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NASA NTRS2000

Modeling the Diurnal Tides in the MLT Region with the Doppler Spread Parameterization of Gravity Waves

Special Session: SA03 The mesosphere/lower thermosphere region: Structure, dynamics, composition, and emission. Ground based and satellite observations in the upper mesosphere and lower thermosphere (MLT) reveal large seasonal variations in the horizontal wind fields of the diurnal and semidiurnal tides. To provide an understanding of the observations, we discuss results obtained with our Numerical Spectral Model (NMS) that incorporates the gravity wave Doppler Spread Parameterization (DSP) of Hines. Our model reproduces many of the salient features observed, and we discuss numerical experiments that delineate the important processes involved. Gravity wave momentum deposition and the seasonal variations in the tidal excitation contribute primarily to produce the large equinoctial amplitude maxima in the diurnal tide. Gravity wave induced variations in eddy viscosity, not accounted for in the model, have been shown by Akmaev to be important too. For the semidiurnal tide, with amplitude maximum observed during winter solstice, these processes also contribute, but filtering by the mean zonal circulation is more important. A deficiency of our model is that it cannot reproduce the observed seasonal variations in the phase of the semidiurnal tide, and numerical experiments are being carried out to diagnose the cause and to alleviate this problem. The dynamical components of the upper mesosphere are tightly coupled through non-linear processes and wave filtering, and this may constrain the model and require it to reproduce in detail the observed phenomenology.

Mayr, H. G.↗
NASA NTRS2000

A Survey and Recent Development of Lunar Gravity Assist

Earth's moon is the largest in the solar system relative to its parent body, the Earth, and can have significant effect on the path of a spacecraft flying close by. This effect, when planned to benefit a specific mission, is called lunar gravity assist (LGA) , and assumes that one aims the spacecraft towards the Moon in such a way that the Moon's gravitational pull will alter the spacecraft's course in a favorable manner. The first application of LGA was in the Apollo program, where the command and lunar modules (and astronauts) were propelled to the Moon such that, if no additional course changes were made, they would swing around the backside of the Moon at a certain altitude and be flung back to Earth to enter the atmosphere at a specified location in the Pacific ocean. This LGA was an essential element saving the lives of the astronauts on Apollo 13. This paper will illustrate the basic mechanics of gravity assist, and list the many applications where it has been used effectively over the past 30 some years. These include missions to the sun and Earth libration points, redirecting a spacecraft from one of these point to a comet encounter, and enhancing payloads by providing an energy boost by the Moon. More recently, studies and actual missions have shown the benefits of LGA in: (1) assisting lunar capture, (2) repositioning geosynchronous communications satellites, (3) boosting spacecraft to Earth escape and departure to planets and other solar system bodies, and (4) allowing small spacecraft to be launched as secondary payloads and released into almost a random orbit from which each may depart and maneuver in space with gravity assists from the Earth and Moon to perform a specific planetary or other mission. This latter application is a recent development by the author and is being applied in 2002 and later years, with piggyback flights on the Ariane 5 which launches comsats to GEO.

Penzo, Paul A.↗
NASA NTRS2000

Structure and Dynamics of the Polar Regions of Mars from MGS Topography and Gravity

The Mars Global Surveyor (MGS) spacecraft has been engaged in systematic mapping of Mars since insertion into Mars orbit in September, 1997. The objectives of the MGS mission are to globally map Mars as well as to quantify seasonal changes on the planet. MGS geophysical/geodetic observations of topography from the Mars Orbiter Laser Altimeter (MOLA) and gravity from the Radio Science investigation are providing significant new insights on both static and time-varying aspects of the polar regions of Mars. These observations have implications for polar processes on diurnal seasonal and climatic timescales. Thus far, MOLA has collected over 300 million precise measurements of Martian topography and cloud heights. The instrument has also provided measurements of the width of the backscattered optical pulse and of the 1064 nm reflectivity of the Martian surface and atmosphere. The along-track resolution of MOLA ground shots is approx. 300 m and the across-track spacing in the polar regions is a maximum of about four kilometers. The vertical accuracy of the topography is determined by the precision recovery of spacecraft orbits from the Radio Science investigation, which includes MOLA altimetry in the form of crossovers. This accuracy is currently approx. one meter. The gravity field is derived from X-band Doppler tracking with typical accuracy of 0.03 to 0.05 mm/s averaged over ten seconds. Current Mars gravity fields are to approximately degree and order 80 but are interpretable to the approximate degree and order 60 (spatial resolution < 180 km), which represents an estimate of the approximate coefficient limit of a field that can be produced without a power law constraint on the gravitational field inversion, which is commonly imposed for solution stability. Additional information is contained in the original extended abstract.

Zuber, Maria T.↗
NASA NTRS1999

Waves in Radial Gravity Using Magnetic Fluid

Terrestrial laboratory experiments studying various fluid dynamical processes are constrained, by being in an Earth laboratory, to have a gravitational body force which is uniform and unidirectional. Therefore fluid free-surfaces are horizontal and flat. Such free surfaces must have a vertical solid boundary to keep the fluid from spreading horizontally along a gravitational potential surface. In atmospheric, oceanic, or stellar fluid flows that have a horizontal scale of about one-tenth the body radius or larger, sphericity is important in the dynamics. Further, fluids in spherical geometry can cover an entire domain without any sidewall effects, i.e. have truly periodic boundary conditions. We describe spherical body-force laboratory experiments using ferrofluid. Ferrofluids are dilute suspensions of magnetic dipoles, for example magnetite particles of order 10 nm diameter, suspended in a carrier fluid. Ferrofluids are subject to an additional body force in the presence of an applied magnetic field gradient. We use this body force to conduct laboratory experiments in spherical geometry. The present study is a laboratory technique improvement. The apparatus is cylindrically axisymmetric. A cylindrical ceramic magnet is embedded in a smooth, solid, spherical PVC ball. The geopotential field and its gradient, the body force, were made nearly spherical by careful choice of magnet height-to-diameter ratio and magnet size relative to the PVC ball size. Terrestrial gravity is eliminated from the dynamics by immersing the "planet" and its ferrofluid "ocean" in an immiscible silicone oil/freon mixture of the same density. Thus the earth gravity is removed from the dynamics of the ferrofluid/oil interface and the only dynamically active force there is the radial magnetic gravity. The entire apparatus can rotate, and waves are forced on the ferrofluid surface by exterior magnets. The biggest improvement in technique is in the wave visualization. Fluorescing dye is added to the oil/freon mixture and an argon ion laser generates a horizontal light that can be scanned vertically. Viewed from above, the experiment is a black circle with wave deformations surrounded by a light background. A contour of the image intensity at any light sheet position gives the surface of the ferrofluid "ocean" at that "latitude". Radial displacements of the waves as a function of longitude are obtained by subtracting the contour line positions from a no-motion contour at that laser sheet latitude. The experiments are run by traversing the forcing magnet with the laser sheet height fixed and images are frame grabbed to obtain a time-series at one latitude. The experiment is then re-run with another laser-sheet height to generate a full picture of the three-dimensional wave structure in the upper hemisphere of the ball as a function of time. We concentrate here on results of laboratory studies of waves that are important in Earth's atmosphere and especially the ocean. To get oceanic scaling in the laboratory, the experiment must rotate rapidly (4-second rotation period) so that the wave speed is slow compared to the planetary rotation speed as in the ocean. In the Pacific Ocean, eastward propagating Kelvin waves eventually run into the South American coast. Theory predicts that some of the wave energy should scatter into coastal-trapped Kelvin waves that propagate north and south along the coast. Some of this coastal wave energy might then scatter into mid-latitude Rossby waves that propagate back westward. Satellite observations of the Pacific Ocean sea-surface temperature and height seem to show signatures of westward propagating mid-latitude Rossby waves, 5 to 10 years after the 1982-83 El Nino. The observational data is difficult to interpret unambiguously owing to the large range of motions that fill the ocean at shorter timescales. This series of reflections giving eastward, north- ward, and then westward traveling waves is observed cleanly in the laboratory experiments, confirming the theoretical expectations.

Ohlsen, D. R.↗
NASA NTRS1999

Decoupling the Roles of Inertia and Gravity on Particle Dispersion

Inertial and gravitational forces determine a particle's motion in a turbulent flow field. Gravity plays the dominant role in this motion by pulling the particles through adjacent regions of fluid turbulence. To better understand and model how a particle's inertia effects its displacement, one must examine the dispersion in a turbulent flow in the absence of gravity. In this paper, we present the particle experiments planned for NASA's KC-135 Reduced-Gravity Aircraft, which generates microgravity conditions for about 20 seconds. We also predict the particle behavior using simulation and ground-based experiments. We will release particles with Stokes numbers of 0.1, 1, and 10 into an enclosed tank of near-isotropic, stationary, and homogenous turbulence. These particle Stoke numbers cover a broad range of flow regimes of interest. Two opposed grids oscillating back and forth generate the turbulent field in the tank with a range of turbulence scales that covers about three orders of magnitude and with turbulence intensities of about ten times the mean velocity. The motion of the particles will be tracked using a stereo image velocimetry technique.

Groszmann, D. E.↗
NASA NTRS1999

Comments on the Operation of Capillary Pumped Loop Devices in Low Gravity

The operation of Capillary Pumped Loops (CPL's) in low gravity has generally been unable to match ground-based performance. The reason for this poorer performance has been elusive. In order to investigate the behavior of a CPL in low-gravity, an idealized, glass CPL experiment was constructed. This experiment, known as the Capillary-driven Heat Transfer (CHT) experiment, was flown on board the Space Shuttle Columbia in July 1997 during the Microgravity Science Laboratory mission. During the conduct of the CHT experiment an unexpected failure mode was observed. This failure mode was a result of liquid collecting and then eventually bridging the vapor return line. With the vapor return line blocked, the condensate was unable to return to the evaporator and dry-out subsequently followed. The mechanism for this collection and bridging has been associated with long wavelength instabilities of the liquid film forming in the vapor return line. Analysis has shown that vapor line blockage in present generation CPL devices is inevitable. Additionally, previous low-gravity CPL tests have reported the presence of relatively low frequency pressure oscillations during erratic system performance. Analysis reveals that these pressure oscillations are in part a result of long wavelength instabilities present in the evaporator pores, which likewise lead to liquid bridging and vapor entrapment in the porous media. Subsequent evaporation to the trapped vapor increases the vapor pressure. Eventually the vapor pressure causes ejection of the bridged liquid. Recoil stresses depress the meniscus, the vapor pressure rapidly increases, and the heated surface cools. The process then repeats with regularity.

Hallinan, K. P.↗
NASA NTRS2001

Normal Gravity Testing of a Microchannel Phase Separator for In Situ Resource Utilization

A microchannel separator, with 2.7 millimeters as the smallest dimension, was tested, and a pore throat structure captured and removed liquid from a gas-liquid stream. The microchannel device was tested over a of gas and liquid flow rates ranging from 0.0005 up to 0. 14 volume fraction of liquid. Four liquids were tested with air. The biggest factor affecting the throughput is the capacity of liquid flow through the pore throat, which is dictated by permeability, liquid viscosity, flow area, pore throat thickness, and pressure difference across the pore throat. Typically, complete separation of gas and liquid fractions was lost when the liquid flow rate reached about 40 to 60% of the pore throat capacity. However, this could occur over a range of 10 to 90% utilization of pore throat capacity. Breakthrough occurs in the microchannel phase separator at conditions similar to the annular to plug flow transition of two-phase microgravity pipe flow implying that operating in the proper flow regime is crucial. Analysis indicates that the Bond number did not affect performance, supporting the premise that hydrodynamic, interfacial, and capillary forces are more important than gravity. However, the relative importance of gravity is better discerned through testing under reduced gravity conditions.

TeGrotenhuis, Ward E.↗
NASA NTRS2001

Large-Scale Flow Structure in Turbulent Nonpremixed Flames under Normal- And Low-Gravity Conditions

It is well known that buoyancy has a major influence on the flow structure of turbulent nonpremixed jet flames. Buoyancy acts by inducing baroclinic torques, which generate large-scale vortical structures that can significantly modify the flow field. Furthermore, some suggest that buoyancy can substantially influence the large-scale structure of even nominally momentum-dominated flames, since the low velocity flow outside of the flame will be more susceptible to buoyancy effects. Even subtle buoyancy effects may be important because changes in the large-scale structure affects the local entrainment and fluctuating strain rate, and hence the structure of the flame. Previous studies that have compared the structure of normal- and micro-gravity nonpremixed jet flames note that flames in microgravity are longer and wider than in normal-gravity. This trend was observed for jet flames ranging from laminar to turbulent regimes. Furthermore, imaging of the flames has shown possible evidence of helical instabilities and disturbances starting from the base of the flame in microgravity. In contrast, these characteristics were not observed in normal-gravity. The objective of the present study is to further advance our knowledge of the effects of weak levels of buoyancy on the structure of transitional and turbulent nonpremixed jet flames. In later studies we will utilize the drop tower facilities at NASA Glenn Research Center (GRC), but the preliminary work described in this paper was conducted using the 1.25-second drop tower located at the University of Texas at Austin. A more detailed description of these experiments can be found in Idicheria et al.

Clemens, N. T.↗
NASA NTRS2001

Combustion of Metals in Carbon Dioxide and Reduced-Gravity Environments

Ongoing exploration and future mission2001110444 s to Mars have given impetus to research on the use of natural resources of the planet. Since carbon dioxide (CO2) constitutes approximately 95% of the Mars atmosphere and since it reacts directly and vigorously with several metals, this investigation focuses on metal-CO2 reactions as a possible combination for rocket-propellant production and energy generation. Magnesium (Mg) has been initially selected as the metal fuel owing to its low ignition temperature and high specific impulse and burning rate in CO2. Our studies in this field started with low gravity (g) combustion tests of Mg in O2, CO2, and CO. Reduced gravity provided a clear picture of the burning phenomena by eliminating the intrusive buoyant flows in high-temperature metal reactions and by removing the destructive effect of gravity on the shape of molten metal samples. Suspended cylindrical metal samples of 2, 3, and 4-mm in diameter and length were radiatively ignited in low-g to generate free-floating samples exhibiting a spherically symmetric flame with increasing metal-oxide accumulation in an outer shell. For the Mg-CO2 combination, burning times twice as long as in normal-g and five times longer than in Mg-O2 flames were observed, revealing a diffusion-controlled reaction. The burning time is proportional to the square of the sample diameter. In tests conducted with pure CO, combustion was not possible without constant heating of the sample due to the formation of a thick carbon-containing coating around the Mg sample generated by surface reactions. The following work presents two new studies that attempt to explain some of the low-g experimental observations. First, a simplified one-dimensional, quasi-steady numerical model is developed to obtain temperature, species concentrations, and burning rates of the spherically symmetric diffusion flame around the Mg sample burning in O2 and CO2. Second, a Planar Laser Induced Fluorescence (PLIF) technique is implemented to provide spatially resolved measurements of magnesium oxide (MgO) in the reaction zone of Mg samples burning in O2 and CO2. These experiments reveal fundamental differences between the two combustion systems.

Branch, M. C.↗
NASA NTRS2000

In Situ Observations of PSCs Generated by Gravity Waves

During SOLVE, the bulk of the in-situ observations of PSCs are of large scale extended structures associated with synoptic scale cooling. The nature of these structures is also determined by layers of high relative NOy that have been stretched into thin layers by advective processes. Some of the in situ observations, however, are clearly correlated with gravity wave signatures. The first goal of this work is to examine these cases and evaluate gravity wave parameters. In particular, we are interested in the intrinsic periods of the waves and their temperature amplitude, which are key ingredients in the nucleation process. Secondly, we will examine some rudimentary properties of the particle size distributions and composition, comparing these with in situ observations of the more extended PSC features. Finally, we will attempt to ascertain the mechanism which generates the gravity waves.

Pfister, Leonhard↗
NASA NTRS2000

Regulation of Osteoblast Survival by the Extracellular Matrix and Gravity

Spaceflight adversely affects the skeleton, posing a substantial risk to astronaut's health during long duration missions. The reduced bone mass observed in growing animals following spaceflight is due at least in part to inadequate bone formation by osteoblasts. Thus, it is of central importance to identify basic cellular mechanisms underlying normal bone formation. The fundamental ideas underlying our research are that interactions between extracellular matrix proteins, integrin adhesion receptors, cytoplasmic signaling and cytoskeletal proteins are key ingredients for the proper functioning of osteoblasts, and that gravity impacts these interactions. As an in vitro model system we used primary fetal rat calvarial cells which faithfully recapitulate osteoblast differentiation characteristically observed in vivo. We showed that specific integrin receptors ((alpha)3(beta)1), ((alpha)5(beta)1), ((alpha)8(betal)1) and extracellular matrix proteins (fibronectin, laminin) were needed for the differentiation of immature osteoblasts. In the course of maturation, cultured osteoblasts switched from depending on fibronectin and laminin for differentiation to depending on these proteins for their very survival. Furthermore, we found that manipulating the gravity vector using ground-based models resulted in activation of key intracellular survival signals generated by integrin/extracellular matrix interactions. We are currently testing the in vivo relevance of some of these observations using targeted transgenic technology. In conclusion, mechanical factors including gravity may participate in regulating survival via cellular interactions with the extracellular matrix. This leads us to speculate that microgravity adversely affects the survival of osteoblasts and contributes to spaceflight-induced osteoporosis.

Globus. Ruth K.↗
NASA NTRS2001

Particle Engulfment and Pushing (PEP): Past Micro-Gravity Experiments and Future Experimental Plan on the International Space Station (ISS)

The interaction of an insoluble particle with a growing solid-liquid interface (SLI) has been a subject of investigation for the four decades. For a metallurgist or a material scientist understanding the fundamental physics of such an interaction is relevant for applications that include distribution of reinforcement particles in metal matrix composites, inclusion management in castings, and distribution of Y2Ba1Cu1O5 (211) precipitates (flux pinning sites) in Y1Ba2Cu3O7 (123) superconducting crystals. The same physics is also applicable to other areas including geological applications (frost heaving in soils) and preservation of biological cells. Experimentally this interaction can be quantified in terms of a critical growth velocity, Vcr, of the SLI below which particles are pushed ahead of the advancing interface, and above which the particles are engulfed. Past experimental evidence suggests that this Vcr is an inverse function of the particle radius, R. In order to isolate the fundamental physics that governs such a relationship it is necessary to minimize natural convection at the SLI that is inherent in ground based experiments. Hence for the purpose of producing benchmark data (Vcr vs. R) PEP is a natural candidate for micro-gravity experimentation. Accordingly, experiments with pure Al containing a dispersion of ZrO2 particles and an organic analogue, succinonitrile (SCN) containing polystyrene particles have been performed on the LMS and USMP-4 mission respectively. In this paper we will summarize the experimental data that was obtained during these two micro-gravity missions and show that the results differ compared to terrestrial experiments. We will also discuss the basic elements of our analytical and numerical model and present a comparison of the predictions of these models against micro-gravity experimental data. Finally. we will discuss our future experimental plan that includes the ISS glovebox and MSRRl.

Sen, Subhayu↗
NASA NTRS1994

Evidence for Gravity as a Cue for Absolute Size and Distance

The constant force of gravity constrains the motion of objects in natural environments. In principle, gravity could provide valuable scaling information about size and distance of unfamiliar objects. We supply empirical evidence that observers use some of the information provided by gravity to scale objects. However, scaling judgments based on retinal size and event duration alone are nearly as accurate as those made with gravitational acceleration information.

Hecht, Heiko↗
NASA NTRS2001

Passive and Active Stabilization of Liquid Bridges in Low Gravity

Tests are planned in the low gravity environment of the International Space Station (ISS) of new methods for the suppression of the capillary instability of liquid bridges. Our suppression methods are unusual in that they are not limited to liquid bridges having very special properties and may impact a variety of low-gravity and earth-based technologies. There are two main approaches to be investigated: (1) Passive Acoustic Stabilization (PAS); and (2) Active Electrostatic Stabilization (AES). In PAS, the suppression of the mode growth is accomplished by placing the bridge in an acoustic field having the appropriate properties such that the acoustic radiation pressure automatically pulls outward on the thinnest portion of the bridge. In AES, the bridge deformation is sensed optically and counteracted by actively adjusting the electrostatic Maxwell stresses via two ring electrodes concentric with the slightly conducting bridge to offset the growth of the unstable mode. While the present work emphasizes cylindrical bridges, the methods need not be restricted to that case. The methods to be explored are relevant to the suppression of capillary instabilities in floating zone crystal growth, breakup of liquid jets and columns, bubbles, and annular films as well as the management of coolants or propellants in low-gravity.

Marston, Philip L.↗
NASA NTRS2000

Comparison of Global Distributions of Zonal-Mean Gravity Wave Variance Inferred from Different Satellite Instruments

Gravity wave temperature fluctuations acquired by the CRISTA instrument are compared to previous estimates of zonal-mean gravity wave temperature variance inferred from the LIMS, MLS and GPS/MET satellite instruments during northern winter. Careful attention is paid to the range of vertical wavelengths resolved by each instrument. Good agreement between CRISTA data and previously published results from LIMS, MLS and GPS/MET are found. Key latitudinal features in these variances are consistent with previous findings from ground-based measurements and some simple models. We conclude that all four satellite instruments provide reliable global data on zonal-mean gravity wave temperature fluctuations throughout the middle atmosphere.

Preusse, Peter↗
NASA NTRS2002

Rayleigh-Taylor Gravity Waves and Quasiperiodic Oscillation Phenomenon in X-ray Binaries

Accretion onto compact objects in X-ray binaries (black hole, neutron star (NS), white dwarf) is characterized by non-uniform flow density profiles. Such an effect of heterogeneity in presence of gravitational forces and pressure gradients exhibits Rayleigh-Taylor gravity waves (RTGW). They should be seen as quasiperiodic wave oscillations (QPO) of the accretion flow in the transition (boundary) layer between the Keplerian disk and the central object. In this paper the author shows that the main QPO frequency, which is very close to the Keplerian frequency, is split into separate frequencies (hybrid and low branch) under the influence of the gravitational forces in the rotational frame of reference. The RTGWs must be present and the related QPOs should be detected in any system where the gravity, buoyancy and Coriolis force effects cannot be excluded (even in the Earth and solar environments). The observed low and high QPO frequencies are an intrinsic signature of the RTGW. The author elaborates the conditions for the density profile when the RTGW oscillations are stable. A comparison of the inferred QPO frequencies with QPO observations is presented. The author finds that hectohertz frequencies detected from NS binaries can be identified as the RTGW low branch frequencies. The author also predicts that an observer can see the double NS spin frequency during the NS long (super) burst events when the pressure gradients and buoyant forces are suppressed. The Coriolis force is the only force which acts in the rotational frame of reference and its presence causes perfect coherent pulsations with a frequency twice of the NS spin. The QPO observations of neutron binaries have established that the high QPO frequencies do not go beyond of the certain upper limit. The author explains this observational effect as a result of the density profile inversions. Also the author demonstrates that a particular problem of the gravity waves in the rotational frame of reference in the approximation of very small pressure gradients is reduced to the problem of the classical oscillator in the rotational frame of reference which was previously introduced and applied for the interpretation of kHZ QPO observation by Osherovich & Titarchuk.

Titarchuk, Lev↗
NASA NTRS2003

Crustal Structure of the Iceland Region from Spectrally Correlated Free-air and Terrain Gravity Data

Seismic refraction studies have provided critical, but spatially restricted constraints on the structure of the Icelandic crust. To obtain a more comprehensive regional view of this tectonically complicated area, we spectrally correlated free-air gravity anomalies against computed gravity effects of the terrain for a crustal thickness model that also conforms to regional seismic and thermal constraints. Our regional crustal thickness estimates suggest thickened crust extends up to 500 km on either side of the Greenland-Scotland Ridge with the Iceland-Faeroe Ridge crust being less extended and on average 3-5 km thinner than the crust of the Greenland-Iceland Ridge. Crustal thickness estimates for Iceland range from 25-35 km in conformity with seismic predictions of a cooler, thicker crust. However, the deepening of our gravity-inferred Moho relative to seismic estimates at the thermal plume and rift zones of Iceland suggests partial melting. The amount of partial melting may range from about 8% beneath the rift zones to perhaps 20% above the plume core where mantle temperatures may be 200-400 C above normal. Beneath Iceland, areally limited regions of partial melting may also be compositionally and mechanically layered

Leftwich, T. E.↗
NASA NTRS2003

Prediction of Bubble Diameter at Detachment from a Wall Orifice in Liquid Cross Flow Under Reduced and Normal Gravity Conditions

Bubble formation and detachment is an integral part of the two-phase flow science. The objective of the present work is to theoretically investigate the effects of liquid cross-flow velocity, gas flow rate embodied in the momentum flux force, and orifice diameter on bubble formation in a wall-bubble injection configuration. A two-dimensional one-stage theoretical model based on a global force balance on the bubble evolving from a wall orifice in a cross liquid flow is presented in this work. In this model, relevant forces acting on the evolving bubble are expressed in terms of the bubble center of mass coordinates and solved simultaneously. Relevant forces in low gravity included the momentum flux, shear-lift, surface tension, drag and inertia forces. Under normal gravity conditions, the buoyancy force, which is dominant under such conditions, can be added to the force balance. Two detachment criteria were applicable depending on the gas to liquid momentum force ratio. For low ratios, the time when the bubble acceleration in the direction of the detachment angle is greater or equal to zero is calculated from the bubble x and y coordinates. This time is taken as the time at which all the detaching forces that are acting on the bubble are greater or equal to the attaching forces. For high gas to liquid momentum force ratios, the time at which the y coordinate less the bubble radius equals zero is calculated. The bubble diameter is evaluated at this time as the diameter at detachment from the fact that the bubble volume is simply given by the product of the gas flow rate and time elapsed. Comparison of the model s predictions was also made with predictions from a two-dimensional normal gravity model based on Kumar-Kuloor formulation and such a comparison is presented in this work.

Nahra, Henry K.↗