling about its centre and going through three basic stages of evolution: (a) the circular horizontal cross-sections of the upper and lower parts of the vortex distort and become elongated; (b) the upper and lower sections then become vertically aligned by reducing their horizontal intercentroid distances; and (c) the distorted horizontal cross-sections relax towards axisymmetry, often through the process of filamentation. For a given vortex height, if the horizontal scale of the flow is close to the internal radius of deformation, or equivalently, the density stratification is not too strong, the processes of filamentation and vertical alignment are enhanced. However, for stronger stratifications, both filamentation and vertical alignment are found to be greatly inhibited. For relatively small initial inclination angles, filamentation only occurs in the upper and lower sections of the vortex. Increasing the angle of tilt also increases the tendency of the surface to steepen and filament in the middle sections of the vortex. For a fixed value of the ratio of horizontal scale of the flow to the deformation radius, taller vortices have an increased tendency to align and axisymmetrize than shorter vortices of equal inclination angle.

Hydrodynamical instabilities of thermocapillary flow in a half-zone

1995 ◽  
Vol 297 ◽  
pp. 357-372 ◽  
Author(s):  
Måarten Levenstam ◽  
Gustav Amberg
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Physical Mechanism ◽  
Three Dimensional ◽  
Thermocapillary Flow ◽  
Vortex Rings ◽  
Crystal Growth Process ◽  
Oscillatory State ◽  
Prandtl Numbers ◽  
The Stability

The stability of the flow in a half-zone configuration is analysed with the aid of direct numerical simulation. The work is concentrated on the small Prandtl numbers relevant for typical semiconductor melts. The axisymmetric thermocapillary flow is found to be unstable to a steady non-axisymmetric state with azimuthal wavenumber 2, for a zone with aspect ratio 1. The critical Reynolds number for this bifurcation is 1960. This three dimensional steady solution loses stability to an oscillatory state at a Reynolds number of 6250. For small Prandtl numbers, both bifurcations are seen to be quite insensitive to changes in the Prandtl number, and are thus hydrodynamic in nature. An analogy to the instability of thin vortex rings is made. This analogy suggests a physical mechanism behind the instability and also gives an explanation of how the azimuthal wavenumber of the bifurcated solution is selected. The implications of this for the floating-zone crystal growth process are discussed.

Applicability of Dynamic Modeling for Turbulent Schmidt and Prandtl Numbers on Density Stratification Breakup in Several Flow Conditions

2020 ◽  
Author(s):  
Satoshi Abe ◽  
Etienne Studer ◽  
Masahiro Ishigaki ◽  
Yasuteru Sibamoto ◽  
Taisuke Yonomoto
Keyword(s):  
Dynamic Modeling ◽  
Density Stratification ◽  
Severe Accident ◽  
Navier Stokes ◽  
Small Scale ◽  
Flow Conditions ◽  
Energy Agency ◽  
Experimental Facilities ◽  
Prandtl Numbers ◽  
Rans Analysis

Abstract Many experiments on density stratification breakup in several flow conditions have been performed with the large- and small-scale experimental facilities to understand the mechanism underlying hydrogen behavior in a nuclear containment vessel during a severe accident. To improve the predictability of the RANS (Reynolds-averaged Navier Stokes) approach, we implemented the dynamic modeling for turbulent Schmidt Sct and Prandtl Prt numbers. In this paper, the capability of the RANS analysis with dynamic Sct modeling is assessed with several experimental data obtained by using the MISTRA (Commissariat à l’énergie atomique et aux énergies alternatives, CEA, France), CIGMA and VIMES (Japan Atomic Energy Agency, Japan). For the quantitative assessment, the completion time of the stratification breakup, defined as when helium concentration in the upper region decreases to the same value in the lower region, is focused. The comparison study shows the good performance of the dynamic modeling for Sct and Prt. Besides, in the case with the low jet Froude number, the CFD accuracy declines significantly, because the jet upward bending is over-estimated.

Three-dimensional instability of viscoelastic elliptic vortices

1997 ◽  
Vol 353 ◽  
pp. 357-381 ◽  
Author(s):  
H. HAJ-HARIRI ◽  
G. M. HOMSY
Keyword(s):  
Plane Waves ◽  
Three Dimensional ◽  
Deborah Number ◽  
Base Flow ◽  
Necessary Condition ◽  
Inertial Instability ◽  
Viscoelastic Instability ◽  
Dimensional Instability ◽  
Model Oscillator ◽  
Vortex Axis

An analysis of the three-dimensional instability of two-dimensional viscoelastic elliptical flows is presented, extending the inviscid analysis of Bayly (1986) to include both viscous and elastic effects. The problem is governed by three parameters: E, a geometric parameter related to the ellipticity; Re, a wavenumber-based Reynolds number; and De, the Deborah number based on the period of the base flow. New modes and mechanisms of instability are discovered. The flow is generally susceptible to instabilities in the form of propagating plane waves with a rotating wavevector, the tip of which traces an ellipse of the same eccentricity as the flow, but with the major and minor axes interchanged. Whereas a necessary condition for purely inertial instability is that the wavevector has a non-vanishing component along the vortex axis, the viscoelastic modes of instability are most prominent when their wavevectors do vanish along this axis. Our analytical and numerical results delineate the region of parameter space of (E, ReDe) for which the new instability exists. A simple model oscillator equation of the Mathieu type is developed and shown to embody the essential qualitative and quantitative features of the secular viscoelastic instability. The cause of the instability is a buckling of the ‘compressed’ polymers as they are perturbed transversely during a particular phase of the passage of the rotating plane wave.

scholarly journals First evidence of inertial modes in γ Doradus stars: The core rotation revealed

2020 ◽  
Vol 640 ◽  
pp. A49
Author(s):  
R.-M. Ouazzani ◽  
F. Lignières ◽  
M.-A. Dupret ◽  
S. J. A. J. Salmon ◽  
J. Ballot ◽  
...  
Keyword(s):  
Main Sequence ◽  
Density Stratification ◽  
Uniform Density ◽  
Main Sequence Stars ◽  
Intermediate Mass ◽  
The Core ◽  
Intermediate Mass Stars ◽  
Convective Core ◽  
Coupling Factors ◽  
Core Rotation

The advent of space photometry with CoRoT and Kepler has allowed for the gathering of exquisite and extensive time series for a wealth of main-sequence stars, including γ Doradus stars, whose detailed seismology was not achievable from the ground. γ Doradus stars present an incredibly rich pulsation spectra, with gravito-inertial modes, in some cases supplemented with δ Scuti-like pressure modes – for the hybrid stars – and, in many cases, with Rossby modes. The present paper aims to show that in addition to these modes which have been established in the radiative envelope, pure inertial modes that are trapped in the convective core can be detected in Kepler observations of γ Doradus stars thanks to their resonance with the gravito-inertial modes. We started by using a simplified model of perturbations in a full sphere of uniform density. Under these conditions, the spectrum of pure inertial modes is known from analytical solutions of the so-called Poincaré equation. We then computed coupling factors, which helped select the pure inertial modes which interact best with the surrounding dipolar gravito-inertial modes. Using complete calculations of gravito-inertial modes in realistic models of γ Doradus stars, we are able to show that the pure inertial and gravito-inertial resonances appear as “dips” in the gravito-inertial mode period spacing series at spin parameters that are close to those predicted by the simple model. We find the first evidence of such dips in the Kepler γ Doradus star KIC 5608334. Finally, using complete calculations in isolated convective cores, we find that the spin parameters of the pure inertial and gravito-inertial resonances are also sensitive to the density stratification of the convective core. In conclusion, we have discovered that certain dips in gravito-inertial mode period spacings that have been observed in some Kepler stars are, in fact, signatures of resonances with pure-inertial modes that are trapped in the convective core. This holds the promise that it would be possible to finally access the central conditions, namely, the rotation and density stratification, of intermediate-mass stars in the main sequence.

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