Absolute and convective instability of a liquid sheet with transverse temperature gradient

The spatial–temporal instability behavior of a viscous liquid sheet with temperature difference between the two surfaces was investigated theoretically. The practical situation motivating this investigation is liquid sheet heated by ambient gas, usually encountered in industrial heat transfer and liquid propellant rocket engines. The existing dispersion relation was used, to explore the spatial–temporal instability of viscous liquid sheets with a nonuniform temperature profile, by setting both the wave number and frequency complex. A parametric study was performed in both sinuous and varicose modes to test the influence of dimensionless numbers on the transition between absolute and convective instability of the flow. For a small value of liquid Weber number, or a great value of gas-to-liquid density ratio, the flow was found to be absolutely unstable. The absolute instability was enhanced by increasing the liquid viscosity. It was found that variation of the Marangoni number hardly influenced the absolute instability of the sinuous mode of oscillations; however it slightly affected the absolute instability in the varicose mode.     

Linear Stability Analysis of a Viscous Liquid Sheet in a High-Speed Viscous Gas

Air-assisted atomizers in which a thin liquid sheet is deformed under the action of a high-speed air flow are extensively used in industrial applications, e.g., in aircraft turbojet injectors. Primary atomization in these devices is a consequence of the onset and growth of instabilities on the air/liquid interfaces. To better understand this process, a temporal linear instability analysis is applied to a thin planar liquid sheet flowing between two semi-infinite streams of a high-speed viscous gas. This study includes the full viscous effects both in the liquid and gas basic states and perturbations. The relevant dimensionless groups entering the non-dimensional Orr–Sommerfeld equations and boundary conditions are the liquid and gas stream Reynolds numbers, the gas to liquid momentum flux ratio, the gas/liquid velocity ratio, the Weber number and the equivalent gas boundary layer to liquid sheet thickness ratio. Growth rates and temporal frequencies as a function of the wave number, varying the different dimensionless parameters are presented, together with neutral stability curves. From the results of this parametric study it is concluded that when the physical properties of gas and liquid are fixed, the momentum flux ratio is especially relevant to determine the instability conditions. It is also observed that the gas boundary layer thickness strongly influences the wave propagation, and acts by damping sheet oscillation frequency and growth. This is especially important because viscosity in the basic gas velocity profile has always been ignored in instability analysis applied to the geometry under study. This revised version was published online in July 2006 with corrections to the Cover Date.     

Instability of a viscoelastic liquid jet with axisymmetric and asymmetric disturbances

The temporal instability behavior of a viscoelastic liquid jet in the wind-induced regime with axisymmetric and asymmetric disturbances moving in an inviscid gaseous environment is investigated theoretically. The corresponding dispersion relation between the wave growth rate and the wavenumber is derived. The linear instability analysis shows that viscoelastic liquid jets are more unstable than their Newtonian counterparts, and less unstable than their inviscid counterparts, for both axisymmetric and asymmetric disturbances, respectively. The instability behavior of viscoelastic jets is influenced by the interaction of liquid viscosity and elasticity, in which the viscosity tends to dampen the instability, whereas the elasticity results in an enhancement of instability. Relatively, the effect of the ratio of deformation retardation to stress relaxation time on the instability of viscoelastic jets is weak. It is found that the liquid Weber number is a key measure that controls the viscoelastic jet instability behavior. At small Weber number, the axisymmetric disturbance dominates the instability of viscoelastic jets, i.e., the growth rate of an axisymmetric disturbance exceeds that of asymmetric disturbances. When the Weber number increases, both the growth rate and the instability range of disturbances increase drastically. The asymptotic analysis shows that at large Weber number, more asymmetric disturbance modes become unstable, and the growth rate of each asymmetric disturbance mode approaches that of the axisymmetric disturbance. Therefore, the asymmetric disturbances are more dangerous than that of axisymmetric disturbances for a viscoelastic jet at large Weber numbers. Similar to the liquid Weber number, the ratio of gas to liquid density is another key measure that affects the viscoelastic jet instability behavior substantially.     

Formation of radially expanding liquid sheet by impinging two round jets

A thin circular liquid sheet can be formed by impinging two identical round jets against each other. The liquid sheet expands to a certain critical radial distance and breaks. The unsteady process of the formation and breakup of the liquid sheet in the ambient gas is simulated numerically. Both liquid and gas are treated as incompressible Newtonian fluids. The flow considered is axisymmetric. The liquid-gas interface is modeled with a level set function. A finite difference scheme is used to solve the governing Navier-Stokes equations with physical boundary conditions. The numerical results show how a thin circular sheet can be formed and break at its circular edge in slow motion. The sheet continues to thin as it expands radially. Hence, the Weber number decreases radially. The Weber number is defined as ρu 2 h/σ, where ρ and σ are, respectively, the liquid density and the surface tension, and u and h are, respectively, the average velocity and the half sheet thickness at a local radial location in the liquid sheet. The numerical results show that the sheet indeed terminates at a radial location, where the Weber number reaches one as observed in experiments. The spatio-temporal linear theory predicts that the breakup is initiated by the sinuous mode at the critical Weber number We c =1, below which the absolute instability occurs. The other independent mode called the varicose mode grows more slowly than the sinuous mode according to the linear theory. However, our numerical results show that the varicose mode actually overtakes the sinuous mode during the nonlinear evolution, and is responsible for the final breakup. The linear theory predicts the nature of disturbance waves correctly only at the onset of the instability, but cannot predict the exact consequence of the instability.     

圆环旋转黏性液体射流空间不稳定性研究

利用线性稳定性理论, 进行了液体黏性对不同旋转强度下圆环旋转液体射流 空间不稳定性影响的研究. 在推导出的三维扰动下具有固体涡核型旋转速度分布的圆环旋转 黏性液体射流色散方程的基础上, 针对中低速射流, 进行了类反对称模式与类对称模式下圆 环旋转黏性液体射流的空间不稳定性分析. 研究结果表明, 对于旋转强度较大的圆环旋转液 体射流, 液体黏性的增加, 不利于射流的破碎; 随着液体黏性的增加, 射流的特征频率和最 不稳定波数减小. 然而, 对于旋转强度较小的圆环旋转液体射流, 液体黏性的增加, 有利于 射流的破碎; 随着液体黏性的增加, 类反对称模式下射流特征频率先减小后增大, 类对称模 式下射流特征频率增大; 随着液体黏性的增加, 类反对称模式下射流最不稳定波数先减小后 增大, 类对称模式下射流最不稳定波数增大.     

Nonlinear instability of an annular liquid sheet exposed to gas flow

Nonlinear instability and breakup of an annular liquid sheet has been modeled in this paper. The liquid sheet is considered to move axially and is exposed to co-flowing inner and outer gas streams. Also, the effect of outer gas swirl on sheet breakup has been studied. In the developed model a perturbation expansion method has been used with the initial magnitude of the disturbance as the perturbation parameter. This is a comprehensive model in that other geometries of planar sheet and a coaxial jet can be obtained as limiting cases of very large inner radius and inner radius equal to zero, respectively. In this temporal analysis, the effect of liquid Weber number, initial disturbance amplitude, inner gas-to-liquid velocity ratio, outer gas-to-liquid velocity ratio and outer gas swirl strength on the breakup time is investigated. The model is validated by comparison with earlier analytical studies for the limiting case of a planar sheet as well as with experimental data of sheet breakup length available in literature. It is shown that the linear theory cannot predict breakup of an annular sheet and the developed nonlinear model is necessary to accurately determine the breakup length. In the limiting case of a coaxial jet, results show that gas swirl destabilizes the jet, makes helical modes dominant compared to the axisymmetric mode and decreases jet breakup length. These results contradict earlier linear analyses and agree with experimental observations. For an annular sheet, it is found that gas flow hastens the sheet breakup process and shorter breakup lengths are obtained by increasing the inner and the outer gas velocity. Axially moving inner gas stream is more effective in disintegrating the annular sheet compared to axially moving outer gas stream. When both gas streams are moving axially, the liquid sheet breakup is quicker compared to that with any one gas stream. In the absence of outer gas swirl, the axisymmetric mode is the dominant instability mode. However, when outer gas flow has a swirl component higher helical modes become dominant. With increasing outer gas swirl strength, the maximum disturbance growth rate increases and the most unstable circumferential wave number increases resulting in a highly asymmetric sheet breakup with shorter breakup lengths and thinner ligaments.     

圆环旋转粘性液膜射流三维不稳定性研究

利用线性稳定性理论进行了射流液体粘性对圆环旋转液膜射流稳定性影响的研究,推导出了三维扰动下具有固体旋涡型速度分布的圆环旋转粘性液膜射流的色散方程;在此基础上进行了类反对称模式与类对称模式下的圆环旋转粘性液膜射流的三维不稳定性分析。研究结果表明,在类反对称模式下,液体粘性超过一定值后,射流最大扰动增长率随液体粘性的增加而迅速减小;轴对称模态的射流特征频率产生一个突降变化;随液体粘性增加,轴对称模态不稳定波数范围减小,非轴对称模态不稳定波数范围呈现出先减小后增大趋势。在类对称模式下,液体粘性对射流最大扰动增长率的影响主要体现在对非轴对称模态的影响上;液体粘性只在粘性较大时才会对非轴对称模态射流特征频率产生一定影响;液体粘性超过一定值后,轴对称模态与非轴对称模态的不稳定波数范围都会快速下降。     

Instability of a planar liquid sheet with surrounding fluids between two parallel walls

We investigate the linear and nonlinear instability of a planar liquid sheet with surrounding fluids between two parallel plane solid walls. Linear analysis shows that the maximum temporal growth rate and unstable wave number region of disturbances increase for the dilational and sinuous modes when the gap between the sheet and the wall decreases. The walls have more influence on the instability when the density ratio of the surrounding fluid to the sheet and/or the Weber number decrease. On the other hand, nonlinear analysis is performed by means of the discrete vortex method, where double vortex rows and their mirror images are placed so as to satisfy the boundary condition on the walls. Numerical results show that the walls enhance nonlinearity, which causes deformation and distortion of the sheet, whereas the nonlinearity diminishes linear growth rates except for long dilational disturbances. In particular, as the walls are placed more closely to the sheet, local sheet thinning becomes more pronounced in the long dilational mode, while the dilational mode is more strongly induced from the sinuous mode through monotonic or periodic energy exchanges between the two modes.     

Effect of streamwise and spanwise electric fields on transient growth in a two-fluid channel flow

A linear model of a two-fluid channel flow under streamwise/spanwise electric field is built. Both the fluids are assumed to be incompressible, viscous and perfectly dielectric. The effect of the streamwise and spanwise electric fields on transient behavior of small three-dimensional disturbances is studied. The numerical result shows that the streamwise electric field suppresses transient growth of the disturbance with spanwise uniform wave number. The spanwise electric field diminishes transient growth of the disturbance with streamwise uniform wave number. Two peaks of optimal growth are detected in the wave number plane. The peak at relatively large spanwise wave number is dominated by the lift-up mechanism and little influenced by electric field. Differently, the peak at relatively small wave number is associated with the characteristic of the interface and possibly influenced by electric field. The effect of the Weber number, the Reynolds number and the relative electrical permittivity on optimal growth is studied as well. A scaling law is obtained for relatively small Weber numbers and relatively large Reynolds numbers.     

Higher‐order‐accurate numerical method for temporal stability simulations of Rayleigh‐Bénard‐Poiseuille flows

For Rayleigh‐Bénard‐Poiseuille flows, thermal stratification resulting from a wall‐normal temperature gradient together with an opposing gravitational field can lead to buoyancy‐driven instability. Moreover, for sufficiently large Reynolds numbers, viscosity‐driven instability can occur. Two higher‐order‐accurate methods based on the full and linearized Navier‐Stokes equations were developed for investigating the temporal stability of such flows. The new methods employ a spectral discretization in the homogeneous directions. In the wall‐normal direction, the convective and viscous terms are discretized with fifth‐order‐accurate biased and fourth‐order‐accurate central compact finite differences. A fourth‐order‐accurate explicit Runge‐Kutta method is employed for time integration. To validate the methods, the primary instability was investigated for different combinations of the Reynolds and Rayleigh number. The results from these primary stability investigations are consistent with linear stability theory results from the literature with respect to both the onset of the instability and the dependence of the temporal growth rate on the wave angle. For the cases with buoyancy‐driven instability, strong linear growth is observed for a broad range of spanwise wavenumbers. The largest growth rates are obtained for a wave angle of 90°. For the cases with viscosity‐driven instability, the linear growth rates are lower and the first mode to experience nonlinear growth is a higher harmonic with half the wavelength of the fundamental.     

横向交流电场下液膜参数不稳定性分析

当将运动的平面液膜置于横向的交流电场之间时会产生参数振荡现象.为了得到交流电场下平面液膜的色散关系并为液膜的破碎行为分析提供理论基础,本文基于漏电介质模型对液体的电学特性进行假设,对平面液膜在直流和交流电场下的参数不稳定性进行了分析.由于主流是基于时间的流动, 在稳定性分析中引入了Floquet理论. 在文中,将电场定义为部分交流电场和部分直流电场共同耦合后的混合型电场. 最后,波数和不稳定增长率之间的无量纲色散方程可由矩阵的形式表示.本文考虑了多种参数对不稳定的影响, 包括气液密度比、韦伯数、雷诺数、欧拉数、 松弛时间以及衡量交流电场占比的参数及频率参数,并得知欧拉数同时影响毛细不稳定性及参数不稳定性,交流电场占比对不稳定性的影响体现在恒定电场力上, 而交流电场频率主要影响参数不稳定性. 为了在实验中更容易地寻求参数振荡现象,增大电欧拉数及减小交流电场频率是有效的方法.      

Flow and instability of capillary jets interacting with the surrounding medium

The flow of an axially symmetric capillary jet of a viscous incompressible liquid in the space occupied by another liquid is investigated. The problem of stationary flow in the jet and in the surrounding medium under the action of viscosity, capillary forces, and gravity was obtained numerically. The instability problem of this flow to small perturbations in the form of running waves is stated and solved numerically. The values of the dimensionless Reynolds, Weber, and Froude numbers are explained, as well as the effect of the initial velocity profile in the jet, its instability, and subsequent jet decay into drops.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–59, November–December, 1978.     

Effect of heat exchange on the interfacial instability of gas-liquid jet

IntroductionInrecentyears,theprosperityofmetalpowdermarket[1] andthedevelopmentofsprayformingtechniquesmakeitnecessarytostudythemechanismofjetatomization .Freelyfallinghightemperaturemetalliquid ,impactedbyhighspeedgasaround ,breaksintodropletsofdifferentsizes,whichprocedureiscalledair_blastatomizationorsprayatomization .Infact,thisprocesshasawiderengineeringapplications,rangingfromfuelinjectorsingasturbinesandjetengines,totwo_phaseflowchemicalreactors ,spraydrying ,andsoon .OnthebasisofKelvi…     

A study on the aerodynamic instability of attenuating liquid sheets

We describe the characteristics of a radially spreading unstable liquid sheet in quiescent air via optical measurement techniques and linear instability theory. A high speed CCD camera system and a complimentary laser refraction method were employed to measure the intact sheet diameter, unstable wave lengths, wave speed, wave frequency spectrum and spatial wave growth rates. Linear instability models for thinning, viscous and inviscid liquid sheets, which are available from the literature, allow for a comparison of experimental data and predicted sheet behaviour. The last section evaluates the differences and similarities between the current liquid sheet experiment and industrial spray applications such as fuel atomisation via pressure-swirl nozzles.     

Instabilities of a gas-liquid flow between two inclined plates analyzed using the Navier–Stokes equations

The paper is devoted to a theoretical analysis of a counter-current gas-liquid flow between two inclined plates. We linearized the Navier–Stokes equations and carried out a stability analysis of the basic steady-state solution over a wide variation of the liquid Reynolds number and the gas superficial velocity. As a result, we found two modes of the unstable disturbances and computed the wavelength and phase velocity of their neutral disturbances varying the liquid and gas Reynolds number. The first mode is a “surface mode” that corresponds to the Kapitza's waves at small values of the gas superficial velocity. We found that the dependence of the neutral disturbance wavelength on the liquid Reynolds number strongly depends on the gas superficial velocity, the distance between the plates and the channel inclination angle for this mode. The second mode of the unstable disturbances corresponds to the transition to a turbulent flow in the gas phase and there is a critical value of the gas Reynolds number for this mode. We obtained that this critical Reynolds number weakly depends on both the channel inclination angle, the distance between the plates and the liquid flow parameters for the conditions considered in the paper. Despite a thorough search, we did not find the unstable modes that may correspond to the instability in frame of the viscous (or inviscid) Kelvin–Helmholtz heuristic analysis.     

Electrohydrodynamic instability of a dielectric compressible liquid sheet streaming into an ambient stationary compressible gas

The effect of compressibility of fluids on the linear electrohydrodynamic instability of a dielectric liquid sheet issued from a nozzle into an ambient dielectric stationary gas in the presence of a horizontal electric field is investigated. It is found that increasing the Mach number from subsonic to transonic causes the maximum growth rate and the dominant wavenumber of the disturbances to increase, and the increase is higher in the presence of the electric field. Liquid compressibility has been found to have a minimal effect on instability. At constant wavenumber and electric field values, the growth rate of disturbances increases as the gas Mach number tends to 1, and then begins to decrease with further increase in the gas Mach number. At small values of wavenumber, antisymmetrical disturbances grow faster than symmetrical ones, while the growth rate of both types of disturbances approach each other at large wavenumbers, which increases by increasing the electric field values. At small Weber numbers, antisymmetrical disturbances exhibit a higher maximum growth rate and a lower dominant wavenumber than symmetrical disturbances. However, the maximum growth rate and dominant wavenumber of the two types of disturbances are almost identical when both Weber number and electric field values become large. An increase in the gas to liquid density ratio enhances the instability, and this effect is enhanced for high electric field values. Surface tension and electric fields always oppose and increase the development of instability, respectively; and they have opposite effects for long wavelengths and high Weber numbers.     

Liquid jet in a high Mach number air stream

The instability of circular liquid jet immersed in a coflowing high velocity air stream is studied assuming that the flow of the viscous gas and liquid is irrotational. The basic velocity profiles are uniform and different. The instabilities are driven by Kelvin–Helmholtz instability due to a velocity difference and neckdown due to capillary instability. Capillary instabilities dominate for large Weber numbers. Kelvin–Helmholtz instability dominates for small Weber numbers. The wavelength for the most unstable wave decreases strongly with the Mach number and attains a very small minimum when the Mach number is somewhat larger than one. The peak growth rates are attained for axisymmetric disturbances (n = 0) when the viscosity of the liquid is not too large. The peak growth rates for the first asymmetric mode (n = 1) and the associated wavelength are very close to the n = 0 mode; the peak growth rate for n = 1 modes exceeds n = 0 when the viscosity of the liquid jet is large. The effects of viscosity on the irrotational instabilities are very strong. The analysis predicts that breakup fragments of liquids in high speed air streams may be exceedingly small, especially in the transonic range of Mach numbers.     

Small Reynolds number instabilities in two-layer Couette flow

Instabilities in two-layer Couette flow are investigated from a small Reynolds number expansion of the eigenvalue problem governing linear stability. The wave velocity and growth rate are given explicitly, and previous results for long waves and short waves are retrieved as special cases. In addition to the inertial instability due to viscous stratification, the flow may be subject to the Rayleigh–Taylor instability. As a result of the competition of these two instabilities, inertia may completely stabilise a gravity-unstable flow above some finite critical Froude number, or conversely, for a gravity-stable flow, inertia may give rise to finite wavenumber instability above some finite critical Weber number. General conditions for these phenomena are given, as well as exact or approximate values of the critical numbers. The validity domain of the many asymptotic expansions is then investigated from comparison with the numerical solution. It appears that the small-Re expansion gives good results beyond Re = 1, with an error less that 1%. For Reynolds numbers of a few hundred, we show from the eigenfunctions and the energy equation that the nature of the instability changes: instability still arises from the interfacial mode (there is no mode crossing), but this mode takes all the features of a shear mode. The other modes correspond to the stable eigenmodes of the single-layer Couette flow, which are recovered when one fluid is rigidified by increasing its viscosity or surface tension.     

The instability of water-mud interface in viscous two-layer flow with large viscosity contrast

The temporal instability of parallel viscous two-phase mixing layers is extended to current-fluid mud by considering a composite error function velocity profile. The influence of viscosity ratio, Reynolds number, and Froude number on the instability of the system are discussed and a new phenomenon never discussed is investigated based on our numerical results. It is shown that viscosity can enlarge the unstable wave number range, cause new instability modes, and certainly reduce the growth rate of Kelvin—Helmholtz (K—H) instability.     

Experimental and analytical investigation of liquid sheet breakup characteristics

The main objective of this research is to study analytically and experimentally the liquid sheet breakup of a flat fan jet nozzle resulting from pressure-swirling. In this study the effects of nozzle shape and spray pressure on the liquid sheet characteristics were investigated for four nozzles with different exit widths (1.0, 1.5, 2.0 and 2.5 mm). The length of liquid sheet breakup, liquid sheet velocity and the size of formed droplets were measured by a digital high speed camera. The breakup characteristics of plane liquid sheets in atmosphere are analytically investigated by means of linear and nonlinear hydrodynamic instability analyses. The liquid sheet breakup process was studied for initial sinuous and also varicose modes of disturbance. The results presented the effect of the nozzle width and the spray pressure on the breakup length and also on the size of the formed droplets. Comparing the experimental results with the theoretical ones for all the four types of nozzles, gives a good agreement with difference ranges from 4% to 12%. Also, the comparison between the obtained results and the results due to others shows a good agreement with difference ranged from 5% to 16%. Empirical correlations have been deduced describing the relation between the liquid sheet breakup characteristics and affecting parameters; liquid sheet Reynolds number, Weber number and the nozzle width.