Newtonian Jet Research Articles

NONSTAGGERED BOUNDARY-FITTED COORDINATE METHOD FOR FREE SURFACE FLOWS

A new approach based on the finite-difference technique has been developed to study the steady incompressible Navier-Stokes equations in the laminar region, where the domain is partially bounded by a free surface. The nonstaggered fractional step method is used to solve the flow equations written in terms of primitive variables. The physical domain is transformed to a rectangle by means of a numerical mapping technique. The location of the phase boundary is accomplished by means of two methods depending on the surface tension effect: the normal-stress boundary condition or the kinematic boundary condition. We have tested the accuracy and efficiency of the numerical method by solving four different test problems: lid-driven flow in an inclined cavity, film in the absence of gravity, the "stick-slip" problem, and the Newtonian jet swell problem.

The Effect of Dynamic Surface Tension on the Oscillation of Slender Elliptical Newtonian Jets

We investigate free-surface oscillating jets with elliptical cross section, focusing on behavior associated with decaying surface tension. Previous one-dimensional equations for an oscillating jet are extended to allow variable surface tension on short space and time scales relevant for surfactant mixtures. We presume the decay of surface tension as a function of surface age, and derive the resulting jet behavior. Three plausible forms of decay are studied: an exponential decay, a diffusion model derived in Brazee et al. (1994), and an algebraic form due to Hua and Rosen (1991). Our simulations suggest both experimental regimes, and measurable jet features in these regimes, which may be exploited in an inverse formulation to deduce the unknown rapid surface tension decay of a given surfactant mixture. In particular, we establish numerical relationships between the amplitude and the wavelength of either a sustained far-field oscillation or oscillation at a fixed downstream location and the entire history of surface tension decay. These numerical relationships are ideal for the inverse formulation, in that the complete surface tension evolution may be deduced solely from far-field or downstream jet measurements, away from the confined part of the jet where the surface tension is rapidly changing.

Morphology and Dynamics of Relativistic Jets

We present a comprehensive analysis of the morphology and dynamics of relativistic pressure-matched axisymmetric jets. The numerical simulations have been carried out with a high-resolution shock-capturing hydrocode based on an approximate relativistic Riemann solver derived from the spectral decomposition of the Jacobian matrices of relativistic hydrodynamics. We discuss the dependence of the jet morphology on several parameters, paying special attention to the relativistic effects caused by high Lorentz factors and large internal energies of the beam flow. The parameter space of our analysis is spanned by the ratio of the beam and ambient medium rest mass density (η), the beam Mach number (Mb), the beam Lorentz factor (Wb), and the adiabatic index (γ) of the equation of state (assuming an ideal gas). Both the ultrarelativistic regime (Wb ≥ 20) and the hypersonic regime (relativistic Mach number greater than 100) have been studied. Our results show that the enhancement of the effective inertial mass of the beam due to relativistic effects (through the specific enthalpy and the Lorentz factor) makes relativistic jets significantly more stable than Newtonian jets. We find that relativistic jets propagate very efficiently through the ambient medium, at speeds that agree very well with those obtained from an estimate based on a one-dimensional momentum balance. The propagation efficiency of a relativistic jet is an increasing function of the beam flow velocity. Relativistic jets seem to give rise to two different morphologies, according to the relevance of relativistic effects. Hot beams (i.e., with internal energies comparable to the beam rest-mass energy) show little internal structure (as they are almost in pressure equilibrium with their surroundings) and relatively smooth cocoons forming lobes near the head of the jet. Highly supersonic models, in which the kinematic relativistic effects due to high beam flow Lorentz factors dominate, display extended cocoons that are overpressured with respect to the environment. The cocoon thickness decreases, and its mean pressure increases with increasing beam Lorentz factor.

General Relativistic Simulation of Jet Formation from Magnetized Accretion Disk

AbstractRecently, superluminal motions are observed not only from active galactic nuclei but also in our Galaxy. These phenomena are explained as relativistic jets propagating almost toward us with Lorentz factor more than 2. For the formation of such a relativistic jet, magnetically driven mechanism around a black hole is most promising. We have extended the 2.5D Newtonian MHD jet model (Shibata & Uchida 1986) to general relativistic regime. For this purpose, we have developed a general relativistic magnetohydrodynamic (GRMHD) numerical code and applied it to the simulation of the magnetized accretion disk around a black hole. We have found the formation of magnetically driven jets with 86 percent of light velocity (i.e. Lorentz factor ~ 2.0).

The motion of a solid sphere suspended by a Newtonian or viscoelastic jet

This paper describes experimental observations of a solid sphere suspended by a vertical or inclined jet. A laminar Newtonian jet is able to suspend a sphere only through viscous entrainment at low Reynolds numbers (Re ~ 10). A turbulent Newtonian jet (Re ~ 104) attracts a sphere that is sufficiently large but rejects smaller ones. The Coanda effect is responsible for steady suspension of solid spheres even in highly slanted jets. Anomalous rotation, opposite to the direction of the local shear, occurs under certain conditions, and its physical mechanism cannot be explained based on available information. A viscoelastic laminar jet is narrower than a comparable Newtonian one and it can suspend spheres at Reynolds numbers in the hundreds, precisely the Re range in which a Newtonian jet fails to suspend a sphere. It is suggested that the contrast between laminar Newtonian and viscoelastic jets may be related to a reversal in the pressure distribution on the surface of the sphere caused by non-Newtonian normal stresses. Flow visualization provides insights into the flow field in the jet and around the solid sphere.

Drop formation in liquid–liquid systems before and after jetting

The formation of drops resulting from the breakup of an axisymmetric Newtonian liquid jet injected vertically into another immiscible Newtonian liquid at various Reynolds numbers is investigated here. The full transient from startup to breakup into drops was simulated numerically by solving the time-dependent axisymmetric equations of motion and continuity using a combination of the volume-of-fluid (VOF) and continuous-surface-force (CSF) methods. The numerical simulation results compare well with previous experimental data and are significantly more accurate than previous simplified analyses based on drop formation before and after jetting over a wide range of conditions.

Quantitative Visualization of a Submerged Pseudoplastic Jet Using Particle Image Velocimetry

A preliminary experimental study of a pseudoplastic jet flow is reported in this paper. The velocity field was measured using Particle Image Velocimetry. Unlike a Newtonian jet, the pseudoplastic jet was observed to experience a sudden drop in its velocity at a reproducible position downstream of the nozzle for the range of velocities examined. This position moved downstream with an increase in the nozzle exit velocity. The center-line streamwise velocity decayed as X–15 to X–30 within the terminating region of the jet for three different nozzle exit velocities of 2.43, 3.17, and 5.42 m/s. This decay is in contrast to X–1 decay for a turbulent or laminar Newtonian jet. The location of the terminating region did not appear to scale with Reynolds number, Plasticity number, or Hedstrom number. At Reynolds numbers of 3000 and 6400, the instantaneous streamwise velocity maps indicated that the flow was fairly laminar, with a sinuous instability appearing at the higher Reynolds number condition. Close observation of the jet indicated that local turbulence could exist within regions of high shear rate. Further detailed study is required to confirm this observation.

A numerical study of the asymptotic evolution and breakup of Newtonian and viscoelastic jets

Numerical and asymptotic methods are used to study the surface-tension driven instability of the jet. For the Newtonian case, we establish a relationship between the initial shape of the jet and the asymptotic approach to breakup. For the Oldroyd-B fluid, no breakup occurs, and we study the evolution of the jet for large time. On the other hand, the numerical solutions show finite time breakup for the Giesekus model. For the upper convected Maxwell model, nonunique discontinuous solutions exist, and the choice of the physically appropriate solution depends on regularizing terms in the equations. We discuss one such regularization involving higher order derivatives resulting from taking into account the axial curvature of the free surface.

The falling laminar jet of semi-dilute polymer solutions

The studies of falling laminar jets have been done mainly on Newtonian jet due to the complexity of non-Newtonian fluid parameter. Numerical solutions of the Newtonian jet equation are in good agreement with the experimental results of polymer solutions in the experimental concentration range. Numerical simulation results suggest that the free falling jet is mostly influenced by the Froude number. The jet radius increases with the Froude number. It was found that surface tension or viscosity does not contribute much to the shapes of the free falling jet.

Prediction of droplet size from the breakup of cylindrical liquid jets

A simple equation to predict the size of droplets formed during the breakup of cylindrical liquid jets is obtained from instability analysis. This droplet-size equation applies to low-velocity, liquid-in-liquid and liquid-in-gas jets involving Newtonian or non-Newtonian fluids that follow power-law shear stress versus deformation rate relationships. The equation is tested by comparing the resultant theoretical predictions for droplet size with experimental data for seventeen Newtonian liquid systems and five power-law non-Newtonian/Newtonian liquid systems (power-law liquid jet in Newtonian liquid and Newtonian liquid jet in power-law liquid), as well as with numerical solutions to Tomotika's equation. Good agreement is observed. The present analysis demonstrates clearly the dependence of droplet size on a modified Ohnesorge number.

Dynamic breakup of liquid–liquid jets

The axisymmetric, dynamic breakup of a Newtonian liquid jet injected vertically into another immiscible Newtonian liquid at various Reynolds numbers is investigated here. The full transient from jet start-up to breakup into drops was simulated numerically by solving the time-dependent axisymmetric equations of motion and continuity using an algorithm based on the Volume of Fluid (VOF) method that was previously proven successful in simulations of steady-state liquid jets (i.e., of the jet region close to the nozzle before breakup). The algorithm has been further refined here based on its performance on transient problems such as the solution of the free liquid–liquid capillary jet breakup problem. The comparison of the simulation results with previous experimental measurements of jet length under conditions where all forces, i.e., viscous, inertial, buoyancy, and surface tension, are important, can be judged satisfactory given the sensitive dependence of the results on details of the experimental setup that are not available. The comparison involves the jet length till breakup as well as the jet and drop shapes, often far from regular. In comparison with experiment, the results of the present numerical method show a greater sensitivity of the jet length to the Reynolds number than the best predictions previously available based on the linear stability analysis of the free liquid–liquid capillary jet breakup problem.

Dynamics of Slender Viscoelastic Free Jets

Previous studies of slender viscoelastic and Newtonian free surface jets have focused mainly on steady state predictions, guided primarily by industrial textile applications. Dynamical analyses and simulations of fiber flows, even the linearized stability analysis of nontrivial steady states, have lagged behind considerable experimental and textile processing observations and advances. The foundational work of Chang and Lodge [Rheol. Acta., 10 (1971), pp. 448–449] and Petrie [Rheol. Acta., 14 (1975), pp. 955–957], [Elongational Flows, Pitman, London, 1979] has been extended by the axisymmetric modeling and simulation of Beris and Liu [J. Non-Newtonian Fluid Mech., 26 (1988), pp. 363–394], [Comput. Methods Appl. Mech. Engrg., 76 (1989), pp. 179–204], for Maxwell fluids, and Markovich and Renardy [J. Non-Newtonian Fluid Mech., 17 (1985), pp. 13–22] for Johnson–Segalman fluids. The model in [Markovich and Renardy] is a single, nonlinear parabolic equation in which elastic retardation provides dominant smooth...

Steady laminar flow of liquid–liquid jets at high Reynolds numbers*

The axisymmetric steady-state laminar flow of a Newtonian liquid jet injected vertically into another immiscible Newtonian liquid is investigated for various Reynolds numbers. The steady-state solution was calculated by solving the axisymmetric transient equations of motion and continuity using a numerical scheme based on the volume of fluid (VOF) method combined with the new continuum surface force (CSF) algorithm. The analysis takes into account pressure, viscous, inertial, gravitational, and surface tension forces. Comparison with previous experimental measurements, performed on a xylene/water system, under conditions where all of these forces are important, shows good agreement over the entire range of conditions studied. Comparisons of the present numerical method with the numerical results of previous boundary-layer methods help establish their range of validity.

A finite difference technique for solving the Newtonian jet swell problem

A finite difference technique that incorporates a numerical mapping has been successfully applied to analyse both planar and axisymmetric Newtonian jets. A pressure gradient equation and a free-surface slope equation have been derived for free-surface iteration. The computation of pressure inside the jet surface using the pressure gradient equation is stable and accurate at high Reynolds numbers. The free-surface slope equation is needed for updating the free surface and is applicable for jets with strong surface tension effects. The present development can simulate the Newtonian jets for Reynolds numbers as high as 2000 and capillary number as low as 10−5. Numerical predictions by the present technique are close to the results of previous finite element simulations.

Numerical solution of a Newtonian jet emanating from a converging channel

Abstract A theoretical analysis has been carried out to find the shape and final thickness of a Newtonian jet emanating from a converging channel. The gravitational force is neglected but the surface tension effect is included in the present analysis. There are four variables, i.e. the contraction ratio L , the converging angle θ, the Reynolds number Re and the capillary number Ca, that completely determine the flow field. The effects of these four variables on the motion of the jet are examined. The mathematical problem of the jet is formulated with stream function and vorticity as dependent variables. The boundary-fitted coordinate transformation method developed by Thompson et al. is adopted to map the flow geometry into a regular domain for numerical integration, and the finite difference method is applied to solve the flow equations in the transformed plane. We have found that the final thickness of the jet will reach its frozen value as L > 15. As Re > 50 , the jet contraction coefficient is very close to that of a potential Helmholtz jet. Surface tension is only important if Re is small. The jet contraction coefficients as functions of Re and θ are presented. We have also found that a vortex may exist in the converging channel if the converging angle θ > 75° .

An improved finite difference scheme for a Newtonian jet swell problem

AbstractThe finite difference scheme developed by Liu et al. for the Newtonian jet swell problem has been improved: an algebraic approach has been adopted for the numerical mapping; a new formulation for free surface iteration has been proposed; the discrete flow equations have been solved by a combination of the successive line underrelaxation method and the Picard method. With these modifications we are capable of achieving more accurate numerical solutions and a substantial saving in computing time.

Deformation Behavior of a Viscoelastic Fluid Jet

When a viscoelastic fluid jet falling vertically encounters a deformation from the horizontal direction, the jet shows an anomalous behavior very different from that of a Newtonian fluid jet. We observe the deformation behavior of the jet in detail and find that the jet behavior is divided into two conditions. Moreover, we derive an empirical formula of the transition point, making use of the dimensional analysis method.

Finite difference solution of a Newtonian jet swell problem

AbstractA finite difference technique has been developed to study the Newtonian jet swell problem. The streamfunction and vorticity were used as dependent variables to describe the jet flow. The boundary‐fitted co‐ordinate transformation method was adopted to map the flow geometry into a rectangular domain. The standard finite difference method was then applied for solving the flow equations. The location of the jet free surface was updated by the kinematic boundary condition, and an adjustable parameter was included in the free‐surface iteration. We could obtain numerical solutions for the Reynolds number as high as 100, and the differences between the present study and previous finite element simulations on the jet swell ratio are less than 5%.

A STUDY OF THE SINGULARITY IN THE DIE-SWELL PROBLEM

The matched-eigenfunction method of Trogdon and Joseph is improved and used to solve the die-swell problem in an axisymmetric Newtonian liquid jet. The improved solution is the first computation that is consistent with the local analysis of Michael at the separation point. The stream function is expanded using two sets of eigenfunctions to describe the flow inside and outside the die, and these are matched at the exit to determine the expansion coefficients

A study on the deformation behavior of a viscoelastic fluid jet.

When a viscoelastic fluid jet falling down vertically inflicts with a deformation from the horizontal direction, the jet shows anomalous behavior which is very different from that of a Newtonian fluid jet. We observe the deformation behavior of the jet in detail and find that the jet behavior is divided into two types. The transition point of deformation, which is the dividing point, is investigated under various flow conditions. Moreover, we find an empirical formula of the position of the transition point making use of the dimensional analysis method.