Weakly nonlinear instability of planar viscous sheets

2013 ◽  
Vol 735 ◽  
pp. 249-287 ◽  
Author(s):  
Lijun Yang ◽  
Chen Wang ◽  
Qingfei Fu ◽  
Minglong Du ◽  
Mingxi Tong
Keyword(s):  
Growth Rate ◽  
Reynolds Number ◽  
Weber Number ◽  
Density Ratio ◽  
Second Order ◽  
Liquid Density ◽  
Interface Deformation ◽  
Stabilizing Effect ◽  
Breakup Time ◽  
Spatial Growth Rate

AbstractA second-order instability analysis has been performed for sinuous disturbances on two-dimensional planar viscous sheets moving in a stationary gas medium using a perturbation technique. The solutions of second-order interface disturbances have been derived for both temporal instability and spatial instability. It has been found that the second-order interface deformation of the fundamental sinuous wave is varicose or dilational, causing disintegration and resulting in ligaments which are interspaced by half a wavelength. The interface deformation has been presented; the breakup time for temporal instability and breakup length for spatial instability have been calculated. An increase in Weber number and gas-to-liquid density ratio extensively increases both the temporal or spatial growth rate and the second-order initial disturbance amplitude, resulting in a shorter breakup time or length, and a more distorted surface deformation. Under normal conditions, viscosity has a stabilizing effect on the first-order temporal or spatial growth rate, but it plays a dual role in the second-order disturbance amplitude. The overall effect of viscosity is minor and complicated. In the typical condition, in which the Weber number is 400 and the gas-to-liquid density ratio is 0.001, viscosity has a weak stabilizing effect when the Reynolds number is larger than 150 or smaller than 10; when the Reynolds number is between 150 and 10, viscosity has a weak destabilizing effect.

Nonlinear instability of plane liquid sheets

2000 ◽  
Vol 406 ◽  
pp. 281-308 ◽  
Author(s):  
SEYED A. JAZAYERI ◽  
XIANGUO LI
Keyword(s):  
Weber Number ◽  
Density Ratio ◽  
Perturbation Expansion ◽  
Liquid Sheet ◽  
Fundamental Wave ◽  
Liquid Density ◽  
Initial Amplitude ◽  
Nonlinear Stability Analysis ◽  
Liquid Sheets ◽  
Breakup Time

A nonlinear stability analysis has been carried out for plane liquid sheets moving in a gas medium at rest by a perturbation expansion technique with the initial amplitude of the disturbance as the perturbation parameter. The first, second and third order governing equations have been derived along with appropriate initial and boundary conditions which describe the characteristics of the fundamental, and the first and second harmonics. The results indicate that for an initially sinusoidal sinuous surface disturbance, the thinning and subsequent breakup of the liquid sheet is due to nonlinear effects with the generation of higher harmonics as well as feedback into the fundamental. In particular, the first harmonic of the fundamental sinuous mode is varicose, which causes the eventual breakup of the liquid sheet at the half-wavelength interval of the fundamental wave. The breakup time (or length) of the liquid sheet is calculated, and the effect of the various flow parameters is investigated. It is found that the breakup time (or length) is reduced by an increase in the initial amplitude of disturbance, the Weber number and the gas-to-liquid density ratio, and it becomes asymptotically insensitive to the variations of the Weber number and the density ratio when their values become very large. It is also found that the breakup time (or length) is a very weak function of the wavenumber unless it is close to the cut-off wavenumbers.

Absolute and convective instability of a liquid sheet

1990 ◽  
Vol 220 ◽  
pp. 673-689 ◽  
Author(s):  
S. P. Lin ◽  
Z. W. Lian ◽  
B. J. Creighton
Keyword(s):  
Weber Number ◽  
Convective Instability ◽  
Density Ratio ◽  
Sheet Thickness ◽  
Surface Tension Force ◽  
Liquid Sheet ◽  
Inertia Force ◽  
Liquid Density ◽  
Gas To Liquid ◽  
Density Ratios

The linear stability of a viscous liquid sheet in the presence of ambient gas is investigated. It is shown that there are two independent modes of instability, sinuous and varicose. The large-time asymptotic amplitude of sinuous disturbances is found to be bounded but non-vanishing for all calculated values of Reynolds numbers and the gas-to-liquid density ratios when the Weber number is greater than one half. The Weber numberWeis defined as the ratio of the surface tension force to the inertia force per unit area of the gas–liquid interface. WhenWeis smaller than one half, the sinuous mode is stable if the gas-to-liquid density ratio is zero, otherwise it is convectively unstable. The varicose mode is always convectively unstable unless the density ratio,Q, is zero. Then it is asymptotically stable. The spatial growth rate of the varicose mode is smaller than that of the sinuous mode for the same flow parameters. The wavelength of the most amplified waves in both modes is found to scale with the product of the sheet thickness andQ/We. It is shown, by use of the energy equation, that the mechanism of instability is a capillary rupture whenWe[ges ] 0.5, and the convective instability is due to the interfacial pressure fluctuation whenWe< 0.5.

scholarly journals Simulation of droplet impacting a square solid obstacle in microchannel with different wettability by using high density ratio pseudopotential multiple-relaxation-time (MRT) lattice Boltzmann method (LBM)

2019 ◽  
Vol 97 (1) ◽  
pp. 93-113 ◽  
Author(s):  
Wandong Zhao ◽  
Ying Zhang ◽  
Wenqiang Shang ◽  
Zhaotai Wang ◽  
Ben Xu ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Relaxation Time ◽  
Liquid Film ◽  
External Force ◽  
Lattice Boltzmann ◽  
Weber Number ◽  
Droplet Diameter ◽  
Density Ratio ◽  
Force Term ◽  
High Density Ratio

In this paper, a pseudopotential high density ratio (DR) lattice Boltzmann model was developed by incorporating multi-relaxation-time collision matrix, large DR external force term, surface tension adjustment external force term, and solid–liquid pseudopotential force. It was found that the improved model can precisely capture the two-phase interface at high DR. Besides, the effects of initial Reynolds number, Weber number, solid wall contact angle (CA), ratio of obstacle size to droplet diameter (χ1), and ratio of channel width to droplet diameter (χ2) on the deformation and breakup of a droplet when impacting on a square obstacle were investigated. The results showed that with the Reynolds number increasing, the droplet will fall along the obstacle and then spread along both sides of the obstacle. Furthermore, by increasing Weber number, the breakup of the liquid film will be delayed and the liquid film will be stretched to form an elongated ligament. With decreasing of the wettability of solid particle (CA → 180°), the droplet will surround the obstacle and then detach from the obstacle. When χ1 is greater than 0.5, the droplet will spread along both sides of the obstacle quickly; otherwise, the droplet will be ruptured earlier. Furthermore, when χ2 decreases, the droplet will spread earlier and then fall along the wall more quickly; otherwise, the droplet will expand along both sides of the obstacle. Moreover, increasing the hydrophilicity of the microchannel, the droplet will impact the channel more rapidly and infiltrate the wall along the upstream and downstream simultaneously; on the contrary, the droplet will wet downstream only.

Investigations on the Droplet Impact onto a Spherical Surface with a High Density Ratio Multi-Relaxation Time Lattice-Boltzmann Model

2014 ◽  
Vol 16 (4) ◽  
pp. 892-912 ◽  
Author(s):  
Duo Zhang ◽  
K. Papadikis ◽  
Sai Gu
Keyword(s):  
Reynolds Number ◽  
Relaxation Time ◽  
Lattice Boltzmann ◽  
Weber Number ◽  
Spherical Surface ◽  
Density Ratio ◽  
Droplet Impact ◽  
High Density ◽  
Lattice Boltzmann Model ◽  
Boltzmann Model

AbstractIn the current study, a two-dimensional multi-relaxation time (MRT) lattice Boltzmann model which can tolerate high density ratios and low viscosity is employed to simulate the liquid droplet impact onto a curved target. The temporal variation of the film thickness at the north pole of the target surface is investigated. Three different temporal phases of the dynamics behavior, namely, the initial drop deformation phase, the inertia dominated phase and the viscosity dominated phase are reproduced and studied. The effect of the Reynolds number, Weber number and Galilei number on the film flow dynamics is investigated. In addition, the dynamic behavior of the droplet impact onto the side of the curved target is shown, and the effect of the contact angle, the Reynolds number and the Weber number are investigated.

Nonlinear dual-mode instability of planar liquid sheets

2015 ◽  
Vol 778 ◽  
pp. 621-652 ◽  
Author(s):  
Chen Wang ◽  
Lijun Yang ◽  
Hanyu Ye
Keyword(s):  
Nonlinear Analysis ◽  
Density Ratio ◽  
Boundary Integral ◽  
Second Order ◽  
Initial Disturbance ◽  
Liquid Density ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis ◽  
Liquid Sheets ◽  
Break Up

The nonlinear temporal instability of gas-surrounded planar liquid sheets, whose linear instability contains both sinuous and varicose modes, is studied. Both the weakly nonlinear analysis using a second-order perturbation expansion and the numerical simulation using a boundary integral method have been applied. Their comparison shows that the weakly nonlinear analysis can precisely predict the shapes of sheets for most of the time of disturbance evolution and qualitatively explain the instability mechanism when sheets break up. Both the first harmonics of the linear sinuous mode and linear varicose mode are varicose; they contribute to the breakup of sheets, but the first harmonic generated by the coupling between the linear sinuous and varicose modes is sinuous; it plays an important role in modulating the wave profile. The instability with various initial phase differences between the upper and lower interfaces is examined. Except for the varicose initial disturbance, the linear sinuous mode dominates in the shapes of sheets when their amplitudes grow large. Within the second-order analysis, the major modes that can cause the breakup include the linear varicose mode, the first harmonic of the linear sinuous mode and the first harmonic of the linear varicose mode. The effects of various flow parameters have been investigated. At relatively large wavenumbers where approximate analytical and numerical results agree well when sheets break up, increasing the wavenumber reduces the wave amplitude. Reducing the initial disturbance amplitude makes the first harmonic of the linear sinuous mode the dominant mode in causing the breakup. Increasing the Weber number or gas-to-liquid density ratio significantly reduces breakup time and enhances instability.

Spatial-Temporal Stability of an Electrified Viscoelastic Liquid Jet

2013 ◽  
Vol 135 (9) ◽  
Author(s):  
Qing-fei Fu ◽  
Li-jun Yang ◽  
Pi-min Chen ◽  
Yu-xin Liu ◽  
Chen Wang
Keyword(s):  
Growth Rate ◽  
Reynolds Number ◽  
Convective Instability ◽  
Density Ratio ◽  
Temporal Stability ◽  
Viscoelastic Liquid ◽  
Liquid Jet ◽  
Absolute Growth Rate ◽  
Temporal Instability ◽  
Absolute Growth

This paper presents theoretically the spatial-temporal instability behavior of an electrified viscoelastic liquid jet. Dimensionless parameters have been tested for their influence on the transition of absolute and convective instability for the electrified viscoelastic liquid jet. The results show that larger electrical Euler and Weber numbers can change the flow to convectively unstable. The increase of Reynolds number can decrease the absolute growth rate. Variations of time constant and density ratio rarely change the spatial-temporal instability behavior of the jet. The disturbance wavelength changes very little with these parameters when the flow is absolutely unstable.

Numerical Investigation on Head-On Collisions of Binary Micro-Droplets by an Improved Multiphase Lattice Boltzmann Flux Solver

2016 ◽  
Author(s):  
Y. Wang ◽  
C. Shu
Keyword(s):  
Reynolds Number ◽  
Lattice Boltzmann ◽  
Weber Number ◽  
Density Ratio ◽  
Distribution Functions ◽  
Time Step ◽  
Large Density ◽  
Wide Range ◽  
Large Density Ratio ◽  
Critical Weber Number

Head-on collisions of binary micro-droplets are of great interest in both academic research and engineering applications. Numerical simulation of this problem is challenging due to complex interfacial changes and large density ratio between different fluids. In this work, the recently proposed lattice Boltzmann flux solver (LBFS) is applied to study this problem. The LBFS is a finite volume method for the direct update of macroscopic flow variables at cell centers. The fluxes of the LBFS are reconstructed at each cell interface through lattice moments of density distribution functions (DDFs). As compared with conventional multiphase lattice Boltzmann method, the LBFS can be easily applied to study complex multiphase flows with large density ratio. In addition, external forces can be implemented more conveniently and the tie-up between the time step and mesh spacing is also removed. Moreover, it can deal with complex boundary conditions directly as those do in the conventional Navier-Stokes solvers. At first, the reliability of the LBFS is validated by simulating a micro-droplet impacting on a dry surface at density ratio 832 (air to water). The obtained result agrees well with experimental measurement. After that, numerical simulations of head-on collisions of two micro droplets are carried out to examine different collisional behaviors in a wide range of Reynolds numbers and Weber numbers of 100 ≤ Re ≤ 2000 and 10 ≤ We ≤ 500. A phase diagram parameterized by these two control parameters is obtained to classify the outcomes of these collisions. It is shown that, at low Reynolds number (Re=100), two droplets will be coalescent into a bigger one for all considered Weber numbers. With the increase of the Reynolds number, separation of the collision into multiple droplets appears and the critical Weber number for separation is decreased. When the Reynolds number is sufficiently high, the critical Weber number for separation is between 20 and 25.

scholarly journals Effects of flexibility on the aerodynamic performance of flapping wings

2011 ◽  
Vol 689 ◽  
pp. 32-74 ◽  
Author(s):  
C.-K. Kang ◽  
H. Aono ◽  
C. E. S. Cesnik ◽  
W. Shyy
Keyword(s):  
Reynolds Number ◽  
Density Ratio ◽  
Flapping Wings ◽  
Mass Force ◽  
Propulsive Force ◽  
Propulsive Efficiency ◽  
Reduced Frequency ◽  
Moving Body ◽  
Order Of Magnitude ◽  
Direct Guidance

AbstractEffects of chordwise, spanwise, and isotropic flexibility on the force generation and propulsive efficiency of flapping wings are elucidated. For a moving body immersed in viscous fluid, different types of forces, as a function of the Reynolds number, reduced frequency (k), and Strouhal number (St), acting on the moving body are identified based on a scaling argument. In particular, at the Reynolds number regime of $O(1{0}^{3} \ensuremath{-} 1{0}^{4} )$ and the reduced frequency of $O(1)$, the added mass force, related to the acceleration of the wing, is important. Based on the order of magnitude and energy balance arguments, a relationship between the propulsive force and the maximum relative wing-tip deformation parameter ($\gamma $) is established. The parameter depends on the density ratio, St, k, natural and flapping frequency ratio, and flapping amplitude. The lift generation, and the propulsive efficiency can be deduced by the same scaling procedures. It seems that the maximum propulsive force is obtained when flapping near the resonance, whereas the optimal propulsive efficiency is reached when flapping at about half of the natural frequency; both are supported by the reported studies. The established scaling relationships can offer direct guidance for micro air vehicle design and performance analysis.

Numerical Simulation of Droplet Generation in Series Co-Flow Capillaries

2009 ◽  
Author(s):  
Yanxi Song ◽  
Jinliang Xu
Keyword(s):  
Reynolds Number ◽  
Disperse Phase ◽  
Weber Number ◽  
Three Dimensional ◽  
Continuous Phase ◽  
Disperse Flow ◽  
Double Emulsions ◽  
Wide Range ◽  
In Series ◽  
Dispersed Phases

We study the production and motion of monodisperse double emulsions in microfluidics comprising series co-flow capillaries. Both two and three dimensional simulations are performed. Flow was determined by dimensionless parameters, i.e., Reynolds number and Weber number of continuous and dispersed phases. The co-flow generated droplets are sensitive to the Reynolds number and Weber number of the continuous phase, but insensitive to those of the disperse phase. Because the inner and outer drops are generate by separate co-flow processes, sizes of both inner and outer drops can be controlled by adjusting Re and We for the continuous phase. Meanwhile, the disperse phase has little effect on drop size, thus a desirable generation frequency of inner drop can be reached by merely adjusting flow rate of the inner fluid, leading to desirable number of inner drops encapsulated by the outer drop. Thus highly monodisperse double emulsions are obtained. It was found that only in dripping mode can droplet be of high mono-dispersity. Flow begins to transit from dripping regime to jetting regime when the Re number is decreased or Weber number is increased. To ensure that all the droplets are produced over a wide range of running parameters, tiny tapered tip outlet for the disperse flow should be applied. Smaller the tapered tip, wider range for Re and we can apply.

A Numerical Simulation Algorithm of the Inviscid Dynamics of Axisymmetric Swirling Flows in a Pipe

2016 ◽  
Vol 138 (9) ◽  
Author(s):  
J. Granata ◽  
L. Xu ◽  
Z. Rusak ◽  
S. Wang
Keyword(s):  
Numerical Simulation ◽  
Reynolds Number ◽  
Difference Scheme ◽  
Second Order ◽  
Swirling Flows ◽  
Inviscid Limit ◽  
Simulation Algorithm ◽  
Spatial Integration ◽  
Order Difference ◽  
Breakdown Process

Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (Re < 6000); however, the characteristic Reynolds number is much higher (Re > 20,000) in most of engineering applications. To address this difficulty, this paper presents a numerical simulation algorithm of the dynamics of incompressible, inviscid-limit, axisymmetric swirling flows in a pipe, including the vortex breakdown process. It is based on an explicit, first-order difference scheme in time and an upwind, second-order difference scheme in space for the time integration of the circulation and azimuthal vorticity. A second-order Poisson equation solver for the spatial integration of the stream function in terms of azimuthal vorticity is used. In addition, when reversed flow zones appear, an averaging step of properties is applied at designated time steps. This adds slight artificial viscosity to the algorithm and prevents growth of localized high-frequency numerical noise inside the breakdown zone that is related to the expected singularity that must appear in any flow simulation based on the Euler equations. Mesh refinement studies show agreement of computations for various mesh sizes. Computed examples of flow dynamics demonstrate agreement with linear and nonlinear stability theories of vortex flows in a finite-length pipe. Agreement is also found with theoretically predicted steady axisymmetric breakdown states in a pipe as flow evolves to a time-asymptotic state. These findings indicate that the present algorithm provides an accurate prediction of the inviscid-limit, axisymmetric breakdown process. Also, the numerical results support the theoretical predictions and shed light on vortex dynamics at high Re.

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