scholarly journals Barotropic Instability at Minimal Resolution

2006 ◽  
Vol 134 (10) ◽  
pp. 2943-2950 ◽  
Author(s):  
Joseph Egger
Keyword(s):  
Initial Conditions ◽  
Baroclinic Instability ◽  
Plane Channel ◽  
Necessary Condition ◽  
Barotropic Instability ◽  
Mean Flow ◽  
Linear Shear ◽  
The Stability ◽  
Basic Features ◽  
Plane Channel Flow

Abstract Two linear minimum-resolution models of β-plane channel flow are presented in analogy to the well-known two-layer model of baroclinic instability in order to see if basic features of barotropic instability can be demonstrated using similarly simple models. Two models are discussed within this pedagogical framework. A spectral model with two wave modes is applied to a cosine jet. Necessary conditions for instability are derived. The stability analysis shows that this simple model captures the shortwave cutoff and the asymmetry of the instability with respect to the direction of the jet quite well. It is demonstrated that the cutoff follows from Fjörtoft’s theorem for wave triads. A gridpoint model with two points in the interior of the channel is discussed as well. An analog to the classical necessary condition for instability is derived. A stability problem in a nonrotating system is discussed where the mean flow velocity is constant near the walls and a linear shear flow is assumed near the channel’s axis. In this case, the stability characteristics of the low-order model come close to those of the full problem where a simple analytic solution is available. Addition of the β term stabilizes the flow. A proper choice of the initial conditions always enables short-term growth of the perturbation energy for stable mean flows. A qualitative interpretation of the instability mechanism is presented for both models, which exploits the fact that the locations of corresponding extrema of streamfunction and vorticity need not coincide. It is concluded that low-resolution models are well suited for a discussion of the basic features of barotropic instability.

Numerical simulations of transition in oscillatory plane channel flow

1989 ◽  
Vol 208 ◽  
pp. 45-66 ◽  
Author(s):  
Bart A. Singer ◽  
Joel H. Ferziger ◽  
Helen L. Reed
Keyword(s):  
Channel Flow ◽  
Positive Curvature ◽  
Plane Channel ◽  
Maximum Growth ◽  
Mean Flow ◽  
Low Frequencies ◽  
Frequency Maximum ◽  
The Mean ◽  
The Stability ◽  
Plane Channel Flow

The effect of flow oscillation on the stability of plane channel flow is studied via numerical simulation. For weak oscillation, the ratio of the Stokes layer thickness to the distance from the wall to the critical layer in steady flow provides the best normalization for the mean-flow frequency. Maximum growth rates occur when the instantaneous velocity profile has large regions of positive curvature. The effect of oscillation is generally stabilizing. However, at low frequencies, TS wave energies may vary by 106 in a cycle and irreversible secondary instability may be produced at the peak.

scholarly journals On strongly nonlinear gravity waves in a vertically sheared atmosphere

2020 ◽  
Vol 6 (1) ◽  
pp. 63-74
Author(s):  
Mark Schlutow ◽  
Georg S. Voelker
Keyword(s):  
Gravity Waves ◽  
Lower Layer ◽  
Vertical Shear ◽  
Horizontal Wind ◽  
Necessary Condition ◽  
Mean Flow ◽  
Individual Layer ◽  
Strongly Nonlinear ◽  
The Mean ◽  
The Stability

Abstract We investigate strongly nonlinear stationary gravity waves which experience refraction due to a thin vertical shear layer of horizontal background wind. The velocity amplitude of the waves is of the same order of magnitude as the background flow and hence the self-induced mean flow alters the modulation properties to leading order. In this theoretical study, we show that the stability of such a refracted wave depends on the classical modulation stability criterion for each individual layer, above and below the shearing. Additionally, the stability is conditioned by novel instability criteria providing bounds on the mean-flow horizontal wind and the amplitude of the wave. A necessary condition for instability is that the mean-flow horizontal wind in the upper layer is stronger than the wind in the lower layer.

Effect of an Azimuthal Mean Flow On the Structure and Stability of Thermoacoustic Modes in an Annular Combustor Model with Electroacoustic Feedback

2020 ◽  
Author(s):  
Sylvain C. Humbert ◽  
Jonas Moeck ◽  
Alessandro Orchini ◽  
Christian Oliver Paschereit
Keyword(s):  
Mach Number ◽  
Initial Conditions ◽  
Mean Flow ◽  
Final State ◽  
Mean Velocity ◽  
Acoustic Damping ◽  
Annular Combustor ◽  
The Mean ◽  
High Mach ◽  
The Stability

Abstract Thermoacoustic oscillations in axisymmetric annular combustors are generally coupled by degenerate azimuthal modes, which can be of standing or spinning nature. Symmetry breaking due to the presence of a mean azimuthal flow splits the degenerate thermoacoustic eigenvalues, resulting in pairs of counter-spinning modes with close but distinct frequencies and growth rates. In this study, experiments have been performed using an annular system where the thermoacoustic feedback due to the flames is mimicked by twelve identical electroacoustic feedback loops. The mean azimuthal flow is generated by fans. We investigate the standing/spinning nature of the oscillations as a function of the Mach number for two types of initial states, and how the stability of the system is affected by the mean azimuthal flow. It is found that spinning, standing or mixed modes can be encountered at very low Mach number, but increasing the mean velocity promotes one spinning direction. At sufficiently high Mach number, spinning modes are observed in the limit cycle oscillations. In some cases, the initial conditions have a significant impact on the final state of the system. It is found that the presence of a mean azimuthal flow increases the acoustic damping. This has a beneficial effect on stability: it often reduces the amplitude of the self-sustained oscillations, and can even suppress them in some cases. However, we observe that the suppression of a mode due to the mean flow may destabilize another one. We discuss our findings in relation with an existing low-order model.

Frontal instabilities and waves in a differentially rotating fluid

2011 ◽  
Vol 685 ◽  
pp. 532-542 ◽  
Author(s):  
J.-B. Flór ◽  
H. Scolan ◽  
J. Gula
Keyword(s):  
Baroclinic Instability ◽  
Phase Speed ◽  
Small Scale ◽  
Immiscible Fluid ◽  
Mean Flow ◽  
Disk Rotation ◽  
Fluid Layers ◽  
The Stability ◽  
Almost All ◽  
Instability Regime

AbstractWe present an experimental investigation of the stability of a baroclinic front in a rotating two-layer salt-stratified fluid. A front is generated by the spin-up of a differentially rotating lid at the fluid surface. In the parameter space set by rotational Froude number, $F$, dissipation number, $d$ (i.e. the ratio between disk rotation time and Ekman spin-down time) and flow Rossby number, a new instability is observed that occurs for Burger numbers larger than the critical Burger number for baroclinic instability. This instability has a much smaller wavelength than the baroclinic instability, and saturates at a relatively small amplitude. The experimental results for the instability regime and the phase speed show overall a reasonable agreement with the numerical results of Gula, Zeitlin & Plougonven (J. Fluid Mech., vol. 638, 2009, pp. 27–47), suggesting that this instability is the Rossby–Kelvin instability that is due to the resonance between Rossby and Kelvin waves. Comparison with the results of Williams, Haines & Read (J. Fluid Mech., vol. 528, 2005, pp. 1–22) and Hart (Geophys. Fluid Dyn., vol. 3, 1972, pp. 181–209) for immiscible fluid layers in a small experimental configuration shows continuity in stability regimes in $(F, d)$ space, but the baroclinic instability occurs at a higher Burger number than predicted according to linear theory. Small-scale perturbations are observed in almost all regimes, either locally or globally. Their non-zero phase speed with respect to the mean flow, cusped-shaped appearance in the density field and the high values of the Richardson number for the observed wavelengths suggest that these perturbations are in many cases due to Hölmböe instability.

Influence of optimally amplified streamwise streaks on the Kelvin–Helmholtz instability

2018 ◽  
Vol 838 ◽  
pp. 478-500 ◽  
Author(s):  
Mathieu Marant ◽  
Carlo Cossu
Keyword(s):  
Initial Conditions ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Basic Flow ◽  
Sensitivity Analyses ◽  
Mean Flow ◽  
Helmholtz Instability ◽  
Flow Distortion ◽  
Streamwise Streaks ◽  
The Stability

The optimal energy amplifications of streamwise-uniform and spanwise-periodic perturbations of the hyperbolic-tangent mixing layer are computed and found to be very large, with maximum amplifications increasing with the Reynolds number and with the spanwise wavelength of the perturbations. The optimal initial conditions are streamwise vortices and the most amplified structures are streamwise streaks with sinuous symmetry in the cross-stream plane. The leading suboptimal perturbations have opposite (varicose) symmetry. When forced with finite amplitudes these perturbations modify the characteristics of the Kelvin–Helmholtz instability. Maximum temporal growth rates are reduced by optimal sinuous perturbations and are slightly increased by varicose suboptimal ones. In contrast, the onset of absolute instability is delayed by varicose suboptimal perturbations and is slightly promoted by sinuous optimal ones. We show that if, instead of the computed fully nonlinear basic-flow distortions, the stability analysis is based on a shape assumption for the flow distortions, then opposite effects on the flow stability are predicted in most of the considered cases. These strong differences are attributed to the spanwise-uniform component of the nonlinear basic-flow distortion which, we conclude, should be systematically included in sensitivity analyses of the stability of two-dimensional basic flows to three-dimensional basic-flow perturbations. We finally show that the leading-order quadratic sensitivity of the eigenvalues to the amplitude of the streaks is preserved if the effects of the mean flow distortion are included in the sensitivity analysis.

scholarly journals Restricted nonlinear model for high- and low-drag events in plane channel flow

2019 ◽  
Vol 864 ◽  
pp. 221-243 ◽  
Author(s):  
Frédéric Alizard ◽  
Damien Biau
Keyword(s):  
Channel Flow ◽  
Plane Channel ◽  
Wall Turbulence ◽  
Asymptotic State ◽  
State Variables ◽  
Mean Flow ◽  
Quadratic Interaction ◽  
Maximum Drag Reduction ◽  
The Mean ◽  
Plane Channel Flow

A restricted nonlinear (RNL) model, obtained by partitioning the state variables into streamwise-averaged quantities and superimposed perturbations, is used in order to track the exact coherent state in plane channel flow investigated by Toh & Itano (J. Fluid Mech., vol. 481, 2003, pp. 67–76). When restricting nonlinearities to quadratic interaction of the fluctuating part into the streamwise-averaged component, it is shown that the coherent structure and its dynamics closely match results from direct numerical simulation (DNS), even if only a single streamwise Fourier mode is retained. In particular, both solutions exhibit long quiescent phases, spanwise shifts and bursting events. It is also shown that the dynamical trajectory passes close to equilibria that exhibit either low- or high-drag states. When statistics are collected at times where the friction velocity peaks, the mean flow and root-mean-square profiles show the essential features of wall turbulence obtained by DNS for the same friction Reynolds number. For low-drag events, the mean flow profiles are related to a universal asymptotic state called maximum drag reduction (Xi & Graham, Phys. Rev. Lett., vol. 108, 2012, 028301). Hence, the intermittent nature of self-sustaining processes in the buffer layer is contained in the dynamics of the RNL model, organized in two exact coherent states plus an asymptotic turbulent-like attractor. We also address how closely turbulent dynamics approaches these equilibria by exploiting a DNS database associated with a larger domain.

scholarly journals Instability of supersonic compression ramp flow

2014 ◽  
Vol 372 (2020) ◽  
pp. 20130342
Author(s):  
R. P. Logue ◽  
J. S. B. Gajjar ◽  
A. I. Ruban
Keyword(s):  
Reynolds Number ◽  
Initial Conditions ◽  
Grid Size ◽  
Numerical Results ◽  
Mean Flow ◽  
Governing Equations ◽  
Global Modes ◽  
The Mean ◽  
Compression Ramp ◽  
The Stability

The instability of supersonic compression ramp flow is investigated. It is assumed that the Reynolds number is large and that the governing equations are the unsteady triple-deck equations. The mean flow is first calculated by solving the steady equations for various scaled ramp angles α , and the numerical results suggest that there is no singularity for increasing ramp angles. The stability of the flow is investigated using two approaches, first by solving the linearized unsteady equations and looking for global modes proportional to e λ t . In the second approach, the linearized unsteady equations are solved numerically with various initial conditions. Whereas no globally unsteady modes could be found for the range of ramp angles studied, the numerical simulations show the formation of wavepacket type disturbances which grow and convect and reach large amplitudes. However, the numerical results show large variations with grid size even on very fine grids.

scholarly journals Dynamics of Eddy-Driven Low-Frequency Dipole Modes. Part IV: Planetary and Synoptic Wave-Breaking Processes during the NAO Life Cycle

2008 ◽  
Vol 65 (3) ◽  
pp. 737-765 ◽  
Author(s):  
Dehai Luo ◽  
Tingting Gong ◽  
Yina Diao
Keyword(s):  
Life Cycle ◽  
Wave Breaking ◽  
Planetary Wave ◽  
Initial Conditions ◽  
Low Frequency ◽  
Phase Evolution ◽  
Life Cycles ◽  
Necessary Condition ◽  
Mean Flow ◽  
The North

Abstract Based on a highly idealized, analytical solution of the North Atlantic Oscillation (NAO) derived in Part III of this series, it is shown that wave breaking is not a necessary condition for the occurrence of NAO events. The breaking of synoptic waves can arise from the interaction between planetary and synoptic waves that gives rise to NAO events, and the type of wave breaking is dominated by the initial conditions of the two waves that determine the phase of the NAO. The planetary wave breaking (PWB) seems to be attributed to an amplification of the NAO amplitude. It is further found that both the planetary wave breaking and the cyclonic (anticyclonic) breaking of synoptic waves undergo an in-phase (out phase) evolution during the life cycles of negative (positive) phase NAO, or NAO− (NAO+), events. An interesting result found is that for NAO− (NAO+) events the breaking of synoptic waves is enhanced (weakened) during the growing phase, but is weakened (enhanced) during the decaying phase. In the absence of a topographic planetary wave (TPW), PWB occurs mainly in the midlatitude regions of the Atlantic basin for NAO− events, but is concentrated in subtropical and subpolar regions for NAO+ events. However, once the TPW is involved, the reversed planetary-scale potential vorticity (PV) gradient that characterizes the PWB exhibits a southwest–northeast (southeast–northwest) tilted tripole for NAO− (NAO+) events, in agreement with the diagnostic results presented herein. The PWB in the subtropical Atlantic is found to occur more frequently for NAO+ events than for NAO− events because the weaker subtropical mean flow is more likely to emerge during the NAO+ life cycle. In conclusion, the results of the highly idealized model used here appear to show that the PWB, synoptic wave breaking, and meridional shift of the westerly jet may be different descriptions of the NAO phenomenon.

Effect of an Azimuthal Mean Flow on the Structure and Stability of Thermoacoustic Modes in an Annular Combustor Model With Electroacoustic Feedback

2020 ◽  
Author(s):  
Sylvain C. Humbert ◽  
Jonas P. Moeck ◽  
Alessandro Orchini ◽  
Christian Oliver Paschereit
Keyword(s):  
Mach Number ◽  
Initial Conditions ◽  
Mean Flow ◽  
Final State ◽  
Mean Velocity ◽  
Acoustic Damping ◽  
Annular Combustor ◽  
The Mean ◽  
High Mach ◽  
The Stability

Abstract Thermoacoustic oscillations in axisymmetric annular combustors are generally coupled by degenerate azimuthal modes, which can be of standing or spinning nature. Symmetry breaking due to the presence of a mean azimuthal flow splits the degenerate thermoacoustic eigenvalues, resulting in pairs of counter-spinning modes with close but distinct frequencies and growth rates. In the present study, experiments have been performed using an annular system where the thermoacoustic feedback due to the flames is mimicked by twelve identical electroacoustic feedback loops. The mean azimuthal flow is generated by fans. We investigate the standing/spinning nature of the oscillations as a function of the azimuthal Mach number for two types of initial states, and how the stability of the system is affected by the mean azimuthal flow. It is found that spinning, standing or mixed modes can be encountered at very low Mach number, but increasing the mean velocity promotes one spinning direction. At sufficiently high Mach number, only spinning modes are observed in the limit cycle oscillations. In some cases, the initial conditions have a significant impact on the final state of the system. It is found that the presence of a mean azimuthal flow increases the acoustic damping. This has a beneficial effect on stability: it often reduces the amplitude of the self-sustained oscillations, and can even suppress them in some cases. However, we observe that the suppression of a mode due to the mean flow may destabilize another one. We discuss our findings in relation with an existing low-order model.

Sign in / Sign up

Export Citation Format

Share Document