The splitting behavior of a bubble in a tip vortex flow is studied using an axisymmetric boundary element method accounting for the effect of surrounding viscous flow. Bubble splitting behavior, bubble sizes before and after splitting, and emitted pressure signals are characterized in a wide range of cavitation numbers. The effect of the Reynolds number and the effect of initial bubble nucleus size are also investigated. These observations lead us to a simple bubble splitting model which can be applied within a practical spherical model to include bubble subdivision into smaller sub-bubbles. INTRODUCTION
The method is based on coupling bubble dynamics models and an unsteady Reynolds Averaged Navier Stokes (RANS) solver. In the coupled approach, surface averaged pressure (SAP) spherical models [4,5] and non-spherical axisymmetric [1-3] or fully three-dimensional models [4] using boundary element methods can be utilized to describe the bubble dynamics. Bubble behavior in a vortex occur in three phases; (a) bubble capture by the vortex, (b) interaction between the vortex and an initially quasi-spherical bubble, and (c) the dynamics of elongated bubbles on the vortex axis [6]. In this study, the SAP spherical bubble dynamics model is used for the two phases while the axisymmetric method is utilized in the third phase.
2DYNAFS, is a boundary element potential flow solver. The code has been verified successfully for diverse types of fluid dynamic problems [7]. Recently, it has been extended to accommodate the ambient vortex flow field and to model the extreme deformation of bubbles including splitting [1-3]. A summary of the numerical method and its validation against laboratory experiments can be found in Choi and Chahine [1,2].
A few past investigations considered bubble splitting between two parallel plates [8-10]. These showed the hourglass-like bubble deformation experimentally and numerically. Ishida et al. [10] reported the existence of a small peak in the pressure signal at the splitting. However, the numerical studies while able to simulate the bubble behavior until spitting, could not predict the dynamics beyond that point.
TIP VORTEX FLOW FIELDS The tip vortex flow field used in this study is computed
with DF_UNCLE, a Reynolds Averaged Navier-Stokes (RANS) flow solver. Two different scales are considered as follows: • Hydrofoils : elliptic plan form; (a) chord = 1 m, half span =
1.5 m, (b) chord = 0.5 m, half span = 0.75 m • Inflow: U∞ =2.88m/s; (a) Re=1.44x106, (b) Re=2.88x106
Figure 1. The axisymmetric tip vortex flow field for Re=2.88x106. (x,r) are made dimensionless by the chord, p and u, by 2Uρ ∞
and U∞, respectively.
REPRESENTATIVE RESULTS In this section, the behavior of a bubble nucleus of radius
50 µm placed at 0.1 m upstream of the tip is used as an example. The cavitation number studied for this bubble condition varies from 2.2 to 2.8. The initial pressure in the nucleus bubble is set to be in equilibrium with the ambient pressure for the corresponding cavitation number σ defined as
21
2
vp p
Uσ
ρ
∞
∞
−= . (1)
A representative bubble splitting behavior from the simulations is shown in Figure 2 for σ =2.50. The figure shows the final stage of the simulation when the elongated bubble splits into two sub-bubbles which then develop reentrant jets. In general, bubbles do not always split. The same size nuclei at a slightly higher σ =2.54 shows no splitting. Instead a single jet from the downstream end of the elongated bubble forms.
Figure 3. Equivalent radius of the bubble as it flows downstream for cavitation numbers, 2.50 and 2.54.
Figure 4, shows the maximum radius the bubble reached for a given σ , as well as its radius at the moment of splitting and the radii of the sub-bubbles generated. The size of the larger sub-bubble after splitting is only slightly smaller than the size of the bubble just before splitting. This difference between the two sizes is even smaller as the cavitation number increases and approaches the inception number. When the cavitation number is very close to inception no splitting occurs.
Time [sec]
Equ
ival
entR
adiu
s[m
]
0.05 0.06 0.07 0.080
0.001
0.002
0.003
0.004
0.005
2DynaFSSAP
Re = 2.88e6, Ro = 50 µm, σ = 2.54
single jet in 2DynaFS
Time [sec]
Equ
ival
entR
adiu
s[m
]
0.05 0.06 0.07 0.080
0.001
0.002
0.003
0.004
0.005
2DynaFSSAP
Re = 2.88e6, Ro = 50 µm, σ = 2.50
3
Figure 5 shows the relative sizes of the bubbles before and after splitting. We can see that the ratio of the maximum equivalent radii to the radius at splitting increases from 1.06 to 1.15 as the cavitation number increases from 2.2 to 2.5. The ratio of the larger sub-bubble generated to the equivalent radius of the bubble before splitting increases from 0.95 to 0.99, while the relative size of the smaller sub-bubble decreases from 0.55 to 0.2. The curves of these ratios flatten as the cavitation number decreases.
Cavitation Number
RE
Q[m
]
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.80
0.0025
0.005
0.0075
Max. 2DynaFS, 128 seg2DynaFS before splitting2DynaFS after splitting2DynaFS after splitting
Equivalent Radius for Re=2.88E06, Ro=50µm
no splitting
Figure 4. Equivalent radii of the bubble at the moment of splitting and radii of the sub-bubbles generated.
minimum pressures during the initial sudden growth phase. Similarly, the collapse peak-to-peak pressure and the jet peak-to-peak pressure are defined respectively for the collapse/rebound phase and for the jet development phase.
Figure 6. Pressure signals at a field point predicted for σ=2.50. The peak-to-peak pressures are defined as the maximum pressure minus the minimum pressure within the indicated portion of the pressure signal.
Cavitation Number
Pre
ssur
e[P
a]
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.810-2
10-1
100
101
102
103
104
SAP (1st)SAP (collapse)2DynaFS (1st) 128 seg2DynaFS (jet) 128 seg
Field Point Pressure for Re=2.88E06, Ro=50µmPeak-to-peak pressure at (r,z)=(1.5297, 0.0) m
Figure 8. Axial locations of the center of the bubble and the formed sub-bubbles for three cavitation number cases; 2.20, 2.40, and 2.50 (from left to right)
The trajectories of the center of the initial bubble and the resulting sub-bubbles are shown in Figure 8. In all three cavitation number cases shown in the figure, the bubble nucleus slows down as it flows downstream and it almost stops traveling downstream at the moment of splitting. Just after the splitting, the center of the larger sub-bubble is located slightly upstream of the center of the pre-split bubble whereas the center of the smaller sub-bubble is located much downstream of the pre-split bubble. The centers of the sub-bubbles for the three cases shown in Figure 8 can be described in terms of the equivalent radius of each case. That is, -0.71 and 4.09 for σ =2.20, -0.51 and 4.09 for σ =2.40, and -0.15 and 3.87 for σ =2.50.
Cavitation Number
RE
Q[m
]
1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.80
0.0025
0.005
0.0075
Max. SAPMax. 2DynaFS
Equivalent Radius, Re=1.44x106 vs. 2.88x106, Ro=50µm
no splitting
Re=1.44x106
Re=2.88x106
no splitting
Figure 9. Comparison of the maximum equivalent radii for two Reynolds numbers, 1.44x106 and 2.88x106.
REYNOLDS NUMBER EFFECT In order to observe the Reynolds number effect on the
bubble size and the pressure peaks, the results in the previous section are compared with the corresponding results from the simulations using the tip vortex flow field at Reynolds number 1.44x106. The maximum equivalent radii of the bubbles for the two Reynolds numbers are compared in Figure 9. The cavitation inception based on the bubble size occurs at σ =2.57 for Re=2.88x106, and at σ =2.10 for Re=1.44x106. The trend of the
maximum bubble size is found to be very similar for the two Reynolds numbers. The bubbles do not split for cavitation numbers just below the inception, however, the relative magnitude of the bubble size is smaller for the lower Reynolds number cases.
Normalized Cavitation Number (σ/-Cpmin)
Nor
mal
ized
Equ
ival
entR
adiu
s(R
EQ/a
c)
0.7 0.8 0.9 1 1.10
0.5
1
1.5
Max. SAPMax. 2DynaFS
Equivalent Radius, Re=1.44x106 vs. 2.88x106, Ro=50µm
Re = 1.44x106
-Cpmin = 2.27ac = 0.003319 m
at Cpmin
Re = 2.88x106
-Cpmin = 2.73ac = 0.00558 m
at Cpmin
no splitting
Figure 10. Comparison of the maximum equivalent radii for the two Reynolds numbers with both axes normalized.
Normalized Cavitation Number (σ/-Cpmin)
Nor
mal
ized
Equ
ival
entR
adiu
s(R
EQ/a
c)
0.7 0.8 0.9 1 1.10
0.5
1
1.5
Max. 2DynaFS2DynaFS before splitting2DynaFS after splitting2DynaFS after splitting
Equivalent Radius, Re=1.44x106 vs. 2.88x106, Ro=50µm
Re = 1.44x106
-Cpmin = 2.27ac = 0.003319 m
at Cpmin
Re = 2.88x106
-Cpmin = 2.73ac = 0.00558 m
at Cpmin
Figure 11. Comparison of the equivalent radii at the splitting for two for the two Reynolds numbers with both axes normalized.
Because the cavitation inception is known to be scaled for different Reynolds numbers by the minimum pressure in the tip vortex core, the abscissa of Figure 9 can be normalized by using the minimum pressures, ,minpC− =2.27 and 2.73, respectively
for Re=1.44x106 and 2.88x106. Also from our previous results [4] the bubble maximum size is scaled by the core radius of the tip vortex. Thus, the ordinate of Figure 9 can be normalized by the corresponding core radius. The tip vortex core radius at the axial location of minimum pressure is found from the RANS equation solutions as ca =0.003319 m and 0.00558 m respectively for Re=1.44x106 and 2.88x106. After normalizing
Time [sec]0.005 0.01 0.01
Time [sec]0.005 0.01 0.015
Time [sec]
Axi
alLo
catio
nof
C.G
.[m
]
0 0.005 0.01 0.015 0.020
0.02
0.04
0.06
0.08
5
both axes, the cavitation inception curves for the two Reynolds numbers nicely collapse near ,min/ pCσ − =0.94, and the
equivalent radius curves from the two Reynolds number cases also collapse nicely as shown in Figure 10. Based on the normalized bubble size, the bubble splitting happens only when
/eq cR a >0.55~0.60. This threshold is based on the 50 µm
bubble nucleus here, but this value does not vary much for other nucleus sizes as will be discussed later.
The equivalent radii just before and after the bubble splitting are compared for the two Reynolds numbers in Figure 11 with the normalization described above. A strong resemblance between the two Reynolds numbers is observed in the bubble and sub-bubble sizes. Notice that the equivalent radii curves also collapse nicely. In Figure 12, the ratios of the equivalent radii are compared for the two Reynolds numbers using normalized cavitation numbers. The ratios look very similar for the two Reynolds numbers. The radius ratios flatten as the cavitation number decreases much below the cavitation inception number.
Ratio of Radii, Re = 1.44x106 vs. 2.88x106, Ro = 50e-6 m
Re=2.88x106Re=1.44x106
Figure 12. Comparison of the equivalent radii ratios for the two Reynolds numbers with normalized cavitation number on the abscissa.
The peak magnitudes of the pressure signals are compared in Figure 13. The pressure is computed at a field point 1.53 m away from the tip of the hydrofoil for the case of Re=2.88x106, but at a field point 0.81 m away for the smaller hydrofoil case of Re=1.44 x106. The normalization of the cavitation number also works for the pressure signal of the two Reynolds numbers. The highest pressure peak predicted by the SAP spherical model for Re=1.44 x106 has an up and down near the flat region of the lower cavitation numbers, which is due to the fact that the highest pressure is observed at the subsequent collapses/rebounds rather than at the usual first one. The magnitude of the pressure does not depend on the Reynolds numbers as expected, but depends on how far the cavitation number deviates from the cavitation inception number.
The axial location of the bubble centers (such as shown earlier in Figure 8) is studied also for the Re=1.44x106. From this numerical study, the observed centers of the bubble and the sub-bubbles can be compared for the two Reynolds numbers as
shown in Figure 14. The dimensional bubble center values of the two Reynolds number cases seem to follow roughly the length scale ratio, 2, of the two cases. It can be concluded that the upstream sub-bubble center just after the splitting approaches -0.8 Req while the downstream sub-bubble center approaches about 4.1 Req as the cavitation number decreases for both Reynolds numbers.
Normalized Cavitation Number (σ/-Cpmin)
Pre
ssur
e[P
a]
0.7 0.8 0.9 1 1.110-2
10-1
100
101
102
103
104
SAP (1st min/max)SAP (max. at collapse)2DynaFS (1st min/max)2DynaFS (jet min/max)
Field Point Pressure with Ro=50µm, Re=1.44x106 and 2.88x106
Peak-to-peak pressure at (r,z)=(1.5297, 0.0) m
Re=2.88x106
Re=1.44x106
Figure 13. Comparison of the pressure signals for the two Reynolds numbers with normalized cavitation number on the abscissa.
Figure 14. Axial location of the bubble center and the sub-bubble centers at the splitting for two Reynolds numbers, 1.44x106 and 2.88x106. Ro=50µm, CGb: center just before the splitting, CGa1 and CGa2: centers of the sub-bubbles just after the splitting, Req: equivalent radius just before the splitting.
BUBBLE NUCLEUS SIZE EFFECT The effect of the initial nucleus size is studied by
initial bubble sizes seem to lie on straight lines extended from the lower cavitation number portion of the radius curves for the larger initial bubble sizes. The threshold cavitation number between bubble splitting and single jet formation decreases as the nucleus size decreases from 100 µm to 50 µm. The threshold in terms of bubble size is eqR 0.0028 m, which can be
normalized as /eq cR a 0.50. This value of the threshold is not
very different from the value /eq cR a 0.55~0.60 found earlier
for the two Reynolds number cases. Note that single jet behaviors cannot be observed for the smaller initial bubble radii of 10 and 20 µm.
In Figure 16, the ratios of the equivalent radii are compared for the four initial bubble radii. The flattening trend prevails for the lower cavitation numbers regardless of the initial bubble sizes. As the cavitation number decreases, the ratio for the maximum equivalent radius approaches 1.06, the ratio for the larger sub-bubble just after the split approaches 0.95, that for the smaller sub-bubble approaches 0.55 regardless of the initial bubble size.
Figure 15. Comparison of the equivalent radii for four initial bubble sizes, 10, 20, 50, and 100 µm.
cavitation numbers. This might be due to the fact that the jet developed from 10 µm initial bubble is much longer (through a long tubular part of the sub-bubble) than the other jets found in larger size bubble simulations.
Figure 16. Comparison of the ratios of the equivalent radii for four initial bubble sizes, 10, 20, 50, and 100 µm.
Cavitation Number
Pre
ssur
e[P
a]
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310-4
10-2
100
102
104
106
SAP (1st min/max)SAP (max. at collapse)2DynaFS (1st min/max)2DynaFS (jet min/max)
Field Point Pressure at Re=2.88x106, with Ro=10, 20, 50, and 100 µmPeak-to-peak pressure at (r,z)=(1.5297, 0.0) m
20 µm
10 µm
50 µm
100 µm
Figure 17. Comparison of pressure signals for four initial bubble sizes: 10, 20, 50, and 100 µm.
The axial location of the bubble center obtained from several computations with different initial bubble nucleus sizes is shown in Figure 18. It is interesting to observe that the sub-bubble centers normalized by the equivalent radius form common curves throughout the cavitation number ranges studied. The common curve for the downstream sub-bubbles is observed to be more or less flat near 4.4 Req for the cavitation numbers between 1.7 and 2.2. The normalized axial location of the upstream sub-bubble varies from approximately -1.0 in the low cavitation numbers to 0.0 at the cavitation number where the splitting does not happen.
Figure 18. Axial location of the bubble center and the sub-bubble centers at the splitting for four initial bubble sizes, 10, 20, 50, and 100 µm. Re=2.88x106, CGb: center just before the splitting, CGa1 and CGa2: centers of the sub-bubbles just after the splitting, Req: equivalent radius just before the splitting.
advantage of predicting non-spherical bubble behaviors, such as elongation, splitting, and reentrant jets, and the resulting pressure signals. However, these simulations are more costly than simulations using the SAP spherical model. In order to perform massive simulations with multiple bubble nuclei, a simpler model is required so that the bubble splitting can be handled within the spherical model. In the upgraded spherical model, once a bubble reaches a condition that it should split, the bubble is replaced with two sub-bubbles and the simulation is continued with the new sub-bubbles. The splitting model will be more useful for cavitation numbers much lower than the inception because the bubble usually does not split very near the cavitation inception. The splitting prediction model should be able to answer the next two questions.
• Criteria of splitting: When does a bubble split? • Initial condition of the sub-bubbles: What is the size of
each sub-bubble? What is the pressure inside each sub-bubble?
From the studies described previously [3], a bubble splits when its elongation reaches a value between 2.8 and 4.8 depending on the cavitation number and the initial bubble size. Since the elongation cannot be calculated within the spherical model, it cannot be used in the spherical model to determine if a bubble should split. The ratio of the maximum equivalent bubble radius relative to the equivalent radius just before the splitting can be used as a criterion. This ratio is flat at 1.05 for lower cavitation numbers and increases up to 1.15 as the cavitation number approaches the inception. In other words, the following criterion can be used to determine when a bubble should split.
Bubble splitting criterion (for low cavitation numbers): When the bubble reaches it maximum radius and the radius decreases to 0.95 of the maximum, the bubble splits.
Time [sec]
Equ
ival
entR
adiu
s[m
]
0.065 0.066 0.067 0.068 0.0690.002
0.004
0.006
0.008Re = 2.88e6, Ro = 50 µm, σ = 2.20
Figure 19. Equivalent radius of the bubble as a function of time predicted with 50 µm nucleus at σ=2.20.
Figure 20. Equivalent radius of the bubble as a function of time predicted with a 10 µm nucleus at σ=1.54.
Once the splitting is detected from the above criteria, the size of sub-bubbles can be determined also from the ratios shown in Figure 16. The equivalent radii of the larger and the smaller sub-bubbles are, respectively, 0.95 and 0.55 of the equivalent radius before the splitting. The gas pressure inside the sub-bubbles can be modeled with the introduction of a pressure reduction factor α such that the pressure after the splitting can be written as
, ,g after g beforep pα= . (2)
In an ideal case of lossless splitting, the factor α is 1.0. The radial velocity of the sub-bubbles can be observed
indirectly from equivalent radius curves such as shown in Figure 3. A magnified view near the splitting is shown in Figure 19 for the case of σ =2.20. The slope of the equivalent radius in the figure can be interpreted as the average radial velocity of the bubble. Just after the split, the larger sub-bubble experiences a very slow radial velocity (nearly zero slope) while the smaller sub-bubble has a slightly faster radial velocity (steeper slope) than before the split. This trend is also found in most of the bubble behaviors simulated with 50 and 100 µm bubble nuclei at relatively higher cavitation numbers. However, in simulations
Time [sec]
Equ
ival
entR
adiu
s[m
]
0.05 0.075 0.1 0.1250
0.005
0.01
0.015
0.02
2DynaFSSAP
Re = 2.88e6, Ro = 10 µm, σ = 1.54
σ
CG
1/Req
,CG
2/Req
CG
b,C
Ga1
,CG
a2[m
]
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8-2
0
2
4
6
0
0.05
0.1
0.15
0.2
CGbCGa1CGa2CG1/ReqCG2/Req
Center of Gravity of Sub-bubbles, Re=2.88x106
Ro=100µm
Ro=10µm
Ro=20µmRo=50µm
8
with 10 and 20 µm bubble nuclei at relatively lower cavitation numbers, the slope just after the split tends to be zero for both sub-bubbles as shown in Figure 20. Therefore, zero radial velocity as the initial condition of the sub-bubbles seems to be appropriate for the lower cavitation number simulations.
The initial location of the sub-bubbles depends weakly on the Reynolds numbers (Figure 14) and more strongly on the cavitation numbers (Figure 18). The location can be read directly from these two figures. For cavitation numbers in the mid-low range (1.7 σ≤ ≤ 2.2), the locations of the sub-bubbles relative to the bubble center just before the splitting are fairly constant and are -1.0 and 4.4, respectively for the upstream and the downstream ones. The initial condition of the new sub-bubbles can be summarized as follows.
Initial condition of the sub-bubbles (for low cavitation numbers): The radii of the larger and smaller sub-bubbles are respectively, 0.95 and 0.55 of the radius just before splitting. The initial gas pressure is determined from equation (2), and the initial radial velocity is zero. The initial locations of the sub-bubbles are determined from Figure 14 or Figure 18. If a simpler approach is desired, it can be assumed that the initial location of the larger sub-bubbles is 1.0 radius upstream of the pre-split bubble center and that of the smaller one is 4.4 radii downstream of the pre-split bubble center.
CONCLUSION Bubble behaviors in two tip vortex flow fields of Reynolds
The effect of the Reynolds number is found to be mainly a shifting of the cavitation inception number both in the bubble radii and in the pressure peaks. With cavitation numbers normalized by the minimum pressure in the tip vortex core, the curves from the two Reynolds numbers collapse very close to each other nicely as expected. The effect of initial bubble nucleus size is found to be such that, if observed from higher to lower cavitation numbers, a smaller initial bubble nucleus brings an abrupt transition from a non-cavitating status to the cavitating status represented by the common curves of the radii (or of the pressure peaks) much later than a larger bubble nucleus case. These common curves seem to be independent of the nucleus size within the range of 10 to 100 µm for Reynolds number 2.88 million.
The observation of the ratios of the equivalent bubble maximum radii, before and after splitting, leads to a simple model of the bubble splitting for low cavitation numbers. This suggested model is based on the ratios of the equivalent radii of the bubble and sub-bubbles. Studies to expand and apply the model are currently underway.
ACKNOWLEDGMENTS This work was conducted at DYNAFLOW, INC.
(www.dynaflow-inc.com). The support of several colleagues including Dr. Chao-Tsung Hsiao is appreciated. The work has been supported by the Office of Naval Research under contract No. N0014-99-C-0369 monitored by Dr. Ki-Han Kim. This support is greatly appreciated. DYNAFLOW’s RANS code DF_UNCLE is based on the UNCLE code originally developed by Mississippi State University and graciously provided to DYNAFLOW.
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in vortex flow fields”, Computational Mechanics, (in print) 2003.
[2] Choi, J.-K., Chahine, G. L., “Noise due to extreme bubble deformation near inception of tip vortex cavitation”, Proc. FEDSM’03, International Symposium on Cavitation Inception, 4th ASME/JSME Joint Fluids Engineering Conference, Honolulu, Hawaii, July 6-10, 2003.
[3] Choi, J.-K., Chahine, G. L., “A numerical study on the bubble noise and the tip vortex cavitation inception”, 8th International Conference on Numerical Ship Hydrodynamics, Busan, Korea, September 22-25, 2003.
[4] Hsiao, C.-T., Chahine, G. L., “Prediction of vortex cavitation inception using coupled spherical and non-spherical models”, Proc. 24th Symposium on Naval Hydrodynamics, ONR, Fukuoka, Japan, July 8-13, 2002.
[5] Hsiao, C.-T., Chahine G. L., Liu, H., “Scaling effects on prediction of cavitation inception in a line vortex flow”, Journal of Fluids Engineering, Vol.125, 2003, pp.53-60.
[6] Chahine, Georges L., Fluid Vortices, S. Green (ed.), Kluwer Academic, Chapter 18, 1995.
[7] Chahine, G. L., Duraiswami, R., Kalumuck, K. H., “Boundary element method for calculating 2-D and 3-D underwater explosion bubble loading on nearby structures including fluid structure interaction effects,” Technical Report NSWC-DD/TR-93/46, Dynaflow, Inc., 1996.
[8] Chahine, G. L., “Experimental and asymptotic study of nonspherical bubble collapse,” Applied Scientific Research, Vol.38, 1982, pp.187-197.
[9] Kucherenko, V. V., Shamko, V. V., 1986, “Dynamics of electric-explosion cavities between two solid parallel walls,” Journal of Applied Mechanics and Technical Physics, Vol.27, 1986, pp.112-115.
[10] Ishida, H., Nuntadusit, C., Kimoto, H., Nakagawa, T., Yamamoto, T., “Cavitation bubble behavior near solid boundaries,” Proc. CAV2001, 4th International Symposium on Cavitation, session A3, California Inst. of Tech., Pasadena, CA. 2001.
[11] Chahine, G. L., Sarkar, K., Duraiswami, R., “Strong bubble/flow interaction and cavitation inception,” Technical Report 94003-1ONR, DYNAFLOW, INC., 1997.
