Weakly Nonlinear Instability of a Liquid Jet in a Viscous Gas

1992 ◽  
Vol 59 (2S) ◽  
pp. S291-S296 ◽  
Author(s):  
E. A. Ibrahim ◽  
S. P. Lin
Keyword(s):  
Growth Rate ◽  
Normal Mode ◽  
Interfacial Stress ◽  
Linear Stability Theory ◽  
Liquid Jet ◽  
Nonlinear Instability ◽  
Weakly Nonlinear ◽  
Viscous Gas ◽  
Jet Breakup ◽  
Stress Fluctuation

The weakly nonlinear instability of a viscous liquid jet emanated into a viscous gas contained in a coaxial vertical circular pipe is investigated as an initial-value problem. The linear stability theory predicted that the jet may become unstable either due to capillary pinching or due to interfacial stress fluctuation. The results of nonlinear stability analysis shows no tendency of supercritical stability for both of the linearly unstable modes. In fact, the nonlinear growth rate of the disturbance is faster than the exponential growth rate of the linear normal mode disturbance for the same flow parameters. Moreover, the most amplified linear normal mode disturbance evolves nonlinearly into a nonsinusoidal wave of shorter wavelength. No nonlinear instability is found for the linearly stable disturbances. Thus, while the linear theory is adequate for the prediction of the onset of jet breakup, nonlinear theory is required to describe the outcome of the jet breakup.

scholarly journals Generation of Wave Groups by Shear Layer Instability

Fluids ◽  
2019 ◽  
Vol 4 (1) ◽  
pp. 39 ◽  
Author(s):  
Roger Grimshaw
Keyword(s):  
Growth Rate ◽  
Group Velocity ◽  
Linear Growth ◽  
Linear Stability Theory ◽  
Linear Growth Rate ◽  
Wave Groups ◽  
Weakly Nonlinear ◽  
Complex Valued ◽  
Fundamental Requirement ◽  
Temporal Growth Rate

The linear stability theory of wind-wave generation is revisited with an emphasis on the generation of wave groups. The outcome is the fundamental requirement that the group move with a real-valued group velocity. This implies that both the wave frequency and the wavenumber should be complex-valued, and in turn this then leads to a growth rate in the reference frame moving with the group velocity which is in general different from the temporal growth rate. In the weakly nonlinear regime, the amplitude envelope of the wave group is governed by a forced nonlinear Schrödinger equation. The effect of the wind forcing term is to enhance modulation instability both in terms of the wave growth and in terms of the domain of instability in the modulation wavenumber space. Also, the soliton solution for the wave envelope grows in amplitude at twice the linear growth rate.

DETECTION OF AERODYNAMIC EFFECTS IN LIQUID JET BREAKUP AND DROPLET FORMATION

1999 ◽  
Vol 9 (4) ◽  
pp. 331-342 ◽  
Author(s):  
Michael P. Moses ◽  
Steven H. Collicott ◽  
Stephen D. Heister
Keyword(s):  
Droplet Formation ◽  
Liquid Jet ◽  
Jet Breakup

Simulation of Laminar Liquid Jets in Gaseous Crossflow Before the Onset of Primary Breakup

2011 ◽  
Author(s):  
C.-L. Ng ◽  
K. A. Sallam
Keyword(s):  
Experimental Data ◽  
Fuel Injection ◽  
Liquid Jets ◽  
Liquid Jet ◽  
Jet Breakup ◽  
Primary Breakup ◽  
Density Ratios ◽  
First Time ◽  
Two Sides ◽  
Reduced Pressures

The deformation of laminar liquid jets in gaseous crossflow before the onset of primary breakup is studied motivated by its application to fuel injection in jet afterburners and agricultural sprays, among others. Three crossflow Weber numbers that represent three different liquid jet breakup regimes; column, bag, and shear breakup regimes, were studied at large liquid/gas density ratios and small Ohnesorge numbers. In each case the liquid jet was simulated from the jet exit and ended before the location where the experimental data indicated the onset of breakup. The results show that in column and bag breakup, the reduced pressures along the sides of the jet cause the liquid to move to the sides of the jet and enhance the jet deformation. In shear breakup, the flattened upwind surface pushes the liquid towards the two sides of the jet and causing the gaseous crossflow to separate near the edges of the liquid jet thus preventing further deformation before the onset of breakup. It was also found out that in shear breakup regime, the liquid phase velocity inside the liquid jet was large enough to cause onset of ligament formation along the jet side, which was not the case in the column and bag breakup regimes. In bag breakup, downwind surface waves were observed to grow along the sides of the liquid jet triggered a complimentary experimental study that confirmed the existence of those waves for the first time.

scholarly journals Stability of a Viscous Liquid Jet in a Coaxial Twisting Compressible Airflow

Processes ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 918
Author(s):  
Li-Mei Guo ◽  
Ming Lü ◽  
Zhi Ning
Keyword(s):  
Viscous Liquid ◽  
Axial Velocity ◽  
Liquid Velocity ◽  
Velocity Ratio ◽  
Dominant Mode ◽  
Liquid Jet ◽  
Axisymmetric Mode ◽  
Jet Instability ◽  
Jet Stability ◽  
Jet Breakup

Based on the linear stability analysis, a mathematical model for the stability of a viscous liquid jet in a coaxial twisting compressible airflow has been developed. It takes into account the twist and compressibility of the surrounding airflow, the viscosity of the liquid jet, and the cavitation bubbles within the liquid jet. Then, the effects of aerodynamics caused by the gas–liquid velocity difference on the jet stability are analyzed. The results show that under the airflow ejecting effect, the jet instability decreases first and then increases with the increase of the airflow axial velocity. When the gas–liquid velocity ratio A = 1, the jet is the most stable. When the gas–liquid velocity ratio A > 2, this is meaningful for the jet breakup compared with A = 0 (no air axial velocity). When the surrounding airflow swirls, the airflow rotation strength E will change the jet dominant mode. E has a stabilizing effect on the liquid jet under the axisymmetric mode, while E is conducive to jet instability under the asymmetry mode. The maximum disturbance growth rate of the liquid jet also decreases first and then increases with the increase of E. The liquid jet is the most stable when E = 0.65, and the jet starts to become more easier to breakup when E = 0.8425 compared with E = 0 (no swirling air). When the surrounding airflow twists (air moves in both axial and circumferential directions), given the axial velocity to change the circumferential velocity of the surrounding airflow, it is not conducive to the jet breakup, regardless of the axisymmetric disturbance or asymmetry disturbance.

Weakly nonlinear instability of planar viscoelastic sheets

2015 ◽  
Vol 27 (1) ◽  
pp. 013103 ◽  
Author(s):  
Chen Wang ◽  
Li-jun Yang ◽  
Luo Xie ◽  
Pi-min Chen
Keyword(s):  
Nonlinear Instability ◽  
Weakly Nonlinear

Suppression of Baroclinic Instabilities in Buoyancy-Driven Flow over Sloping Bathymetry

2017 ◽  
Vol 47 (1) ◽  
pp. 49-68 ◽  
Author(s):  
Robert D. Hetland
Keyword(s):  
Growth Rate ◽  
Linear Instability ◽  
Linear Stability Theory ◽  
Buoyancy Frequency ◽  
Bottom Slope ◽  
Coriolis Parameter ◽  
Plume Front ◽  
Instability Theory ◽  
Instability Growth ◽  
Flow Configurations

AbstractBaroclinic instabilities are ubiquitous in many types of geostrophic flow; however, they are seldom observed in river plumes despite strong lateral density gradients within the plume front. Supported by results from a realistic numerical simulation of the Mississippi–Atchafalaya River plume, idealized numerical simulations of buoyancy-driven flow are used to investigate baroclinic instabilities in buoyancy-driven flow over a sloping bottom. The parameter space is defined by the slope Burger number S = Nf−1α, where N is the buoyancy frequency, f is the Coriolis parameter, and α is the bottom slope, and the Richardson number Ri = N2f2M−4, where M2 = |∇Hb| is the magnitude of the lateral buoyancy gradients. Instabilities only form in a subset of the simulations, with the criterion that SH ≡ SRi−1/2 = Uf−1W−1 = M2f−2α 0.2, where U is a horizontal velocity scale and SH is a new parameter named the horizontal slope Burger number. Suppression of instability formation for certain flow conditions contrasts linear stability theory, which predicts that all flow configurations will be subject to instabilities. The instability growth rate estimated in the nonlinear 3D model is proportional to ωImaxS−1/2, where ωImax is the dimensional growth rate predicted by linear instability theory, indicating that bottom slope inhibits instability growth beyond that predicted by linear theory. The constraint SH 0.2 implies a relationship between the inertial radius Li = Uf−1 and the plume width W. Instabilities may not form when 5Li > W; that is, the plume is too narrow for the eddies to fit.

scholarly journals Effects of Asymmetric Gas Distribution on the Instability of a Plane Power-Law Liquid Jet

Energies ◽  
2018 ◽  
Vol 11 (7) ◽  
pp. 1854 ◽  
Author(s):  
Jin-Peng Guo ◽  
Yi-Bo Wang ◽  
Fu-Qiang Bai ◽  
Fan Zhang ◽  
Qing Du
Keyword(s):  
Power Law ◽  
Liquid Velocity ◽  
Linear Instability ◽  
Liquid Jets ◽  
Liquid Jet ◽  
Critical Curves ◽  
Jet Breakup ◽  
Plane Jets ◽  
Gas Space ◽  
Aerodynamic Asymmetry

As a kind of non-Newtonian fluid with special rheological features, the study of the breakup of power-law liquid jets has drawn more interest due to its extensive engineering applications. This paper investigated the effect of gas media confinement and asymmetry on the instability of power-law plane jets by linear instability analysis. The gas asymmetric conditions mainly result from unequal gas media thickness and aerodynamic forces on both sides of a liquid jet. The results show a limited gas space will strengthen the interaction between gas and liquid and destabilize the power-law liquid jet. Power-law fluid is easier to disintegrate into droplets in asymmetric gas medium than that in the symmetric case. The aerodynamic asymmetry destabilizes para-sinuous mode, whereas stabilizes para-varicose mode. For a large Weber number, the aerodynamic asymmetry plays a more significant role on jet instability compared with boundary asymmetry. The para-sinuous mode is always responsible for the jet breakup in the asymmetric gas media. With a larger gas density or higher liquid velocity, the aerodynamic asymmetry could dramatically promote liquid disintegration. Finally, the influence of two asymmetry distributions on the unstable range was analyzed and the critical curves were obtained to distinguish unstable regimes and stable regimes.

scholarly journals Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2. Stability of cnoidal waves

2007 ◽  
Vol 73 (6) ◽  
pp. 933-946
Author(s):  
S. PHIBANCHON ◽  
M. A. ALLEN ◽  
G. ROWLANDS
Keyword(s):  
Growth Rate ◽  
Nonlinear Waves ◽  
Electron Distribution ◽  
Magnetized Plasma ◽  
Weakly Nonlinear ◽  
Long Wavelength ◽  
First Order ◽  
Wave Solutions ◽  
Linear Waves ◽  
Weakly Nonlinear Waves

AbstractWe determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the Schamel–Korteweg–de Vries–Zakharov–Kuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to a minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero, and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio b increases as the electron distribution becomes increasingly flat-topped. As b and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle θ at which the perturbation is applied. Solutions whose minimum values are zero and which travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of θ for which the first-order growth rate is not zero.

Long-wave instability of a helical vortex

2015 ◽  
Vol 780 ◽  
pp. 687-716 ◽  
Author(s):  
Hugo Umberto Quaranta ◽  
Hadrien Bolnot ◽  
Thomas Leweke
Keyword(s):  
Growth Rate ◽  
Water Channel ◽  
Vortex Filament ◽  
Linear Stability Theory ◽  
Wave Instability ◽  
Characteristic Distance ◽  
Long Wave ◽  
Instability Modes ◽  
Small Pitch ◽  
Helical Vortex

We investigate the instability of a single helical vortex filament of small pitch with respect to displacement perturbations whose wavelength is large compared to the vortex core size. We first revisit previous theoretical analyses concerning infinite Rankine vortices, and consider in addition the more realistic case of vortices with Gausssian vorticity distributions and axial core flow. We show that the various instability modes are related to the local pairing of successive helix turns through mutual induction, and that the growth rate curve can be qualitatively and quantitatively predicted from the classical pairing of an array of point vortices. We then present results from an experimental study of a helical vortex filament generated in a water channel by a single-bladed rotor under carefully controlled conditions. Various modes of displacement perturbations could be triggered by suitable modulation of the blade rotation. Dye visualisations and particle image velocimetry allowed a detailed characterisation of the vortex geometry and the determination of the growth rate of the long-wave instability modes, showing good agreement with theoretical predictions for the experimental base flow. The long-term (downstream) development of the pairing instability leads to a grouping and swapping of helix loops. Despite the resulting complicated three-dimensional structure, the vortex filaments surprisingly remain mostly intact in our observation interval. The characteristic distance of evolution of the helical wake behind the rotor decreases with increasing initial amplitude of the perturbations; this can be predicted from the linear stability theory.

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