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Simulating magnetic positive positioning of cryogenic propellants in a transient acceleration field
Computers & Fluids 38 (2009) 843–850
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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
Simulating magnetic positive positioning of cryogenic propellants in a transient acceleration field Jeffrey G. Marchetta a,*, Kevin M. Roos b a b
Department of Mechanical Engineering, 322D Engineering Sciences Building, University of Memphis, Memphis, TN 38152, United States Department of Mechanical Engineering, 312 Engineering Sciences Building, University of Memphis, Memphis, TN 38152, United States
a r t i c l e
i n f o
Article history: Received 22 August 2007 Received in revised form 30 June 2008 Accepted 11 September 2008 Available online 25 September 2008
a b s t r a c t A computational simulation of magnetic positive positioning (MP2) is developed to model cryogenic propellant reorientation in reduced gravity. Previous efforts have successfully incorporated an electromagnetic field model into an axisymmetric, two-dimensional, incompressible fluid flow model yielding accurate predictions of fluid motion induced by a magnetic field. To simulate MP2, a three-dimensional magnetic field and magnetic force model was developed as a feature of a commercially available fluid flow model which has been well validated. The computational tool was then improved upon to model magnetically induced flows in a transient acceleration field. Simulation predictions obtained with the enhanced model are compared to available reduced gravity experiment data. Evidence is presented and conclusions are drawn that support the continued use of the simulation as viable modeling and predictive tool in the continuing study of MP2. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Analytical [1], experimental [2], and computational [3] studies have been performed in order to seek reliable and efficient technologies for the management of cryogenic propellants in a reduced gravity environment. Passive and active systems for propellant management have been utilized on small satellites, but are not without their disadvantages. Passive systems, such as liquid acquisition devices (LADS), rely upon surface tension forces generated by specific geometries existing inside the tank, such as screens, channels, or vanes in order to supply liquid to the requisite tank location [4,5]. However, these systems result in increased weight of the spacecraft. Active systems can reorient propellant through the use of external thrusters that maneuver the tank relative to the location of the liquid [3,6]. Such impulsive methods hold the advantage over passive systems because the propellant can be repositioned to meet changing mission requirements. Unfortunately, this method increases system complexity because of the additional thrusters, controls, and fuel. Although both methods are an effective means to manage liquid propellants in reduced gravity, the viability of using these systems for managing cryogenic propellants for long duration missions is uncertain. In addition, the aforementioned disadvantages suggest that investigations of other, potentially more efficient, methods for propellant management are justified. Magnetic positive positioning (MP2) is a potential third candidate for reorientation. MP2 * Corresponding author. Tel.: +1 9016783268; fax: +1 9016785459. E-mail address: [email protected] (J.G. Marchetta). 0045-7930/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2008.09.005
utilizes a magnetic field and the magnetic susceptibility of the liquid propellant to achieve reorientation in space tanks. Paramagnetic fluids, such as liquid oxygen (LOX), are attracted to regions of high magnetic field strength, while diamagnetic fluids, like liquid hydrogen (LH2), are repelled from regions of high magnetic field strength. The concept of using magnetic fields to influence the flow of susceptible liquids has been considered for over 40 years [7]. For terrestrial applications, the effects of a magnetic field on a paramagnetic fluid are negated by the dominant gravitational forces. Fundamental hydrodynamic equations for the coupling of the magnetic force with other forces that would induce flow of a Newtonian fluid were developed by Neuringer and Rosenweig [8] in 1964. The stabilizing influence of a magnetic force on a Rayleigh–Taylor instability was demonstrated by Zelazo and Melcher [9] in 1969. Likewise, Bashtovoy and Krakov [10] showed that an imposed magnetic field can have a stabilizing influence on a jet of magnetically susceptible fluid. In 1988, Berkovsky and Smirnov [11] explored sessile drop stability and shape in the presence of a magnetic field. Later on, Basaran and Wohlhuter [12] looked into the shape and stability of both pendant and sessile drops in similar conditions. Finally, Bashtovoy, Berkovsky, and Vislovich [13] concluded that the free surface of a magnetically susceptible fluid will conform to a contour of constant magnetic field intensity in the presence of a magnetic field, and neglecting gravitational and capillary forces. Early investigations into the plausibility of using electromagnetic fields to manage liquid propellants were dismissed due to the unacceptable risk of arcing inside the propellant tank due to
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Nomenclature a b B C e E g h H l m M
minor axis of fluid profile major axis of fluid profile magnetic induction constant unit vector energy acceleration due to gravity height magnetic intensity characteristic length magnetic dipole strength magnetization
the magnetic field itself [7]. Improvements in high temperature superconducting material formed a new interest in the application of magnetic positive positioning. In 1995, The NASA Marshall Space Flight Center conducted a series of flight experiments to study magnetically actuated propellant orientation, or MAPO [14]. The objective of the experiments was to investigate the feasibility of using a magnetic field to positively reorient liquid propellant within a tank while maintaining its position during propellant expulsion and tank filling operations. The experiments utilized ferrofluids to simulate liquid oxygen, and a video camera was used to capture liquid motion within a 10 cm diameter transparent cylindrical tank during the reduced gravity phase of the parabolic flight pattern of the NASA KC-135 aircraft. Several difficulties arise from the experiment itself. Only 20–30 s of reduced gravity flight time were available for each experiment, and the flows were influenced by unrepeatable initial conditions. Furthermore, the ferrofluids used serve only as an approximation for the properties of liquid oxygen. A purely experimental approach for evaluating the feasibility of MP2 would be limited by the ability to collect sufficient data for full scale tanks and would be greatly hindered by expense and time constraints. A computational simulation is an attractive alternative for assessing the feasibility of magnetic positive positioning of propellants for future spacecraft. Hochstein et al. [15] developed a computational simulation of the MAPO experiment using a magnetic dipole to represent the magnetic field, and presented pre-flight flow field predictions. Marchetta and Hochstein [16,17] integrated the MAPO magnetic field configuration into the simulation and assessed the fidelity of the computational model by comparing flow field predictions to available flight images. Marchetta and Winter [18] incorporated a electromagnetic field model and an incompressible flow model to simulate realistic magnetic fields. This paper presents results obtained using a three-dimensional simulation of MP2. Most of the simulations presented herein of the MAPO experiment show good agreement with the flow behavior observed in photographs taken during the flight experiment [14]. It is shown that the current three-dimensional model of MP2 exceeds the capabilities of its predecessor in its ability to model asymmetric as well as fluctuating acceleration environments. Simulation results were reviewed to assess the utility of the new simulation for future studies and experiment design.
2. Mathematical models At the temperatures and pressures associated with cryogenic propellant storage tanks, LOX and LH2 are well characterized as incompressible, constant property, Newtonian fluids. The Navier
n q RMS s R, H, U R, h, z x, y, z V
lo q r vm
coordinate normal to surface quantity component of energy root mean square surface direction spherical coordinates with H in x–y plane cylindrical coordinates with h in x–y plane cartesian coordinate directions volume permeability of free space density surface tension coefficient magnetic susceptibility
Stokes momentum equation, which describes the unsteady flow of such a fluid, is given by *
q
* * * * * @V þ qðV rÞ V ¼ rp þ lr2 V þq g þF M : @t
ð1Þ
Neuringer and Rosensweig [7] present a development of the magnetic force per unit volume in mks units: *
*
*
F M ¼ l0 ðM rÞ H :
ð2Þ
Assuming the fluid to be nonconducting, the magnetization to be in the direction of the local magnetic field, and the displacement current to be negligible, the magnetic force per unit volume becomes *
F M ¼ l0 MrH:
ð3Þ
For low intensity magnetic fields associated with permanent magnets and small electromagnets, and the assumption that the magnetization is temperature independent, the magnetization is defined as
M ¼ vH
ð4Þ
For some liquids, such as a ferrofluid, the magnetic susceptibility is not constant and increases non-linearly with an increase in the magnetic field strength. The magnetic intensity in mks units is given as
H¼
B
l0 ð1 þ vÞ
:
ð5Þ
3. Computational modeling A computational simulation is used to model unsteady, incompressible flows with deforming free surfaces in which surface tension forces are significant was enhanced to simulate MP2 in reduced gravity. The axisymmetric model used by Hochstein, Marchetta et al. [15–20], a variant of the RIPPLE code, has been extensively validated in modeling free surface flows with surface tension and the propellant management applications, including MP2. The commercial software, FLUENT [21], is a state-of-the-art computer program for modeling fluid flow and heat transfer in complex geometries. Among its many capabilities, FLUENT can model transient, three-dimensional, multiphase, incompressible flows and is well suited for modeling MP2. For all flows, FLUENT solves conservation equations for mass and momentum sequentially. FLUENT uses a control volume-based technique and upwind differencing to convert the governing equations to algebraic equations that
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M¼
187H 13224 þ H
ð6Þ
where the numbers 187 and 13,224 are coefficients of the curve-fit. Similarly, for the simulations of the MAPO experiments containing a 133:1 water/ferro-fluid mixture, the expression for magnetization can also be modeled as a growth saturation function. The magnetization is again a function of magnetic field intensity such that
M¼
187H 13108 þ H
ð7Þ
dius of 0.0762 m and a height 0.381 m and was subjected only to capillary forces. The tank was filled with liquid oxygen, such that 40% of tank volume was full. The simulation was initialized with a flat 1 g interface. A small contact angle of 0.1 ° was specified to approximate a 0 ° contact angle, at the boundaries, and the surface tension coefficient was assumed to be constant. For solid wall boundary conditions, the impermeable no-slip condition is imposed. Three non-uniform hexahedral meshes of increasing mesh density were simulated and results obtained using a mesh with 174,336 cells, produced a mesh independent solution. Fig. 1 illustrates the simulated and analytic normalized free surface shape for a plane 2-D cross-section of the cylinder for the first benchmark case. It has been shown that an equilibrium zero gravity meniscus with a 0 ° contact angle will conform to a half sphere in a cylindrical container, such that the normalized meniscus height is equal to normalized radius of the cylinder. As shown in Fig. 1, the simulated zero-gravity meniscus is in good qualitative agreement with the analytic solution and the RMS error is 0.01. Another case was devised to determine whether liquid oxygen inside a tank subjected to the dipole magnetic field would conform to a contour of constant magnetic field intensity as predicted by theory. The dipole model was used to validate the implementation of a UDF in FLUENT, specifically the magnetic force source term model. Marchetta [23] shows that a contour of constant magnetic field intensity can be obtained from Eq. (6). To simplify the simulation setup, a square box of 0.1 m length was partially filled with LOX. The LOX is initialized a quarter sphere positioned centered in one corner of the box. The computational mesh consisted of 64,000 uniform hexahedral cells. The dipole magnet was shifted away from the box at a distance of 0.05 m to avoid having a singularity in the computational domain. A constant dipole strength was specified and the simulation was iterated until an equilibrium free surface was achieved. Fig. 2 illustrates the simulation prediction for the equilibrium free surface shape for LOX under the influence of
1.0
Normalized Meniscus Height
can be solved numerically. Second order upwind differencing is utilized for all FLUENT simulation predictions presented herein. FLUENT uses a co-located scheme; whereby pressure and velocity are both stored at cell centers. Therefore, an interpolation scheme is required to compute the face values of pressure from the cell values. For the following simulations, the body force weighted scheme is utilized to compute the face pressure by assuming that the normal gradient of the difference between pressure and body forces is constant. The pressure-implicit with splitting of operators (PISO) pressure–velocity coupling scheme is used to obtain a semi-implicit pressure correction equation. The pressure-correction equation is subsequently solved using the algebraic multigrid (AMG) method. Temporal discretization is accomplished using implicit time integration, which is unconditionally stable with respect to time step size. Solutions are subsequently iterated at each time level until the convergence criteria are met. The location of multiple phases within the mesh is determined using the volume-of-fluid (VOF) function, f. The VOF formulation relies on the fact that two or more phases are not interpenetrating. The geometric reconstruction scheme is used to reconstruct the interface between fluids using a piecewise-linear approach. Surface tension forces are computed using the Continuum Surface Force Model and a static contact angle can be specified to model wall adhesion. The addition of the magnetic force source term is achieved through use of a user defined function [22] (UDF) in FLUENT. It was necessary to define a new variable to introduce the magnetic field intensity, H. For the preliminary simulations of the MAPO experiment, a UDF was written which interpolates measured magnetic field data on to a computational mesh. In addition, the magnetic field intensity was utilized to compute the gradient of the magnetic field intensity. Martin and Holt [14] provide tables of magnetization measurements, taken from a magnetometer, for the ferrofluids used in the experiment. Coefficients for a growth saturation function were sought using regression of magnetization measurements. For the 30:1 mixture, the regression analysis yields the magnetization (A/m) as a function of the magnetic field intensity such that
Analytic
0.8
Simulation 0.6 0.4 0.2 0.0 -1
-0.5 0 0.5 Normalized Radius
1
Fig. 1. Normalized, steady-state, and free surface shape for LOX under the influence of capillary forces.
M¼
149H 6755 þ H
ð8Þ
In order to simulate the MAPO experiment, it was necessary to include the transient acceleration fields experienced by the tanks on board the KC-135 aircraft. The previous UDF was improved upon to linearly interpolate accelerometer data taken during the MAPO experiments, and provide the computational tool with the actual transient acceleration environment measured. 4. Model verification and validation Three benchmark cases were chosen to verify and validate the MP2 model. The first simulation was a cylindrical tank with a ra-
Distance along radial axis, m
The magnetization for the 10:1 water/ferro-fluid mixture is a function of magnetic field intensity such that
0.10 0.08
Analytic Simulation
0.06 0.04 0.02 0.00 0.00
0.02 0.04 Distance along z axis, m
0.06
Fig. 2. Steady-state, free surface shape for LOX under the influence of a dipole magnetic field and a contour of constant H, A/m.
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the dipole magnetic field. The RMS error between the predicted free surface shape and the analytic solution of a contour of constant H as predicted by theory was 0.01. It can be concluded that the dipole magnetic force model UDF was properly implemented in FLUENT. The final benchmark validation case, a cylindrical tank with a 10% fill of LOX was subjected to the magnetic force exerted by the MAPO experiment permanent ring magnet. The rare Earth permanent magnet has a thickness of 0.0381 m, and inner and outer diameters of 0.1016 m and 0.254 m, respectively. This case was chosen to validate the implementation of a second FLUENT UDF, which uses the measured magnetic data from the experiment and interpolates the data onto a computational mesh. Again, theory states the steady-state interface should conform to a contour of constant magnetic field intensity. The MAPO tank geometry, a cylindrical tank with a radius of 0.0762 m and a height 0.254 m, was chosen for this simulation. Noting that a mesh independent flow field was observed in the first benchmark case and the similarity in the cylindrical geometry of both cases, an equivalent mesh density was specified for this study. The surface of the ring magnet, which has a maximum flux of about 0.5 T, is positioned 0.0175 m below the tank. Fig. 3 shows the contours of constant magnetic field intensity for a plane 2-D cross-section of the tank. The simulated equilibrium free surface shape qualitatively conformed to the contours of constant H as depicted in Fig. 3. It can be concluded that the influence of the MAPO ring magnet is properly accounted for in the 2nd FLUENT UDF. 5. MAPO experiment predictions The only data available for code validation directly related to MP2 in reduced gravity is the preliminary data from the MAPO experiment [14]. For the MAPO experiments, the only flow field data available is for a 50% tank fill and it is in the form of stillframes extracted from those recordings. It should be noted that quantitative comparison between computational flow field predictions and experiment data is complicated by several factors. First, the ferrofluid/water mixture is very opaque so many of the flow
details internal to the tank surface cannot be seen once the liquid has coated the outer tank wall. Further, the angle of view for the recordings is not normal to the cylinder so problems with data interpretation due to the refraction of light as it passes through the cylindrical wall of the acrylic tank are exacerbated. In addition to those difficulties, there is an important difference between the simulation and experiment. The computational simulation begins with a perfectly flat initial interface and instantly changes from a normal 1 g environment to the transient acceleration environment. The experiment initial free surface shape is not precisely measured. Therefore, it is justifiable to assume that some disparity will be observed between the simulation and experiment due to the variations in initial free surface conditions, and due to the differences between the abrupt change of the computational simulation acceleration environment and the more gradual change of the experiment environment. As such, it is only reasonable to draw conclusions which are based on qualitative comparisons of the surface shape. For the simulations that follow, a flat 1 g interface is used as the initial free surface. After the initialization, the transient measured background acceleration is included as a source in the momentum equation. Fig. 4 shows the three tank configurations used in the experiment. To simplify further discussion, the magnet end of the tank will be called the bottom of the tank and the end opposite to the magnet will be called the top. Three meshes of increasing mesh density, 49,728, 71,232, and 174,336 cells, respectively were utilized to simulate the 50% bottom fill case. The study was judged to have demonstrated a mesh independent simulation when the predicted flow fields for the 50% bottom fill cases were unchanged by additional mesh refinement. Fig. 5 illustrates that the differences in the free surface shape for the three meshes are negligible after 30 s of simulation time. To reduce computational time, the 71,232 cell mesh, shown in Fig. 6, was selected for the remaining simulations of the experiment. The first experiment test considered is a bottom fill tank of 30:1 water/ferrofluid mixture. Fig. 7 presents a sequence of flow fields recorded for a configuration in which liquid is collected at the magnet end of the tank prior to entering the reduced gravity phase of the KC-135 flight. In each picture, the red area contains the ferro-
Free Surface Height (m)
Fig. 4. MAPO experiment tank configurations [14].
0.20 0.16 0.12 0.08
49728 cells 71232 cells
0.04
174336 cells
0.00 -0.08
-0.04
0.00
0.04
0.08
Radial Position (m) Fig. 3. Steady-state, free surface shape for a 10% fill of LOX under the influence of the MAPO magnetic field and contours of constant H, A/m, inside the tank.
Fig. 5. Free surface shape for varying mesh densities.
J.G. Marchetta, K.M. Roos / Computers & Fluids 38 (2009) 843–850
Fig. 6. MAPO experiment tank topology with 71,232 cell computational mesh.
fluid and the blue region contains air. The shaded areas in the simulation pictures indicate regions filled with liquid on the farthest wall of the tank which would not appear in the photographs due to the opacity of the ferrofluid. At 1.5 s, there has already been considerable displacement of the initial flat interface and a general sweeping motion of fluid from the centerline toward the tank wall can be observed. After 1.5 s, the interface near the tank centerline appears to have taken on a shape corresponding to a classical meniscus due to surface tension. In contrast, near the tank wall it appears that momentum imparted by the initial sweep of fluid toward the wall is causing a thin layer of fluid near the wall to climb toward the top of the tank. From this time onward, there is little displacement of the free surface in the vicinity of the tank centerline but a continuing evolution of the thin fluid layer near the wall can be observed. A sequence of flow fields predicted by the computational simulations for the 50% bottom fill of 30:1 water/ ferro-fluid mixture is also presented in Fig. 7. In the simulation predictions, the transition from 1 g to low gravity is observed at 1.5 s. The meniscus shape is evident, however quite asymmetric due to the fluctuating background accelerations. At the 14 s photo, a sloshing effect is noted in both the experimental photos and
847
simulation predictions. Some discrepancy exists past 18 s in the simulation predictions. A small layer of fluid appears to be coating the top of the tank in the simulation as a consequence of the initial sloshing motion, yet this coating is not observed in the experiment. The majority has settled back into the asymmetric surface shape similar to that observed in the photos. Fig. 8 presents a flow field sequence corresponding to a configuration in which the 30:1 water/ferro-fluid mixture is initially collected in the end of the tank opposite to the magnet. Fig. 8 also presents simulation predictions for this 50% top fill case. At 6.0 s in the experiment, it is apparent that the magnet is pulling the fluid along the wall toward the bottom of the tank. After 15 s, a small pool of liquid has collected at the bottom outside corner of the tank. It is interesting to note that although fluid momentum induces flow along the bottom toward the centerline, the magnetic force is strong enough to capture the liquid and hold it in the corner. Although the shape of the free surface of this pool appears to be dominated by surface tension near the tank wall, it appears to conform to a surface of constant magnetic field strength. At 17 s, the fluid appears to separate into two distinct pools at opposing ends of the tank. This is likely due to the combined influence of sloshing due to the acceleration field and the magnetic field at the bottom of the tank. While not easily discerned in the simulation pictures presented above, a sample 2-D cross-section of the tank at 17 s confirms that the liquid is being influenced by the magnet as the shape of the free surface closely resembles a contour of constant magnetic field intensity. It is difficult to verify this behavior with the corresponding experimental photo due to the water/ferro-fluid mixture coating the tank walls. Because of this, comparison between the remaining experimental photos and simulation predictions is not possible. Fig. 9 presents a series of flow fields where the 30:1 water/ferro-fluid is initially positioned along the length of the cylindrical tank, such that the interface is perpendicular to the magnet face. This case is of particular interest in this study because the axisymmetric predecessor to the current simulation was unable to model the asymmetry of the configuration. As the experiment progress, the liquid is drawn up the opposing tank heads, a result consistent with the influence of surface tension. It is evident from the series of photos and simulation results shown in Fig. 9 that more fluid collects at the bottom of the tank as opposed to the top. The additional accumulation of the liquid at the bottom of the tank is likely due to the influence of the magnetic field. Further, as time increases, more
Fig. 7. Sequence of 30:1 bottom fill predictions and photos of the MAPO experiment.
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Fig. 8. Sequence of 30:1 top fill simulation predictions and photos of the MAPO experiment
Fig. 9. Sequence of 30:1 side fill simulation predictions and photos of the MAPO experiment.
fluid accumulation at the bottom of the tank is observed as opposed to the top. 6. Constant background acceleration MP2 predictions The motions observed in the MAPO experiment provide evidence of the influence of the magnet on the water/ferro-fluid mixture. Yet, as noted the flow details are masked by the large three-dimensional fluctuations in the acceleration environment and the opacity of the water/ferro-fluid mixture. To better under-
stand the MP2 process in reduced gravity, additional simulations are performed with a constant background acceleration of 103 g specified along the axis of symmetry of the cylindrical tank. Neglecting the influence of g-jitter, this background acceleration is comparable to average acceleration environment experienced in low earth orbit. A sequence of flow fields predicted by the computational simulations for the 50% bottom fill for constant background acceleration is presented in Fig. 10. In the simulation predictions, the transition from 1 g to low gravity is observed at 1.5 s. Physically, the flow
Fig. 10. Sequence of bottom fill simulation predictions for constant background acceleration.
J.G. Marchetta, K.M. Roos / Computers & Fluids 38 (2009) 843–850
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Fig. 11. Sequence of top fill simulation predictions for constant background acceleration.
Fig. 12. Sequence of side fill simulation predictions for constant background acceleration.
fields are similar to the experiment photos presented in Fig. 7 through 9.5 s, although a difference in transition times is again noted. The free surface reaches equilibrium within 14–18 s. In contrast, the asymmetric free surface in the experiment photos are a result of an asymmetric background acceleration, which may have induced the sloshing observed in the 14 s photo. Fig. 11 presents the simulation predictions for the 50% top fill case for constant background acceleration. This simulation more clearly demonstrates that the magnet is influencing the motion of the ferrofluid down the outer tank wall. The liquid near the bottom of the tank appears to conform to a contour of constant H. In contrast, the liquid near the top of the tank assumes a more spherical shape consistent with capillary forces. The ferrofluid is continuously reoriented throughout the 26.5 s of simulation time as demonstrated by the accumulation of fluid at the bottom of the tank. Fig. 12 presents a series of simulations predictions for constant background acceleration where the ferrofluid is initially positioned along the length of the cylindrical tank, such that the interface is perpendicular to the magnet face. More fluid collects at the bottom of the tank as opposed to the top, a consequence which again demonstrates the influence of the magnetic field on the motion of fluid. In contrast to the top fill case, the free surface shape appears more spherical as time progresses, an indication that surface tension is strongly influencing the liquid near the middle and top of the top. 7. Summary and conclusions In summary, qualitative comparisons of the free surface shapes between MAPO experiment data and simulation predictions showed as good agreement in the majority of cases given the quality of the experiment environment. Alluded to in previous works, the uncertainty for promising applications like MP2 and many others, emphasizes the need for more reduced gravity experiments specifically intended for simulation validation. Although the computational results presented herein appear to be promising, considerable research remains to be performed before the feasibility of
MP2 can be clearly demonstrated. For future work, the discrepancies that exist between the initial conditions experienced by the tanks onboard the KC-135 aircraft and those specified in the simulations need to be addressed. Furthermore, there are several test cases done in the MAPO experiments that are yet to be simulated. In addition, the feasibility of using an electromagnet in full scale tanks needs to be investigated in future work. Acknowledgement The authors are thankful to the Tennessee Space Grant Consortium for supporting this effort. References [1] Sumner IE. ‘‘Liquid Propellant Reorientation in a Low Gravity Environment”, NASA TM-78969, 1978. [2] Aydelott JC. ‘‘Axial Jet Mixing of Ethanol in Cylindrical Containers During Weightlessness”, NASA TP-1487, 1982. [3] Hochstein JI, Patag AE, Chato DJ. Modeling of Impulsive Propellant Reorientation. J Propul Power 1991;7(6):938–45. [4] Schmidt GR, Chung TJ, Nadarajah A. Thermocapillary flow with evaporation and condensation at low gravity. Part 1. Non-deforming surface. J Fluid Mech 1995;295:323–47. [5] Schmidt GR, Chung TJ, Nadarajah A. Thermocapillary flow with evaporation and condensation at low gravity. Part 2. Deformable surface. J Fluid Mech 1995;295:349–66. [6] Hochstein JI, Chato DJ. Pulsed thrust propellant reorientation: concept and modeling. J Propulsion Power 1992;8(4):770–7. [7] Chipchak D. Development of Expulsion and Orientation Systems for Advanced Liquid Rocket Propulsion Systems”, USAF Technical Report RTD-TDR-63-1048, Contract AF04(611)-8200, July, 1963. [8] Neuringer JL, Rosensweig RE. Ferrohydrodynamics. Physics of Fluids 1964;7(12):1927. [9] Zelazo RE, Melcher JR. Dynamics and stability of ferrofluids: surface interactions. J Fluid Mech 1969;39:1. [10] Boshtovoy VG, Krakov MS. Stability of an axisymmetric jet of magnetizable fluid. Translated for Z. Prik. Mekh. I Tekh Fiz. 1978:147. [11] Berkovsky BM, Smirnov NN. Capillary hydrodynamic effects in high magnetic fields. J Fluid Mech 1988;187:319. [12] Basaran OA, Wohlhuter FK. Effect of nonlinear polarization on shapes and stability of pendant and sessile drops in an electric (magnetic) field. J Fluid Mech 1992;244:1.
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[13] Bashtovoy VG, Berkovsky BM, Vislovich AN. Introduction to thermomechanics of magnetic fluids. Berlin: Hemisphere Publishing Co; 1988. [14] Martin JJ, Holt JB. ‘‘Magnetically Actuated Propellant Orientation Experiment, Controlling Fluid Motion With Magnetic Fields in a Low-Gravity Environment”, NASA TM 210129, 2000. [15] Hochstein JI, Warren RT, Schmidt GR. Magnetically actuated propellant orientation (MAPO) experiment: pre-flight flow field predictions. AIAA Paper 97-0570, Jan. 1997. [16] Marchetta JG, Hochstein JI. Fluid Capture by a Permanent Ring Magnet in Reduced Gravity. AIAA Paper 99-0845, Jan. 1999. [17] Marchetta JG, Hochstein JI. A computational model of magnetic positive positioning in reduced gravity. IAF Paper ST-99-W.210, Oct. 1999.
[18] Winter AP, Marchetta JG, Hochstein JI. Simulation and Prediction of Realistic Magnetic Positive Positioning for Space Based Fluid Management Systems. AIAA Paper 2004-1151, 2004. [19] Marchetta JG, Hochstein JI, Simmons BD, Sauter DR.Modeling and Prediction of Magnetic Storage and Reorientation of Lox in Reduced Gravity. AIAA Paper 2002-1005, Jan. 2002. [20] Kothe DB, Mjolsness RC, Torrey MD. RIPPLE: a computer program for incompressible flows with free surfaces, LANL Report LA-12007-MS, April, 1991. [21] Fluent Inc.. Fluent 6.1 User’s Guide. Lebanon, NH: Fluent Inc.; 2003. [22] Fluent Inc.. Fluent 6.1 UDF Manual. Lebanon, NH: Fluent Inc.; 2003. [23] Marchetta JG, Hochstein JI, Simmons BD. On-Orbit Positive Positioning of LOX: Simulation and Correlation. AIAA Paper 2003-1153, Jan. 2003.
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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
Simulating magnetic positive positioning of cryogenic propellants in a transient acceleration field Jeffrey G. Marchetta a,*, Kevin M. Roos b a b
Department of Mechanical Engineering, 322D Engineering Sciences Building, University of Memphis, Memphis, TN 38152, United States Department of Mechanical Engineering, 312 Engineering Sciences Building, University of Memphis, Memphis, TN 38152, United States
a r t i c l e
i n f o
Article history: Received 22 August 2007 Received in revised form 30 June 2008 Accepted 11 September 2008 Available online 25 September 2008
a b s t r a c t A computational simulation of magnetic positive positioning (MP2) is developed to model cryogenic propellant reorientation in reduced gravity. Previous efforts have successfully incorporated an electromagnetic field model into an axisymmetric, two-dimensional, incompressible fluid flow model yielding accurate predictions of fluid motion induced by a magnetic field. To simulate MP2, a three-dimensional magnetic field and magnetic force model was developed as a feature of a commercially available fluid flow model which has been well validated. The computational tool was then improved upon to model magnetically induced flows in a transient acceleration field. Simulation predictions obtained with the enhanced model are compared to available reduced gravity experiment data. Evidence is presented and conclusions are drawn that support the continued use of the simulation as viable modeling and predictive tool in the continuing study of MP2. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Analytical [1], experimental [2], and computational [3] studies have been performed in order to seek reliable and efficient technologies for the management of cryogenic propellants in a reduced gravity environment. Passive and active systems for propellant management have been utilized on small satellites, but are not without their disadvantages. Passive systems, such as liquid acquisition devices (LADS), rely upon surface tension forces generated by specific geometries existing inside the tank, such as screens, channels, or vanes in order to supply liquid to the requisite tank location [4,5]. However, these systems result in increased weight of the spacecraft. Active systems can reorient propellant through the use of external thrusters that maneuver the tank relative to the location of the liquid [3,6]. Such impulsive methods hold the advantage over passive systems because the propellant can be repositioned to meet changing mission requirements. Unfortunately, this method increases system complexity because of the additional thrusters, controls, and fuel. Although both methods are an effective means to manage liquid propellants in reduced gravity, the viability of using these systems for managing cryogenic propellants for long duration missions is uncertain. In addition, the aforementioned disadvantages suggest that investigations of other, potentially more efficient, methods for propellant management are justified. Magnetic positive positioning (MP2) is a potential third candidate for reorientation. MP2 * Corresponding author. Tel.: +1 9016783268; fax: +1 9016785459. E-mail address: [email protected] (J.G. Marchetta). 0045-7930/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2008.09.005
utilizes a magnetic field and the magnetic susceptibility of the liquid propellant to achieve reorientation in space tanks. Paramagnetic fluids, such as liquid oxygen (LOX), are attracted to regions of high magnetic field strength, while diamagnetic fluids, like liquid hydrogen (LH2), are repelled from regions of high magnetic field strength. The concept of using magnetic fields to influence the flow of susceptible liquids has been considered for over 40 years [7]. For terrestrial applications, the effects of a magnetic field on a paramagnetic fluid are negated by the dominant gravitational forces. Fundamental hydrodynamic equations for the coupling of the magnetic force with other forces that would induce flow of a Newtonian fluid were developed by Neuringer and Rosenweig [8] in 1964. The stabilizing influence of a magnetic force on a Rayleigh–Taylor instability was demonstrated by Zelazo and Melcher [9] in 1969. Likewise, Bashtovoy and Krakov [10] showed that an imposed magnetic field can have a stabilizing influence on a jet of magnetically susceptible fluid. In 1988, Berkovsky and Smirnov [11] explored sessile drop stability and shape in the presence of a magnetic field. Later on, Basaran and Wohlhuter [12] looked into the shape and stability of both pendant and sessile drops in similar conditions. Finally, Bashtovoy, Berkovsky, and Vislovich [13] concluded that the free surface of a magnetically susceptible fluid will conform to a contour of constant magnetic field intensity in the presence of a magnetic field, and neglecting gravitational and capillary forces. Early investigations into the plausibility of using electromagnetic fields to manage liquid propellants were dismissed due to the unacceptable risk of arcing inside the propellant tank due to
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Nomenclature a b B C e E g h H l m M
minor axis of fluid profile major axis of fluid profile magnetic induction constant unit vector energy acceleration due to gravity height magnetic intensity characteristic length magnetic dipole strength magnetization
the magnetic field itself [7]. Improvements in high temperature superconducting material formed a new interest in the application of magnetic positive positioning. In 1995, The NASA Marshall Space Flight Center conducted a series of flight experiments to study magnetically actuated propellant orientation, or MAPO [14]. The objective of the experiments was to investigate the feasibility of using a magnetic field to positively reorient liquid propellant within a tank while maintaining its position during propellant expulsion and tank filling operations. The experiments utilized ferrofluids to simulate liquid oxygen, and a video camera was used to capture liquid motion within a 10 cm diameter transparent cylindrical tank during the reduced gravity phase of the parabolic flight pattern of the NASA KC-135 aircraft. Several difficulties arise from the experiment itself. Only 20–30 s of reduced gravity flight time were available for each experiment, and the flows were influenced by unrepeatable initial conditions. Furthermore, the ferrofluids used serve only as an approximation for the properties of liquid oxygen. A purely experimental approach for evaluating the feasibility of MP2 would be limited by the ability to collect sufficient data for full scale tanks and would be greatly hindered by expense and time constraints. A computational simulation is an attractive alternative for assessing the feasibility of magnetic positive positioning of propellants for future spacecraft. Hochstein et al. [15] developed a computational simulation of the MAPO experiment using a magnetic dipole to represent the magnetic field, and presented pre-flight flow field predictions. Marchetta and Hochstein [16,17] integrated the MAPO magnetic field configuration into the simulation and assessed the fidelity of the computational model by comparing flow field predictions to available flight images. Marchetta and Winter [18] incorporated a electromagnetic field model and an incompressible flow model to simulate realistic magnetic fields. This paper presents results obtained using a three-dimensional simulation of MP2. Most of the simulations presented herein of the MAPO experiment show good agreement with the flow behavior observed in photographs taken during the flight experiment [14]. It is shown that the current three-dimensional model of MP2 exceeds the capabilities of its predecessor in its ability to model asymmetric as well as fluctuating acceleration environments. Simulation results were reviewed to assess the utility of the new simulation for future studies and experiment design.
2. Mathematical models At the temperatures and pressures associated with cryogenic propellant storage tanks, LOX and LH2 are well characterized as incompressible, constant property, Newtonian fluids. The Navier
n q RMS s R, H, U R, h, z x, y, z V
lo q r vm
coordinate normal to surface quantity component of energy root mean square surface direction spherical coordinates with H in x–y plane cylindrical coordinates with h in x–y plane cartesian coordinate directions volume permeability of free space density surface tension coefficient magnetic susceptibility
Stokes momentum equation, which describes the unsteady flow of such a fluid, is given by *
q
* * * * * @V þ qðV rÞ V ¼ rp þ lr2 V þq g þF M : @t
ð1Þ
Neuringer and Rosensweig [7] present a development of the magnetic force per unit volume in mks units: *
*
*
F M ¼ l0 ðM rÞ H :
ð2Þ
Assuming the fluid to be nonconducting, the magnetization to be in the direction of the local magnetic field, and the displacement current to be negligible, the magnetic force per unit volume becomes *
F M ¼ l0 MrH:
ð3Þ
For low intensity magnetic fields associated with permanent magnets and small electromagnets, and the assumption that the magnetization is temperature independent, the magnetization is defined as
M ¼ vH
ð4Þ
For some liquids, such as a ferrofluid, the magnetic susceptibility is not constant and increases non-linearly with an increase in the magnetic field strength. The magnetic intensity in mks units is given as
H¼
B
l0 ð1 þ vÞ
:
ð5Þ
3. Computational modeling A computational simulation is used to model unsteady, incompressible flows with deforming free surfaces in which surface tension forces are significant was enhanced to simulate MP2 in reduced gravity. The axisymmetric model used by Hochstein, Marchetta et al. [15–20], a variant of the RIPPLE code, has been extensively validated in modeling free surface flows with surface tension and the propellant management applications, including MP2. The commercial software, FLUENT [21], is a state-of-the-art computer program for modeling fluid flow and heat transfer in complex geometries. Among its many capabilities, FLUENT can model transient, three-dimensional, multiphase, incompressible flows and is well suited for modeling MP2. For all flows, FLUENT solves conservation equations for mass and momentum sequentially. FLUENT uses a control volume-based technique and upwind differencing to convert the governing equations to algebraic equations that
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M¼
187H 13224 þ H
ð6Þ
where the numbers 187 and 13,224 are coefficients of the curve-fit. Similarly, for the simulations of the MAPO experiments containing a 133:1 water/ferro-fluid mixture, the expression for magnetization can also be modeled as a growth saturation function. The magnetization is again a function of magnetic field intensity such that
M¼
187H 13108 þ H
ð7Þ
dius of 0.0762 m and a height 0.381 m and was subjected only to capillary forces. The tank was filled with liquid oxygen, such that 40% of tank volume was full. The simulation was initialized with a flat 1 g interface. A small contact angle of 0.1 ° was specified to approximate a 0 ° contact angle, at the boundaries, and the surface tension coefficient was assumed to be constant. For solid wall boundary conditions, the impermeable no-slip condition is imposed. Three non-uniform hexahedral meshes of increasing mesh density were simulated and results obtained using a mesh with 174,336 cells, produced a mesh independent solution. Fig. 1 illustrates the simulated and analytic normalized free surface shape for a plane 2-D cross-section of the cylinder for the first benchmark case. It has been shown that an equilibrium zero gravity meniscus with a 0 ° contact angle will conform to a half sphere in a cylindrical container, such that the normalized meniscus height is equal to normalized radius of the cylinder. As shown in Fig. 1, the simulated zero-gravity meniscus is in good qualitative agreement with the analytic solution and the RMS error is 0.01. Another case was devised to determine whether liquid oxygen inside a tank subjected to the dipole magnetic field would conform to a contour of constant magnetic field intensity as predicted by theory. The dipole model was used to validate the implementation of a UDF in FLUENT, specifically the magnetic force source term model. Marchetta [23] shows that a contour of constant magnetic field intensity can be obtained from Eq. (6). To simplify the simulation setup, a square box of 0.1 m length was partially filled with LOX. The LOX is initialized a quarter sphere positioned centered in one corner of the box. The computational mesh consisted of 64,000 uniform hexahedral cells. The dipole magnet was shifted away from the box at a distance of 0.05 m to avoid having a singularity in the computational domain. A constant dipole strength was specified and the simulation was iterated until an equilibrium free surface was achieved. Fig. 2 illustrates the simulation prediction for the equilibrium free surface shape for LOX under the influence of
1.0
Normalized Meniscus Height
can be solved numerically. Second order upwind differencing is utilized for all FLUENT simulation predictions presented herein. FLUENT uses a co-located scheme; whereby pressure and velocity are both stored at cell centers. Therefore, an interpolation scheme is required to compute the face values of pressure from the cell values. For the following simulations, the body force weighted scheme is utilized to compute the face pressure by assuming that the normal gradient of the difference between pressure and body forces is constant. The pressure-implicit with splitting of operators (PISO) pressure–velocity coupling scheme is used to obtain a semi-implicit pressure correction equation. The pressure-correction equation is subsequently solved using the algebraic multigrid (AMG) method. Temporal discretization is accomplished using implicit time integration, which is unconditionally stable with respect to time step size. Solutions are subsequently iterated at each time level until the convergence criteria are met. The location of multiple phases within the mesh is determined using the volume-of-fluid (VOF) function, f. The VOF formulation relies on the fact that two or more phases are not interpenetrating. The geometric reconstruction scheme is used to reconstruct the interface between fluids using a piecewise-linear approach. Surface tension forces are computed using the Continuum Surface Force Model and a static contact angle can be specified to model wall adhesion. The addition of the magnetic force source term is achieved through use of a user defined function [22] (UDF) in FLUENT. It was necessary to define a new variable to introduce the magnetic field intensity, H. For the preliminary simulations of the MAPO experiment, a UDF was written which interpolates measured magnetic field data on to a computational mesh. In addition, the magnetic field intensity was utilized to compute the gradient of the magnetic field intensity. Martin and Holt [14] provide tables of magnetization measurements, taken from a magnetometer, for the ferrofluids used in the experiment. Coefficients for a growth saturation function were sought using regression of magnetization measurements. For the 30:1 mixture, the regression analysis yields the magnetization (A/m) as a function of the magnetic field intensity such that
Analytic
0.8
Simulation 0.6 0.4 0.2 0.0 -1
-0.5 0 0.5 Normalized Radius
1
Fig. 1. Normalized, steady-state, and free surface shape for LOX under the influence of capillary forces.
M¼
149H 6755 þ H
ð8Þ
In order to simulate the MAPO experiment, it was necessary to include the transient acceleration fields experienced by the tanks on board the KC-135 aircraft. The previous UDF was improved upon to linearly interpolate accelerometer data taken during the MAPO experiments, and provide the computational tool with the actual transient acceleration environment measured. 4. Model verification and validation Three benchmark cases were chosen to verify and validate the MP2 model. The first simulation was a cylindrical tank with a ra-
Distance along radial axis, m
The magnetization for the 10:1 water/ferro-fluid mixture is a function of magnetic field intensity such that
0.10 0.08
Analytic Simulation
0.06 0.04 0.02 0.00 0.00
0.02 0.04 Distance along z axis, m
0.06
Fig. 2. Steady-state, free surface shape for LOX under the influence of a dipole magnetic field and a contour of constant H, A/m.
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the dipole magnetic field. The RMS error between the predicted free surface shape and the analytic solution of a contour of constant H as predicted by theory was 0.01. It can be concluded that the dipole magnetic force model UDF was properly implemented in FLUENT. The final benchmark validation case, a cylindrical tank with a 10% fill of LOX was subjected to the magnetic force exerted by the MAPO experiment permanent ring magnet. The rare Earth permanent magnet has a thickness of 0.0381 m, and inner and outer diameters of 0.1016 m and 0.254 m, respectively. This case was chosen to validate the implementation of a second FLUENT UDF, which uses the measured magnetic data from the experiment and interpolates the data onto a computational mesh. Again, theory states the steady-state interface should conform to a contour of constant magnetic field intensity. The MAPO tank geometry, a cylindrical tank with a radius of 0.0762 m and a height 0.254 m, was chosen for this simulation. Noting that a mesh independent flow field was observed in the first benchmark case and the similarity in the cylindrical geometry of both cases, an equivalent mesh density was specified for this study. The surface of the ring magnet, which has a maximum flux of about 0.5 T, is positioned 0.0175 m below the tank. Fig. 3 shows the contours of constant magnetic field intensity for a plane 2-D cross-section of the tank. The simulated equilibrium free surface shape qualitatively conformed to the contours of constant H as depicted in Fig. 3. It can be concluded that the influence of the MAPO ring magnet is properly accounted for in the 2nd FLUENT UDF. 5. MAPO experiment predictions The only data available for code validation directly related to MP2 in reduced gravity is the preliminary data from the MAPO experiment [14]. For the MAPO experiments, the only flow field data available is for a 50% tank fill and it is in the form of stillframes extracted from those recordings. It should be noted that quantitative comparison between computational flow field predictions and experiment data is complicated by several factors. First, the ferrofluid/water mixture is very opaque so many of the flow
details internal to the tank surface cannot be seen once the liquid has coated the outer tank wall. Further, the angle of view for the recordings is not normal to the cylinder so problems with data interpretation due to the refraction of light as it passes through the cylindrical wall of the acrylic tank are exacerbated. In addition to those difficulties, there is an important difference between the simulation and experiment. The computational simulation begins with a perfectly flat initial interface and instantly changes from a normal 1 g environment to the transient acceleration environment. The experiment initial free surface shape is not precisely measured. Therefore, it is justifiable to assume that some disparity will be observed between the simulation and experiment due to the variations in initial free surface conditions, and due to the differences between the abrupt change of the computational simulation acceleration environment and the more gradual change of the experiment environment. As such, it is only reasonable to draw conclusions which are based on qualitative comparisons of the surface shape. For the simulations that follow, a flat 1 g interface is used as the initial free surface. After the initialization, the transient measured background acceleration is included as a source in the momentum equation. Fig. 4 shows the three tank configurations used in the experiment. To simplify further discussion, the magnet end of the tank will be called the bottom of the tank and the end opposite to the magnet will be called the top. Three meshes of increasing mesh density, 49,728, 71,232, and 174,336 cells, respectively were utilized to simulate the 50% bottom fill case. The study was judged to have demonstrated a mesh independent simulation when the predicted flow fields for the 50% bottom fill cases were unchanged by additional mesh refinement. Fig. 5 illustrates that the differences in the free surface shape for the three meshes are negligible after 30 s of simulation time. To reduce computational time, the 71,232 cell mesh, shown in Fig. 6, was selected for the remaining simulations of the experiment. The first experiment test considered is a bottom fill tank of 30:1 water/ferrofluid mixture. Fig. 7 presents a sequence of flow fields recorded for a configuration in which liquid is collected at the magnet end of the tank prior to entering the reduced gravity phase of the KC-135 flight. In each picture, the red area contains the ferro-
Free Surface Height (m)
Fig. 4. MAPO experiment tank configurations [14].
0.20 0.16 0.12 0.08
49728 cells 71232 cells
0.04
174336 cells
0.00 -0.08
-0.04
0.00
0.04
0.08
Radial Position (m) Fig. 3. Steady-state, free surface shape for a 10% fill of LOX under the influence of the MAPO magnetic field and contours of constant H, A/m, inside the tank.
Fig. 5. Free surface shape for varying mesh densities.
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Fig. 6. MAPO experiment tank topology with 71,232 cell computational mesh.
fluid and the blue region contains air. The shaded areas in the simulation pictures indicate regions filled with liquid on the farthest wall of the tank which would not appear in the photographs due to the opacity of the ferrofluid. At 1.5 s, there has already been considerable displacement of the initial flat interface and a general sweeping motion of fluid from the centerline toward the tank wall can be observed. After 1.5 s, the interface near the tank centerline appears to have taken on a shape corresponding to a classical meniscus due to surface tension. In contrast, near the tank wall it appears that momentum imparted by the initial sweep of fluid toward the wall is causing a thin layer of fluid near the wall to climb toward the top of the tank. From this time onward, there is little displacement of the free surface in the vicinity of the tank centerline but a continuing evolution of the thin fluid layer near the wall can be observed. A sequence of flow fields predicted by the computational simulations for the 50% bottom fill of 30:1 water/ ferro-fluid mixture is also presented in Fig. 7. In the simulation predictions, the transition from 1 g to low gravity is observed at 1.5 s. The meniscus shape is evident, however quite asymmetric due to the fluctuating background accelerations. At the 14 s photo, a sloshing effect is noted in both the experimental photos and
847
simulation predictions. Some discrepancy exists past 18 s in the simulation predictions. A small layer of fluid appears to be coating the top of the tank in the simulation as a consequence of the initial sloshing motion, yet this coating is not observed in the experiment. The majority has settled back into the asymmetric surface shape similar to that observed in the photos. Fig. 8 presents a flow field sequence corresponding to a configuration in which the 30:1 water/ferro-fluid mixture is initially collected in the end of the tank opposite to the magnet. Fig. 8 also presents simulation predictions for this 50% top fill case. At 6.0 s in the experiment, it is apparent that the magnet is pulling the fluid along the wall toward the bottom of the tank. After 15 s, a small pool of liquid has collected at the bottom outside corner of the tank. It is interesting to note that although fluid momentum induces flow along the bottom toward the centerline, the magnetic force is strong enough to capture the liquid and hold it in the corner. Although the shape of the free surface of this pool appears to be dominated by surface tension near the tank wall, it appears to conform to a surface of constant magnetic field strength. At 17 s, the fluid appears to separate into two distinct pools at opposing ends of the tank. This is likely due to the combined influence of sloshing due to the acceleration field and the magnetic field at the bottom of the tank. While not easily discerned in the simulation pictures presented above, a sample 2-D cross-section of the tank at 17 s confirms that the liquid is being influenced by the magnet as the shape of the free surface closely resembles a contour of constant magnetic field intensity. It is difficult to verify this behavior with the corresponding experimental photo due to the water/ferro-fluid mixture coating the tank walls. Because of this, comparison between the remaining experimental photos and simulation predictions is not possible. Fig. 9 presents a series of flow fields where the 30:1 water/ferro-fluid is initially positioned along the length of the cylindrical tank, such that the interface is perpendicular to the magnet face. This case is of particular interest in this study because the axisymmetric predecessor to the current simulation was unable to model the asymmetry of the configuration. As the experiment progress, the liquid is drawn up the opposing tank heads, a result consistent with the influence of surface tension. It is evident from the series of photos and simulation results shown in Fig. 9 that more fluid collects at the bottom of the tank as opposed to the top. The additional accumulation of the liquid at the bottom of the tank is likely due to the influence of the magnetic field. Further, as time increases, more
Fig. 7. Sequence of 30:1 bottom fill predictions and photos of the MAPO experiment.
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Fig. 8. Sequence of 30:1 top fill simulation predictions and photos of the MAPO experiment
Fig. 9. Sequence of 30:1 side fill simulation predictions and photos of the MAPO experiment.
fluid accumulation at the bottom of the tank is observed as opposed to the top. 6. Constant background acceleration MP2 predictions The motions observed in the MAPO experiment provide evidence of the influence of the magnet on the water/ferro-fluid mixture. Yet, as noted the flow details are masked by the large three-dimensional fluctuations in the acceleration environment and the opacity of the water/ferro-fluid mixture. To better under-
stand the MP2 process in reduced gravity, additional simulations are performed with a constant background acceleration of 103 g specified along the axis of symmetry of the cylindrical tank. Neglecting the influence of g-jitter, this background acceleration is comparable to average acceleration environment experienced in low earth orbit. A sequence of flow fields predicted by the computational simulations for the 50% bottom fill for constant background acceleration is presented in Fig. 10. In the simulation predictions, the transition from 1 g to low gravity is observed at 1.5 s. Physically, the flow
Fig. 10. Sequence of bottom fill simulation predictions for constant background acceleration.
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Fig. 11. Sequence of top fill simulation predictions for constant background acceleration.
Fig. 12. Sequence of side fill simulation predictions for constant background acceleration.
fields are similar to the experiment photos presented in Fig. 7 through 9.5 s, although a difference in transition times is again noted. The free surface reaches equilibrium within 14–18 s. In contrast, the asymmetric free surface in the experiment photos are a result of an asymmetric background acceleration, which may have induced the sloshing observed in the 14 s photo. Fig. 11 presents the simulation predictions for the 50% top fill case for constant background acceleration. This simulation more clearly demonstrates that the magnet is influencing the motion of the ferrofluid down the outer tank wall. The liquid near the bottom of the tank appears to conform to a contour of constant H. In contrast, the liquid near the top of the tank assumes a more spherical shape consistent with capillary forces. The ferrofluid is continuously reoriented throughout the 26.5 s of simulation time as demonstrated by the accumulation of fluid at the bottom of the tank. Fig. 12 presents a series of simulations predictions for constant background acceleration where the ferrofluid is initially positioned along the length of the cylindrical tank, such that the interface is perpendicular to the magnet face. More fluid collects at the bottom of the tank as opposed to the top, a consequence which again demonstrates the influence of the magnetic field on the motion of fluid. In contrast to the top fill case, the free surface shape appears more spherical as time progresses, an indication that surface tension is strongly influencing the liquid near the middle and top of the top. 7. Summary and conclusions In summary, qualitative comparisons of the free surface shapes between MAPO experiment data and simulation predictions showed as good agreement in the majority of cases given the quality of the experiment environment. Alluded to in previous works, the uncertainty for promising applications like MP2 and many others, emphasizes the need for more reduced gravity experiments specifically intended for simulation validation. Although the computational results presented herein appear to be promising, considerable research remains to be performed before the feasibility of
MP2 can be clearly demonstrated. For future work, the discrepancies that exist between the initial conditions experienced by the tanks onboard the KC-135 aircraft and those specified in the simulations need to be addressed. Furthermore, there are several test cases done in the MAPO experiments that are yet to be simulated. In addition, the feasibility of using an electromagnet in full scale tanks needs to be investigated in future work. Acknowledgement The authors are thankful to the Tennessee Space Grant Consortium for supporting this effort. References [1] Sumner IE. ‘‘Liquid Propellant Reorientation in a Low Gravity Environment”, NASA TM-78969, 1978. [2] Aydelott JC. ‘‘Axial Jet Mixing of Ethanol in Cylindrical Containers During Weightlessness”, NASA TP-1487, 1982. [3] Hochstein JI, Patag AE, Chato DJ. Modeling of Impulsive Propellant Reorientation. J Propul Power 1991;7(6):938–45. [4] Schmidt GR, Chung TJ, Nadarajah A. Thermocapillary flow with evaporation and condensation at low gravity. Part 1. Non-deforming surface. J Fluid Mech 1995;295:323–47. [5] Schmidt GR, Chung TJ, Nadarajah A. Thermocapillary flow with evaporation and condensation at low gravity. Part 2. Deformable surface. J Fluid Mech 1995;295:349–66. [6] Hochstein JI, Chato DJ. Pulsed thrust propellant reorientation: concept and modeling. J Propulsion Power 1992;8(4):770–7. [7] Chipchak D. Development of Expulsion and Orientation Systems for Advanced Liquid Rocket Propulsion Systems”, USAF Technical Report RTD-TDR-63-1048, Contract AF04(611)-8200, July, 1963. [8] Neuringer JL, Rosensweig RE. Ferrohydrodynamics. Physics of Fluids 1964;7(12):1927. [9] Zelazo RE, Melcher JR. Dynamics and stability of ferrofluids: surface interactions. J Fluid Mech 1969;39:1. [10] Boshtovoy VG, Krakov MS. Stability of an axisymmetric jet of magnetizable fluid. Translated for Z. Prik. Mekh. I Tekh Fiz. 1978:147. [11] Berkovsky BM, Smirnov NN. Capillary hydrodynamic effects in high magnetic fields. J Fluid Mech 1988;187:319. [12] Basaran OA, Wohlhuter FK. Effect of nonlinear polarization on shapes and stability of pendant and sessile drops in an electric (magnetic) field. J Fluid Mech 1992;244:1.
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[13] Bashtovoy VG, Berkovsky BM, Vislovich AN. Introduction to thermomechanics of magnetic fluids. Berlin: Hemisphere Publishing Co; 1988. [14] Martin JJ, Holt JB. ‘‘Magnetically Actuated Propellant Orientation Experiment, Controlling Fluid Motion With Magnetic Fields in a Low-Gravity Environment”, NASA TM 210129, 2000. [15] Hochstein JI, Warren RT, Schmidt GR. Magnetically actuated propellant orientation (MAPO) experiment: pre-flight flow field predictions. AIAA Paper 97-0570, Jan. 1997. [16] Marchetta JG, Hochstein JI. Fluid Capture by a Permanent Ring Magnet in Reduced Gravity. AIAA Paper 99-0845, Jan. 1999. [17] Marchetta JG, Hochstein JI. A computational model of magnetic positive positioning in reduced gravity. IAF Paper ST-99-W.210, Oct. 1999.
[18] Winter AP, Marchetta JG, Hochstein JI. Simulation and Prediction of Realistic Magnetic Positive Positioning for Space Based Fluid Management Systems. AIAA Paper 2004-1151, 2004. [19] Marchetta JG, Hochstein JI, Simmons BD, Sauter DR.Modeling and Prediction of Magnetic Storage and Reorientation of Lox in Reduced Gravity. AIAA Paper 2002-1005, Jan. 2002. [20] Kothe DB, Mjolsness RC, Torrey MD. RIPPLE: a computer program for incompressible flows with free surfaces, LANL Report LA-12007-MS, April, 1991. [21] Fluent Inc.. Fluent 6.1 User’s Guide. Lebanon, NH: Fluent Inc.; 2003. [22] Fluent Inc.. Fluent 6.1 UDF Manual. Lebanon, NH: Fluent Inc.; 2003. [23] Marchetta JG, Hochstein JI, Simmons BD. On-Orbit Positive Positioning of LOX: Simulation and Correlation. AIAA Paper 2003-1153, Jan. 2003.