INFLUENCE OF TURBULENCE ON THE RESTRAINT OF LIQUID JETS BY SURFACE TENSION IN MICROGRAVITY

An axisymetric phase field model is developed and used to model surface tension forces on liquid jets in microgravity. The previous work in this area is reviewed and a baseline drop tower experiment selected for model comparison. A mathematical model is developed which includes a free surface, a symmetric centerline, and wall boundaries with given contact angles. The model is solved numerically with a compact fourth order stencil on an equally spaced axisymetric grid. After grid convergence studies, a grid is selected and all drop tower tests modeled. Agreement was assessed by comparing predicted and measured free surface rise. Trend wise agreement is good but agreement in magnitude is only fair. Suspected sources of disagreement are the simple turbulence model and the existence of slosh baffles in the experiment that were not included in the model. Nomenclature We = Weber number p = density u = velocity r =jet radius (T= surface tension /= free energy a = constant 1 ^= constant 2 C = phase distribution ^barrier function V = volume ^ = potential function P = pressure jc = distance g = wall function Mradial correction factor Oc = contact angle a =height d=jet diameter D = tank diameter L = tank length Re = Reynolds Number Ka = momentum integral A = spread rate coefficient Introduction Microgravity poses many challenges to the designer of fluid storage for spacecraft. Chief among these are the lack of phase separation in the fluid and the need to supply vapor-free liquid or liquid free vapor to the required spacecraft processes. One of the principal causes of lack of phase separation is the creation of liquid jets. A jet can be created by liquid filling, settling of the fluid to one end of the tank, or even closing a valve to stop the liquid outflow. In normal gravity the gravitational force controls and restricts the liquid jet flow, but in microgravity jets must be contained by surface tension forces. Recent NASA experiments in microgravity" have brought a wealth of data of jet behavior in microgravity. The Vented Tank RessupplyExperiment was surprising in that although it contained a complex geometry of baffles and vanes the limit on liquid inflow was the emergence of a liquid jet from the top of the vane structure. Clearly understanding the restraint of liquid jets by surface tension is key to managing fluids in low gravity. Flow of a submerged axial jet constrained by surface tension in low gravity is similar to stagnation flow against a plate in that the jet hits the constraining surface and is deflected radially out. However, in low garvity the ability of the constraining surface to move in response to the exerted force is unique. In fact to increase the restraining force on the jet as flow rate increases, the surface must deform to decrease the radius of curvature of the free surface, thereby increasing the surface tension force. Eventually the limit is reached where the radius curvature required is sufficiently smaller than the jet diameter that it can no longer restrains the jet. When the deformation of the free surface is large the restraining bulge is long and slender. At this point several other mechanisms act to break down the jet, such as columnar buckling or the //, = turbulent viscosity Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by copyright owner. (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. Taylor Instability where surface waves grow to such amplitude that they pinch a droplet off from the jet. To model this low gravity phenomenon a numerical method that tracks the fluid motion and the surface tension forces is required. Jacqmin has developed a phase-field model that converts the delta-function surface tension force into a continuum function that peaks at the free surface and decays rapidly away. Previous attempts at this formulation have been criticized for smearing the interface but by sharpening the phase function, double gridding the fluid function and using a higher order solution for the fluid function these concerns have been ameliorated. Review of Literature NASA Drop Tower Data is found in references 8-19. Symons" and Spuckler studied the liquid inflow via axial jet into a broad range of tank shapes both empty and partially full. Symons work establishes an empirical limit for jets of Weber number (We) equal to 1.3-1.5 depending on jet velocity profile, where: puav r (1) We = p = density Uw = average jet velocity r -jet radius a ^surface tension Staskus extends the work of Symons by placing baffles in front of the jet. However, no attempt is made to analyze these complex flows. Instead results are reported as a ratio of improvement to the unbaffled jet Weber number. Labus also studies the effect of baffles including ones that break the central jet into several small jets. Aydelott' ~ looks at the problem of a recirculating jet where the liquid level is held constant. Results are classified into four flow patterns, dissipation, geyser formation, aft collection, and circulation. It is the geyser formation/aft collection we concern ourselves with in this paper. Aydelott's assessment that a drop in mixing accompanies this transition indicates the transition's importance. Labus studies both stagnation flow and free surface shape, but js concerned with the back of a flow stagnated against a flat plate in microgravity. Shuttle based experiments in references 1-6 provide valuable low gravity data. Video of Plexiglas tanks during shuttle orbital flight provide several improvements over drop tower tests; including increasing the scale from 4" tanks to 12" tanks and extending the duration of test from 5 seconds to halfan-hour. Tank Pressure Control Experiment has flown three times. The first flight focused on the mixing studies of Aydelott. Advances over Aydelott included actual heat transfer data by using a condensing fluid (refrigerant 113) and longer duration. Bentz" was able to confirm the geysering and circulating regimes of Aydelott, but encountered an asymmetric regime between the two that was even lower heat transfer than aft collection, the lowest heat transfer regime of Aydelott. The second flight of TPCE focused mostly on rapid boiling phenomena, but contains some further tests on mixing. Hasan confirms the findings of Bentz. The third flight was done at a lower fill level but confirms the results of the other flights. The Vented Tank Resupply Experment was designed to look at vanes rather than axial jets, but as noted previously exhibits the classic gesyering behavior. Analytical work is listed in reference 7 and references 20-31. Concus" provides differential equations of the free surface problem, but analyzes only static cases. Nickell analyzes flow from a jet into a liquid and the resultant free surface shape for a normal gravity application, but removes all surface tension from the analysis as secondary. Hochstein" analyzes the microgravity mixing with a volume of fluid approach, but uses only a limited approximation to model the surface tension. Aydelott and Der both analyze the motion of a bubble in the oxygen tank during separation of a Centaur stage with VOF models; noteworthy in these is again the appearance of a geyser. Tegart shows the application of the surface Evolver code of Brakke to actual tank shapes. Brackbill develops an improved surface tension model for VOF codes, but only shows one example of its use for axial jets. Shrader uses a Runga-Kutta scheme to solve the differential equation of free surface deformation in response to an imposed pressure field. This approach is quite promising but does not always converge and limits the interaction between the flow field and the free surface. Jacqmin developed a phase field model of surface tension and implemented as a fourth order accurate scheme using a compact 9-point stencil. Although Jacqmin lays out the basic axisymmetric scheme the computer code and all the examples in his paper are planar. The Jacqmin model will serve as the basis for the present analysis. Note: work with the Jacqmin model has been published previously by Chato and Jaqmin. However, an error in the viscosity resulted in erroneous values for geyser height. (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s) Sponsoring Organization.