Nonlinear spatial instability of an annular swirling viscous liquid sheet

Instability and breakup of a viscous annular liquid sheet that is exposed to co-flowing inner and outer gas streams have been investigated using a nonlinear spatial stability analysis. A perturbation expansion method is used with the initial amplitude of the disturbance as the perturbation parameter. The evolution of the two gas–liquid interfaces is tracked until the sheet breaks up and the breakup length is determined. The model is validated by comparison with available experimental data. The effects of liquid swirl strength, gas-to-liquid density ratio, radius of curvature ratio, and liquid viscosity on the sheet instability and breakup have been studied. The results show that at very low values of liquid swirl, it has a stabilizing effect on sheet breakup, but as the swirl strength increases, it strongly destabilizes the sheet. Also, with increasing swirl strength, the occurrence of the large surface deformations moves from the inner interface to the outer interface. The sheet breakup length increases slightly and then decreases rapidly with an increase in liquid swirl strength. Without liquid swirl, the axisymmetric mode is the dominant instability mode. However, with increasing liquid swirl strength, the higher helical modes become dominant and the breakup becomes increasingly asymmetric. When the undisturbed liquid sheet has a purely axial motion, the inner gas stream is more effective in sheet breakup than the outer gas stream. In the presence of liquid swirl, the outer gas stream is more disruptive than the inner gas stream. The breakup length becomes shorter as gas-to-liquid density ratio and the radius of curvature ratio increases. Increase in liquid viscosity tends to slow the disturbance growth and increases the sheet breakup length.     

Instability of cylindrical compressible gas jets in viscous liquid streams

Summary A linear stability study of cylindrical compressible gas jets in a moving incompressible viscous liquid medium subject to varicose disturbances is described. It was found that the gas jet is always unstable for a range of wavenumbers at any flow condition. When the gas and liquid velocity are not equal, temporal instability is enhanced by surface tension effects for small Weber numbers, and by aerodynamic interaction between the gas and liquid phase for high Weber numbers (where surface tension has a stabilizing influence). Increasing liquid viscosity always reduces the growth rate and the dominant wavenumber, whereas increasing gas density always increases gas jet instability. It was also found that the relative, rather than the absolute, velocity of the gas and liquid controls temporal instability. Increasing gas compressibility always increases the maximum growth rate and dominant wavenumber. On the other hand, for equal gas and liquid velocities, increasing surface tension always destabilizes, while increasing gas density always stabilizes, the gas jet. For absolute and spatial (or convective) instability, it was shown that the critical Weber number, separating the region of absolute from that of spatial instability, decreases monotonically as the liquid velocity is increased. For a stationary liquid medium, the gas jet is always absolutely unstable, and spatial instability does not exist, in contrast to liquid jets in a stationary gas medium. For sufficiently large liquid velocities, the gas jet is spatially unstable, whereas absolute instability disappears. Further, the absolute velocity of gas and liquid flow controls not only the growth of unstable disturbances, but also the characteristics of the instability. Increasing viscous effects tends to suppress absolute instability, while increasing both gas density and compressibility promotes absolute instability for small liquid velocities (however, their effect diminishes as liquid velocity is increased).     

Effect of a gas stream on the wave flow of thin layers of a viscous liquid

An analytical study was undertaken of the effect of a gas stream on the wave flow characteristics of thin layers of a viscous liquid moving over a vertical surface. The results of the investigations are compared with experimental data.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 6, pp. 989–994, December, 1969.     

Shape instability in thin viscous films and jets

Summary The theory of viscous liquid films and thin jets as two-and one-dimensional continua is examined. Theoretical results are presented concerning a special type of instability which leads to the loss of smoothness of the shape (wrinkling) and associated with the failure of hyperbolicity of the governing equations. The conditions for different types of such an instability are formulated in the closed analytical form.     

Instability of an annular viscous liquid jet

Summary A linear analysis has been carried out for the temporal instability of an annular viscous liquid jet moving in an inviscid gas medium, which includes three limiting cases of a round liquid jet, a gas jet and a plane liquid sheet. It is found that there exist two independent unstable modes, which become the well-known sinuous and varicose modes for plane liquid sheets as annular jet radii approach to infinity. Hence, they are named as para-sinuous and para-varicose. It is shown that an ambient gas medium always enhances the annular jet instability. The curvature effects in general increase the disturbance growth rate, and may not be neglected for the breakup process of an annular or conical liquid sheet. An annular jet with a sufficiently small thickness tends to break up much faster than the corresponding plane liquid sheet, in accordance with existing experimental observations. Liquid viscosity has complicated dual effects on the instability. It is also found that there exists a critical Weber number below which surface tension is the source of instability. Whereas above it, instability is suppressed by surface tension effect and it promoted by aerodynamic interaction between the liquid and gas phase. For the practical importance of large Weber numbers such as related to liquid atomization, the para-sinuous mode is always predominant.     

Response of annular liquid jets to mass loading

An adaptive, block-bidiagonal finite difference method is used to study the response of annular liquid jets to the injection of mass into the volume enclosed by the annular jet. It is shown that the annular jet's response is characterized by damped oscillations in both the convergence length and the pressure of the gases enclosed by the jet, and that these oscillations are similar to those of an underdamped mass-spring-dashpot system. It is also shown that the maximum amplitude and the number of these oscillations increase as the nozzle exit angle, the initial pressure ratio across the annular jet, the pressure of the gases surrounding the jet, the Weber number, and the rate and duration of the mass injection are increased, are nearly independent of the annular jet's thickness-to-radius ratio at the nozzle exit, and decrease as the Froude number is increased. The numerical calculations presented in this paper indicate that the pressure of the gases enclosed by the annular liquid jet responds instantaneously to the mass injection, whereas there is a lag in the response of the convergence length. This lag is due to the inertia of the jet and assumption that the gases enclosed by the jet are isothermal, and decreases as the injection duration is increased. The calculations also indicate that the critical pressure coefficient of unity determined from the solution of the steady state governing equations can be exceeded without affecting the stability of the annular jet.     

Nonlinear viscous liquid jets from a rotating orifice

A liquid jet follows a curved trajectory when the orifice from which the jet emerges is rotating. Surface-tension-driven instabilities cause the jet to lose coherence and break to form droplets. The sizes of the drops formed from such jets are in general not uniform, ranging from drops with diameters of the order of the jet diameter to droplets with diameters which are several orders of magnitude smaller. This presentation details a theoretical investigation of the effects of changing operating parameters on the break-up of curved liquid jets in stagnant air at room temperature and pressure. The Navier–Stokes equations are solved in this system with the usual viscous free-surface boundary conditions, using an asymptotic method based upon a slender-jet assumption, which is clearly appropriate from experimental observations of the jet. Nonlinear temporal simulations of the break-up of the liquid jets using slender theory are also presented. These simulations based upon both a steady-trajectory assumption, and the more general equations which allow for an unsteady trajectory, show all the break-up modes viewed in experiments. Satellite-droplet formation is also considered. A. C. King deceased.     

Domain-adaptive finite difference methods for collapsing annular liquid jets

A domain-adaptive technique which maps a time-dependent, curvilinear geometry into a unit square is used to determine the steady state mass absorption rate and the collapse of annular liquid jets. A method of lines is used to solve the one-dimensional fluid dynamics equations written in weak conservation-law form, and upwind differences are employed to evaluate the axial convective fluxes. The unknown, time-dependent, axial location of the downstream boundary is determined from the solution of an ordinary differential equation which is nonlinearly coupled to the fluid dynamics and gas concentration equations. The equation for the gas concentration in the annular liquid jet is written in strong conservation-law form and solved by means of a method of lines at high Peclet numbers and a line Gauss-Seidel method at low Peclet numbers. The effects of the number of grid points along and across the annular jet, time step, and discretization of the radial convective fluxes on both the steady state mass absorption rate and the jet's collapse rate have been analyzed on staggered and non-staggered grids. The steady state mass absorption rate and the collapse of annular liquid jets are determined as a function of the Froude, Peclet and Weber numbers, annular jet's thickness-to-radius ratio at the nozzle exit, initial pressure difference across the annular jet, nozzle exit angle, temperature of the gas enclosed by the annular jet, pressure of the gas surrounding the jet, solubilities at the inner and outer interfaces of the annular jet, and gas concentration at the nozzle exit. It is shown that the steady state mass absorption rate is proportional to the inverse square root of the Peclet number except for low values of this parameter, and that the possible mathematical incompatibilities in the concentration field at the nozzle exit exert a great influence on the steady state mass absorption rate and on the jet collapse. It is also shown that the steady state mass absorption rate increases as the Weber number, nozzle exit angle, gas concentration at the nozzle exit, and temperature of the gases enclosed by the annular liquid jet are increased, but it decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-to-radius ratio at the nozzle exit are increased. It is also shown that the annular liquid jet's collapse rate increases as the Weber number, nozzle exit angle, temperature of the gases enclosed by the annular liquid jet, and pressure of the gases which surround the jet are increased, but decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-toradius ratio at the nozzle exit are increased. It is also shown that both the ratio of the initial pressure of the gas enclosed by the jet to the pressure of the gas surrounding the jet and the ratio of solubilities at the annular liquid jet's inner and outer interfaces play an important role on both the steady state mass absorption rate and the jet collapse. If the product of these ratios is greater or less than one, both the pressure and the mass of the gas enclosed by the annular liquid jet decrease or increase, respectively, with time. It is also shown that the numerical results obtained with the conservative, domain-adaptive method of lines technique presented in this paper are in excellent agreement with those of a domain-adaptive, iterative, non-conservative, block-bidiagonal, finite difference method which uncouples the solution of the fluid dynamics equations from that of the convergence length.     

Effect of liquid and air swirl strength and relative rotational direction on the instability of an annular liquid sheet

Summary Instability of a swirling annular liquid sheet in swirling inner and outer air streams has been investigated by a temporal linear stability analysis. The effects of the swirling and axial motion of the liquid and the air streams, as well as the effects of relative inner and outer air swirl orientation with respect to the liquid swirl direction on the instability have been investigated. Results show that for a non-swirling liquid sheet axial inner air stream is more effective than axial outer air stream in enhancing the sheet instability. This is opposite of a swirling liquid sheet where axial outer air is more effective in promoting sheet instability compared to axially moving inner air stream. The liquid swirl has a destabilizing effect at the outer interface but has a stabilizing effect at the inner interface. At high liquid swirl Weber number, the outer air (with axial and swirl velocity components) is more effective in enhancing sheet instability compared to the inner air (with axial and swirl velocity components). To understand the effect of air swirl orientation with respect to liquid swirl direction, four possible combinations with both swirling air streams with respect to the liquid swirl direction have been considered. Results show that at high liquid swirl Weber number a combination of counter-rotating-inner air stream and co-rotating-outer air stream has the largest most unstable wave number. However, at low liquid swirl, co-inner/counter-outer combination has the largest most unstable wave number. The combination of inner and the outer air stream co-rotating with the liquid has the highest growth rate. In many combustion applications, the liquid sheet is injected in high pressure environment where the effect of high ambient pressure results in increased aerodynamic interaction due to high air density. Hence the effect of high ambient pressure is studied in terms of the dimensionless parameter of air-to-liquid density ratio. Results show significantly higher disturbance growth rates at high air pressure. However, the qualitative sheet stability behavior is similar to that at atmospheric pressure.     

Numerical calculation of opposing jets of a viscous incompressible liquid

The collision of two equal coaxial jets is studied. A numerical solution of the exact equations of motion and of continuity is obtained on the assumption that the isothermal flow is stationary and the properties of the liquid are constant.

     

Stability of nonisothermal flow of a viscous liquid in an annular channel

In nonisothermal flow of a viscous liquid in an annular channel between coaxial cylinders where the outer cylinder has finite dimensions and is stationary, and the inner cylinder infinitely moves along the axis, the central position of the latter is unstable. When superimposing a thermal field, principally it is possible to create as large a force as required which holds the inner cylinder exactly on center.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 63, No. 1, pp. 32–37, July, 1992.     

Laws of propagation of turbulent jets of a viscous liquid of variable composition

The results of an experimental investigation of the propagation of turbulent isothermal jets of helium and argon discharging into an air-flooded space are presented.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 36, No. 1, pp. 75–79, January, 1979.     

The problem of the trajectories of circular gas jets in a nonuniform side stream

We describe an engineering method for constructing the trajectories of circular gas jets in a side drift stream with an arbitrary velocity field along the jet.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 19, No. 6, pp. 1104–1109, December, 1970.     

Capillary instability of liquid jets in the case of heat exchange with the surrounding medium

We consider the effect of heat exchange on the capillary instability of a liquid jet. It is shown that the instability parameters differ from the standard values only when the rate of heat exchange is large.Translated from Inzhenerno-fizicheskii Zhurnal, Vol. 60, No. 4, pp. 537–544, April, 1991.     

Characteristics of the propagation of free annular turbulent jets

Experimental data are presented relating to the propagation of an axially symmetric submerged jet flowing from an annular nozzle formed by two coaxial cylinders.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 21, No. 2, pp. 238–241, August, 1971.     

On the capillary instability of a dielectric viscous liquid cylinder surrounded by another viscous fluid in an electric field

Summary The capillary instability of a long, dielectric, incompressible, viscous liquid cylinder when subjected to small axisymmetric disturbances in the presence of a uniform axial electric field is investigated by making a normal mode analysis of the equations describing the perturbed fluid flow. The resulting characteristic equation, which includes the effect of a surrounding unbounded viscous fluid, is solved in order to express graphically the governing dispersion relation as a function of the relative viscous and dielectric properties of the two media when the viscosities of both are dominant. The influence of the various parameters on the modes of capillary instability as well as their effect on the modes of maximum instability are exhibited for this and other special cases.

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Zusammenfassung Die kapillare Instabilität eines langen, dielektrischen, inkompressiblen, zähen Flüssigkeitszylinders wird untersucht, wenn letzterer kleinen axialsymmetrischen Störungen in Gegenwart eines gleichförmigen, elektrischen Axialfeldes ausgesetzt ist. Das zur Behandlung des vorliegenden Problems herangezogene Verfahren macht Gebrauch von einem Normalschwingungsansatz für die das gestörte Strömungsfeld beschreibenden Gleichungen. Die resultierende charakteristische Gleichung, welche den Einfluß der umgebenden, unendlich ausgedehnten, zähen Flüssigkeit berücksichtigt, wird gelöst. Die grundlegende Dispersionsbeziehung wird graphisch als Funktion der relativen zähen und dielektrischen Eigenschaften beider Medien dargestellt, falls der Zähigkeitseinfluß in beiden Medien dominiert. Der Einfluß verschiedener Parameter auf das Schwingungsverhalten der Kapillarinstabilität, ebenso ihr Einfluß auf das Schwingungsverhalten der maximalen Instabilität, wird sowohl für den vorliegenden Fall als auch für verschiedene Sonderfälle dargestellt.

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With 7 Figures

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Presented at the Canadian Congress of Applied Mechanics, Université Laval, Québec, Canada, May 1967; an abstract appears in the proceedings.     

Effect of a gas stream on the efflux of loose material

The article reveals the nature of the effect a gas stream has on a dynamically unstable arch and correspondingly on the efflux of loose material.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 38, No. 6, pp. 976–982, June, 1980.     

Kinetics of liberation of dissolved gas in a liquid stream

The results are presented for an experimental study of the rate of formation of the gaseous phase during the turbulent flow along a cylindrical channel of a liquid supersaturated with a gas.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 28, No. 2, pp. 282–285, February, 1975.     

Qualitative motion of a liquid in an accelerating gas stream

Experimental data are examined on the interaction of liquid drops with the gas stream in a Venturi scrubber, the most promising available equipment.     

Numerical simulation of the flow of a supersonic stream of viscous compressible gas over a cavity

Numerical simulation of the flow of a supersonic stream of viscous, compressible, heat-conducting gas over a cavity is carried out on the basis of kinetically consistent difference schemes. Different types of flow (open and closed cavities) are considered, the heat fluxes to the walls of the recess are determined, and a nonsteady regime of flow over a cavity is simulated. The results obtained are compared with known experimental relationships.Translated from Inzhenerno-fizicheskii Zhurnal, Vol. 61, No. 4, pp. 570–577, October, 1991.