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At least 415 records · Page 23

Temperature Correction of Pressure-Sensitive Paints Simplified

Pressure-sensitive paint (PSP) has become a useful tool to augment conventional pressure taps in measuring the surface pressure distribution of aerodynamic components in wind tunnel testing. Although PSP offers the advantage of nonintrusive global mapping of the surface pressure, one prominent drawback to the accuracy of this technique is the inherent temperature sensitivity of PSP's luminescent intensity. Typical aerodynamic surface PSP tests rely on the coated surface to be both spatially and temporally isothermal, along with conventional instrumentation, to yield the highest accuracy pressure mappings. In some tests, however, spatial and temporal thermal gradients are generated by the nature of the test, as in a blowing jet impinging on a surface. In these cases, high accuracy and reliable data cannot be obtained unless the temperature variations on the painted surface are accounted for. A new temperature-correction technique was developed at the NASA Glenn Research Center at Lewis Field to collapse a "family" of PSP calibration curves to a single curve of intensity ratio versus pressure. This correction allows a streamlined procedure to be followed whether or not temperature information is used in the data reduction of the PSP.

Bencic, Timothy J.↗

Pressure-Sensitive Paint Measurements on Surfaces with Non-Uniform Temperature

Pressure-sensitive paint (PSP) has become a useful tool to augment conventional pressure taps in measuring the surface pressure distribution of aerodynamic components in wind tunnel testing. While the PSP offers the advantage of a non-intrusive global mapping of the surface pressure, one prominent drawback to the accuracy of this technique is the inherent temperature sensitivity of the coating's luminescent intensity. A typical aerodynamic surface PSP test has relied on the coated surface to be both spatially and temporally isothermal, along with conventional instrumentation for an in situ calibration to generate the highest accuracy pressure mappings. In some tests however, spatial and temporal thermal gradients are generated by the nature of the test as in a blowing jet impinging on a surface. In these cases, the temperature variations on the painted surface must be accounted for in order to yield high accuracy and reliable data. A new temperature correction technique was developed at NASA Lewis to collapse a "family" of PSP calibration curves to a single intensity ratio versus pressure curve. This correction allows a streamlined procedure to be followed whether or not temperature information is used in the data reduction of the PSP. This paper explores the use of conventional instrumentation such as thermocouples and pressure taps along with temperature-sensitive paint (TSP) to correct for the thermal gradients that exist in aeropropulsion PSP tests. Temperature corrected PSP measurements for both a supersonic mixer ejector and jet cavity interaction tests are presented.

Bencic, Timothy J.↗

Wettability Effects on Falling Film Flow and Heat Transfer Over Horizontal Tubes in Jet Flow Mode

The performance of a falling-film heat exchanger is strongly linked to the surface characteristics and the heat transfer processes that take place over the tubes. The primary aim of this numerical study is to characterize the influence of surface wettability on the film flow behavior and its associated surface heat transfer in the jet-flow mode. Volume of fluid (VOF) based simulations are carried out for horizontal tubes with different surface wettabilities. The wettability of the tube surfaces is represented using the Kistler's dynamic contact angle model. Surface wettability effects ranging from superhydrophilic to superhydrophobic are studied by varying the equilibrium contact angle from 2 deg to 175 deg. Two different liquid mass flow rates of 0.06 and 0.18 kg/m-s corresponding to the inline and staggered jet flow modes are studied. Results are presented in terms of the liquid film thickness, the contact areas between the different phases (solid–liquid and liquid–air), and the heat transfer coefficient or Nusselt number. The resistance imposed by the increasing contact angles inhibits the extent of the liquid spreading over the tube surface, and this, in turn, influences the liquid film thickness, and the wetted area of the tube surface. A significant decrement in the heat transfer rate from the tube surfaces was observed as the equilibrium contact angle increased from 2 deg to 175 deg. The local distributions of the Nusselt number over the tube surface are strongly influenced by the flow recirculation in the liquid bulk.

Engineering↗

Attached Jet Velocity Profiles in Mixing Tanks

Free jets have been studied in detail over much of the last century, but the theory for offset and attached jets remains incomplete. Attached jets differ from free jets in that they lose momentum to nearby surfaces, attenuating their velocities. The velocity profiles of free circular jets are nearly Gaussian, with quantitative mathematical descriptions derived from first principles by Goertler and Tollmien (Rajaratnam, 1976). In contrast, mathematical descriptions of three-dimensional attached jets from circular nozzles remain much less mature. Agelin-Chaab and Tachie (2011) used particle imaging velocimetry of a three-dimensional attached jet to show that the scaled velocity decays with scaled distance from the nozzle with a power law exponent between -1.15 and -1.21, which is larger in magnitude than that of a free jet. However, quantitative analytical expressions for the velocity profiles of attached jets similar to those of free jets remain elusive. This paper addresses this critical gap. Here we evaluate the velocity profiles of three-dimensional offset jets emerging from circular nozzles that that become attached jets. These jets lose momentum due to interactions with nearby surfaces and are important to evaluating flows in mixing vessels and to suspending solids and trapped gases in radioactive waste tanks. Despite the importance of attached jets, prior insight has been purely experimental, limited to overly simplistic analytical models, or restricted to computationally expensive computational fluid dynamics case studies. We compare the expression of Verhoff (1963) to experimental results to find reasonable quantitative agreement. As stated by Agelin-Chaab and Tachie (2011), “detailed velocity measurements of 3D offset jets are rare.” Such remains the case. This study adds to the literature by providing information at two additional Reynolds numbers (1.43 106 and 1.87 106) and evaluating simple but accurate expressions for velocity profiles. These Reynolds numbers are an order of magnitude higher than prior reports. The semi-empirical stream wise velocity profile perpendicular to the surface proposed by Verhoff (1963) is in approximate agreement with these velocity profiles, which is surprising because these attached jets are three-dimensional instead of two-dimensional as evaluated by Verhoff. However, additional work is necessary to fully describe these profiles quantitatively.

jet, attached jet, offset jet↗

Acoustics of attached and partially attached flow for simplified OTW configurations with 5:1 slot nozzle

The acoustics of simple engine over-the-wing configurations, with complete and partially attached jet flow to the shielding surface were studied with scale models. The nozzle used consisted of a 5:1 slot nozzle operated at a nominal jet Mach number of 0.6, with the flow directed parallel to and at angles up to 10 deg toward the shielding surface. The flow field at the trailing edge of each nozzle/surface configuration was mapped. the results indicate that, with attached flow, the jet flow field is stretched in the flow direction resulting in locally higher velocities than those for partially attached flow or nozzle only flow. The stretching of the flow field increases the noise levels for the attached flow cases compared to those with only partially attached flow. With attached flow, the shielding benefits are substantially reduced compared with fully detached flow.

Vonglahn, U.↗

Experiments on Exhaust Noise of Tightly Integrated Propulsion Systems

A wide-ranging series of tests have been completed that seek to map the effects of installation, including jet by jet interaction effects, on exhaust noise from various nozzles in forward flight. The primary data was far-field acoustic spectral directivity. The goals of the test series were (i) to generate enough data for empirical models of the different effects, and (ii) to provide data for advanced computational noise predictions methods applied to simplified yet realistic configurations. Data is presented that demonstrate several checks on data quality and that provide an overview of trends observed to date. Among the findings presented here: (i) Data was repeatable between jet rigs for single nozzles with and without surfaces to within +/- 0.5 dB. (ii) The presence of a second jet caused a strong reduction of the summed noise in the plane of the two plumes and an increase over the expected source doubling in most other azimuthal planes. (iii) The impact of the second jet was reduced when the jets were unheated. (iv) The impact of adding a second isolated rectangular jet was relatively independent of the nozzle aspect ratio up to aspect ratio 8:1. (v) Forward flight had similar impact on a high aspect ratio (8:1) jet as on an axisymmetric jet, except at the peak noise angle where the impact was less. (vi) The effect of adding a second round jet to a tightly integrated nozzle where the nozzle lip was less than a diameter from the surface was very dependent upon the length of the surface downstream of the nozzle. (vii) When the nozzles were rectangular and tightly integrated with the airframe surface the impact of a second jet was very dependent upon how close together the two jets were. This paper serves as an overview of the test; other papers presented in the same conference will give more detailed analysis of the results.

Noise↗

Air-coupled tsunamis generated from impacts and airbursts: Our understanding before Hunga-Tonga Hunga-Ha'apai

The effort to prevent or mitigate the effects of an impact on Earth is known as planetary defense. A significant component of planetary defense research involves risk assessment. Much of our understanding of the risk from near-Earth objects comes from the geologic record in the form of impact craters, but not all asteroid impacts are crater-forming events. Small asteroids explode before reaching the surface, generating an airburst, and most impacts into the ocean do not penetrate the water to form a crater in the sea floor. The risk from these non-crater-forming ocean impacts and airbursts is difficult to quantify and represents a significant uncertainty in our assessment of the overall threat. We are currently working to better understand impact scenarios that can generate dangerous tsunamis. One of the suggested mechanisms for the production of asteroid–generated tsunamis is by direct coupling of the pressure wave to the water, analogous to the means by which a moving weather front can generate a meteotsunami. To test this hypothesis, we ran a series of airburst simulations and provided time-resolved pressure and wind profiles to use as source functions for tsunami models. We used the CTH hydrocode to model the various airburst scenarios to compare to the results of other simulations and provide time dependent boundary conditions as input to shallow-water wave propagation codes. The strongest and most destructive meteotsunamis are generated by atmospheric pressure oscillations with amplitudes of only a few hPa1 (mbar), corresponding to changes in sea level of a few cm. The resulting wave is strongest when there is a resonance between the ocean and the atmospheric forcing. A Proudman resonance takes place when the atmospheric disturbance’s translational speed (U) equals the longwave phase speed $\sqrt{gh}$ of shallow water wave. Coupling is strongest when the Froude number (Fr=U/c) is unity. A weather front propagates much slower than the speed of sound, so meteotsunamis are most common and dangerous in shallow bodies of water such as the Mediterranean Sea or Lake Michigan. By contrast, the blast wave from an airburst or crater-forming impact propagates at a speed faster than a tsunami in the deepest ocean, and a Proudman resonance cannot be achieved even though the overpressures are orders of magnitude greater. However, blast wave profiles are N-waves in which a sharp shock wave leading to overpressure is followed by a more gradual rarefaction to a much longer-duration underpressure phase. Even though the blast outruns the water wave it is forcing, the tsunami should continue to be driven by the out-of-resonance gradient associated with the suction phase, which may depend strongly on the details of the airburst or impact scenario. The open question is whether there are any conditions under which such an airburst-driven tsunami can be dangerous enough to contribute to the overall impact risk. We have also identified other potential mechanisms for airburst-generated tsunamis: 1) reaction force at the surface from the plume ejected into space, which carries significant momentum, 2) expanding toroidal vortices at the surface, which travel more slowly than the shock wave and can generate a Proudman resonance in relatively shallow ocean (such as continental shelf), and 3) steam explosion from seawater ablation by a “Type II” (Libyan Desert Glass-type) airburst in which the hot vapor jet descends to the surface. On January 15, 2022, the Hunga-Tonga Hunga-Ha’apai volcano, located approximately 60 km north of Tongatapu, the main island of Tonga, violently erupted with a powerful explosion, culminating the period of volcanic activity that started in December of 2021. This event and resulting tsunamis provided an existence proof for the air pressure wave coupling mechanism we proposed. It also suggests that it can be stronger and more significant over much greater distances than we contemplated, leading to global tsunamis associated with impact events on land as well as in the water. Large atmospheric explosions generate global Lamb waves with larger amplitudes, longer periods, and slower speeds than the local and regional blast waves we modeled prior to that event. This paper reviews our analysis and modeling of airburst-driven tsunamis prior to the 2022 Hunga-Tonga Hunga-Ha’apai tsunami, which was the subject of two presentations at the 2023 Planetary Defense Conference and is the subject of another paper currently in preparation.

54 ENVIRONMENTAL SCIENCES↗

Analytical and experimental investigation of circulation control by means of a turbulent Coanda jet

An analytical and experimental investigation of circulation control on a circular cylinder by means of tangential blowing (Coanda effect) is presented. The analytical method developed has also been used to estimate the blowing coefficients required for achieving potential flow on airfoils with flaps. The analysis is presented for conditions for which the flow in the boundary layer ahead of the jet exit is turbulent. The turbulent boundary layer and the jet layer on the upper surface, and the turbulent boundary layer on the lower surface are computed by a multi-strip integral method. The region of integration is between the correponding transition and separation points on each surface. Longitudinal curvature effects, which give rise to a radial pressure gradient across the jet layer and to an additional adverse tangential pressure gradient just upstream of the separation point, are included in the jet layer analysis in an approximate manner. The longitudinal curvature effect is found to have a pronounced influence on the separation of the jet layer.

Levinsky, E. S.↗

Nucleus morphology of Comet Halley

Images obtained by the Halley multicolor camera were used to determine the projected size and shape of the nucleus. The location of the terminator and numerous surface features were determined. There is good correlation between the brightest surface features and the dust jets; however, many bright features are seen which are not associated with jets. Most of the observed features are circular and appear to be related to surface elevation. The angularity of the terminator gives an indication of the three-dimensional structure of the face which was observed.

Reitsema, H. J.↗

The Onset of Oscillations in Non-Premixed Combustion

In a microgravity environment, molecular diffusion is the primary mechanism by which fuel and oxidizer that are initially separated are brought together to the reaction zone. Combustion systems in microgravity are therefore primed to diffusive-thermal instabilities. One such instability appears in the form of spontaneous oscillations. Oscillations were observed in condensed-phase fuels and gas-jet diffusion flames, and in microgravity jet-flames, candle flames and spherical flames surrounding large fiber-supported fuel droplets. The nature of oscillations is quite different in each of these cases: the droplet flame exhibits radial oscillations, the edge of the candle flame is seen to move back and forth along the hemispherical flame surface and the jet-flame oscillations are primarily up and down along the axis. Despite these differences, associated mainly with the mode of oscillation, one may identify some common factors: in all cases the flame exhibits low-frequency oscillations, oscillations are only observed in special mixtures and their onset occur only at near-extinction conditions.

Matalon, Moshe↗

Simulation of Flow Through Breach in Leading Edge at Mach 24

A baseline solution for CFD Point 1 (Mach 24) in the STS-107 accident investigation was modified to include effects of holes through the leading edge into a vented cavity. The simulations were generated relatively quickly and early in the investigation by making simplifications to the leading edge cavity geometry. These simplifications in the breach simulations enabled: 1) A very quick grid generation procedure; 2) High fidelity corroboration of jet physics with internal surface impingements ensuing from a breach through the leading edge, fully coupled to the external shock layer flow at flight conditions. These simulations provided early evidence that the flow through a 2 inch diameter (or larger) breach enters the cavity with significant retention of external flow directionality. A normal jet directed into the cavity was not an appropriate model for these conditions at CFD Point 1 (Mach 24). The breach diameters were of the same order or larger than the local, external boundary-layer thickness. High impingement heating and pressures on the downstream lip of the breach were computed. It is likely that hole shape would evolve as a slot cut in the direction of the external streamlines. In the case of the 6 inch diameter breach the boundary layer is fully ingested.

Gnoffo, Peter A.↗

Evolution of the Scope and Capabilities of Uplink Support Software for Mars Surface Operations

In January of 2004 both of the Mars Exploration Rover spacecraft landed safely, initiating daily surface operations at the Jet Propulsion Laboratory for what was anticipated to be approximately three months of mobile exploration. The longevity of this mission, still ongoing after ten years, has provided not only a tremendous return of scientific data but also the opportunity to refine and improve the methodology by which robotic Mars surface missions are commanded. Since the landing of the Mars Science Laboratory spacecraft in August of 2012, this methodology has been successfully applied to operate a Martian rover which is both similar to, and quite different from, its predecessors. For MER and MSL, daily uplink operations can be most broadly viewed as converting the combined interests of both the science and engineering teams into a spacecraft-safe set of transmittable command files. In order to accomplish these ends a discrete set of mission-critical software tools were developed which not only allowed for conformation to established JPL standards and practices but also enabled innovative technologies specific to each mission. Although these primary programs provided the requisite capabilities for meeting the high-level goals of each distinct phase of the uplink process, there was little in the way of secondary software to support the smooth flow of data from one phase to the next. In order to address this shortcoming a suite of small software tools was developed to aid in phase transitions, as well as to automate some of the more laborious and error-prone aspects of uplink operations. This paper describes the evolution of this software suite, from its initial attempts to merely shorten the duration of the operator's shift, to its current role as an indispensable tool enforcing workflow of the uplink operations process and agilely responding to the new and unexpected challenges of missions which can, and have, lasted many years longer than originally anticipated.

CoUGAR↗

CHESS 2025: Crown polygons and extracted reflectance for field sampling sites

This dataset contains (1) crown polygons for each tree, meadow, and shrub site sampled in the 2025 Colorado Headwaters Ecological Spectroscopy Study (CHESS) campaign (in geojson format, .geojson) and (2) extracted reflectance, uncertainty, and shade estimates for each crown polygon from the 2018 National Ecological Observatory Network (NEON) and 2025 CHESS campaigns. (in CSV format, .csv). Additional metadata are provided in a data dictionary describing column names and definitions (dd.csv), and in a file-level metadata file (flmd.csv). Crown polygons were manually delineated for each site in the 2025 campaign using a combination of field-collected GPS data (doi:10.15485/3022418), RGB (red, green, blue) and false color reflectance mosaics (doi:10.15485/3013535), and LiDAR-derived (Light Detection and Ranging) canopy height (CHM) and digital surface (DSM) models (DOI and citation to be added upon publication). Where there was misalignment between the spectrometer- and LiDAR-derived data products, polygons prioritized alignment with the spectrometer-derived data products. Polygons were delineated conservatively to only select pixels representative of vegetation samples collected in the field. Crown polygons for 2018 are published at (doi:10.15485/1618130) and were developed using the same protocol. For each polygon, all pixels from all flightlines were extracted where the pixel centroid was contained within the polygon. For each pixel, we extracted the surface reflectance, uncertainty, and shade estimates. Details on the extracted datasets are available at (doi:10.15485/3013527, doi:10.15485/3013535). CHESS Project Description: The Colorado Headwaters Ecological Spectroscopy Study (CHESS) comprised a multi-week airborne remote sensing and field observation campaign in the Upper Gunnison Basin, Colorado, conducted in June and July of 2025. Airborne remote sensing was conducted by the National Ecological Observatory Network Airborne Observation Platform (NEON AOP), concurrent with a field campaign run by the Rocky Mountain Biological Laboratory (RMBL), the Lawrence Berkeley National Laboratory (LBNL) and SLAC National Accelerator Laboratory Watershed Function Science Focus Area (SFA), and NASA-JPL (Jet Propulsion Laboratory) Earth Surface Mineral Dust Source Investigation (EMIT) program. Between June 10 and July 18, 2025, the NEON AOP flight team collected high-resolution aerial imaging spectroscopy and Light Detection and Ranging (LiDAR) data over three domains: the Upper East River (CRBU), Almont Triangle (ALMO), and the Upper Taylor Basin (UPTA). In coordination with the flights, a field campaign acquired ground-truth observations, including observations of vegetation composition, foliar traits, forest demography, and subsurface properties in 18 core sampling areas within the domains. Additional surface water observations were taken at over 380 point locations. All CHESS campaign datasets can be found within the CHESS ESS-DIVE data portal: https://data.ess-dive.lbl.gov/portals/chess. Funding Acknowledgment: This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004) and was funded by EMIT Extended Mission Phase E Science.

2018 NEON and 2025 CHESS Campaigns↗

Measurements of large scale disturbances in rectangular wall jets

Detailed measurements of the velocity field and wall surface pressure distribution of a tone-generating wall jet were obtained in the Ames 7- by 10-foot Wind Tunnel and anechoeic chamber. The jet issued from a 0.508 cm by 10.16 cm nozzle adjacent to a 3.81 cm wall, with an initially parabolic velocity profile at the exit. The jet was operated at Reynolds numbers, R, of 4910, 7740, and 9410. At R = 4910 and 7740, the flow appeared as a double-rowed vortex array which formed upstream of the edge and convected downstream without change in structure. At R = 9410, apparent vortex pairing occurred at the edge location. These flow characteristics were observed in both velocity field measurements and flow visualization records. The unsteady vorticity field of the jet was computed from phase-averaged hot-wire anemometer measurements of the velocity field. The associated acoustic field was estimated by the method of Green's functions, and was found to be similar to measurements of the sound field. These observations tend to support theoretical predictions that vortex shedding at the edge tends to cancel the effect of acoustic sources in other parts of the flow field.

Horne, W. C.↗

Atmospheric Turbulent Intermittency Over the Arctic Sea‐Ice Surface During the MOSAiC Expedition

Turbulent motions in the Arctic stable boundary layer are characterized by intermittency, but they are rarely investigated due to limited observations, in particular over the sea‐ice surface. In the present study, we explore the characteristics of turbulent intermittency over the Arctic sea‐ice surface using data collected during the Multidisciplinary drifting Observation for the Study of Arctic Climate expedition from October 2019 to September 2020. We first develop a new algorithm, which performs well in identifying the spectral gap over the Arctic sea‐ice surface. Then the characteristics of intermittency are investigated. It is found that the strength of intermittency increases under the conditions of light surface wind speed, small surface wind speed gradient, and strong surface air temperature gradient. The momentum flux, sensible heat flux, and latent heat flux calculated by raw eddy‐covariance fluctuations are overestimated by 3%, 10%, and 24%, respectively, because submesoscale motions are included. Furthermore, the characteristics of the atmospheric boundary layer structure under various intermittency conditions reveal that strong low‐level jets are favorable to surface turbulent motions that result in weak intermittency, while strong temperature inversions above the surface layer suppress surface turbulent motions and lead to strong intermittency.

54 ENVIRONMENTAL SCIENCES↗

An experimental study of the flow field surrounding a subsonic jet in a cross flow

An experimental investigation of the flow interaction of a 5.08 cm (2.00 in.) diameter round subsonic jet exhausting perpendicularly to a flat plate in a subsonic cross flow was conducted in the NASA Ames 7x1O ft. Wind Tunnel Number One. Flat plate surface pressures were measured at 400 locations in a 30.48 cm (12.0 in.) concentric circular array surrounding the jet exit. Results from these measurements are provided in tabular and graphical form for jet-to-crossflow velocity ratios ranging from 4 to 12, and for jet exit Mach numbers ranging from 0.50 to 0.93. Laser doppler velocimeter (LDV) three component velocity measurements were made in selected regions in the developed jet plume and near the flat plate surface, at a jet Mach number of 0.50 and jet-to-crossflow velocity ratios of 6 and 8. The results of both pressure and LDV measurements are compared with the results of previous experiments. In addition, pictures of the jet plume shape at jet velocity ratios ranging from 4 to 12 were obtained using schleiren photography. The LDV measurements are consistent with previous work, but more extensive measurements will be necessary to provide a detailed picture of the flow field. The surface pressure results compare closely with previous work and provide a useful characterization of jet induced surface pressures. The results demonstrate the primary influence of jet velocity ratio and the secondary influence of jet Mach number in determining such surface pressures.

Dennis, Robert Foster↗

Utilization of satellite and surface observations in a gravity wave study

The possibility of detecting gravity waves in the upper troposphere by means of satellite imagery is examined. Attention is focused on the gravity wave event of March 27, 1975, for which the surface wave is analyzed by a pressure perturbation analysis based on barograph traces and a surface wind divergence analysis. IR observations were also made of the event with instrumentation on board the synchronous meteorological satellite (SMS-1). Strong wind gusts were reported at surface stations immediately following severe drops in the surface pressures, events which were found not to be connected with convective activity. The wave was concluded to extend throughout the troposphere with an axis tilted with height, indicating the wave was generated from an upper level energy source. It is suggested that the gravity wave was an adjustment to ageostrophic motions within the jet stream entering the Midwest. The finding is in concert with previous observations of jet streaks upstream from surface gravity waves.

Pecnick, M. J.↗

Contact Angle Influence on Geysering Jets in Microgravity Investigated

Microgravity poses many challenges to the designer of spacecraft tanks. Chief among these are the lack of phase separation and the need to supply vapor-free liquid or liquid-free vapor to the spacecraft processes that require fluid. One of the principal problems 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 flow. Anyone who has seen a fountain knows that jets occur in normal gravity also. However, in normal gravity, the gravity controls and restricts the jet flow. In microgravity, with gravity largely absent, surface tension forces must be used to contain jets. To model this phenomenon, a numerical method that tracks the fluid motion and the surface tension forces is required. Jacqmin has developed a phase model that converts the discrete surface tension force into a barrier function that peaks at the free surface and decays rapidly away. Previous attempts at this formulation were criticized for smearing the interface. This can be overcome by sharpening the phase function, double gridding the fluid function, and using a higher-order solution for the fluid function. The solution of this equation can be rewritten as two coupled Poisson equations that also include the velocity.

Chato, David J.↗