An Overview of Computational Results at General FusionInc. with Focus on Hydrodynamics

Victoria Suponitsky1, Sandra Barsky1, Dustin Oliphant1, and Peter Kostka1

General Fusion Inc., 108-3680 Bonneville Place, Burnaby, BC V3N 4T5,Canada

Email: victoria.suponitsky@generalfusion.com

ABSTRACT

Fusion as a way to generate energy is currently un-dergoing a renaissance in scientific interest. The ap-proach called Magnetized Target Fusion, a hybrid be-tween magnetic fusion and inertial confinement fusion,is currently pursued by General Fusion. At GeneralFusion, our scheme is to inject plasma into an evacu-ated cylindrical cavity surrounded by rotating moltenlead-lithium which is contained within a steel ves-sel, and then to compress it to thermonuclear condi-tions by a shock wave generated by pneumatic pistonsplaced on the vessel. This scheme was inspired froma project named ‘LINUS’, first developed by the USNaval Weapons Research Lab in Washington DC inthe late 1970s.Hydrodynamic aspects of the project include but arenot limited to (i) designing a lead pumping scheme thatassures formation of a stable cylindrical void cavityinside the vessel, and (ii) propagation of the pressurewaves followed by the cavity collapse.This overview describes the results of computationalfluid dynamics (CFD) and finite element analysis(FEA) simulations that were carried out to deepen ourunderstanding of mentioned above points (i) and (ii).Numerical results with regard to the cavity formationare qualitatively compared to our in-house experimen-tal data, while for the pressure propagation and cav-ity collapse the results of CFD were compared to FEAsimulations.

1 INTRODUCTION

Recently there has been a revival of interest in con-trolled fusion as a source of energy. A number ofprojects worldwide, from both public and private in-dustry, are currently examining ways to create a com-mercial source of fusion energy. The approach pur-sued by General Fusion is first to merge two hydrogenspheromaks inside the central evacuated cavity sur-

rounded by molten lead-lithium (PbLi) and then to suf-ficiently compress the spheromaks by pressure wavesin order to satisfy thermonuclear conditions and ob-tain fusion. A cavity is formed as a result of tangentialpumping of PbLi into a vessel having a drainage holeon the bottom. The pressure waves are initiated bystriking the outer surface of the PbLi with pneumaticsteel pistons. The amplitude of the pressure wavesand shape of the wave front in proximity of the lead-lithium/vacuum interface are determined by initial ve-locity of the pistons, their arrangement at the surfaceof the vessel and time of firing. Geometry and size ofthe piston also influence the shape and strength of thepressure wave resulting from the impact.In a first step towards our final goal, we are in the pro-cess of constructing a prototype sphere consisting of asmall steel spherical vessel with two rings of pistons.The sphere is1m in diameter and the rings consist of7 pistons each and are symmetrically placed on eitherside of the equator, Fig. 1. The working fluid is lead(Pb). The piston assemblies consist of an outer pistonwhich is accelerated by compressed air, and an innerpiston which directly contacts the lead. The pistonsare placed so that in the case of simultaneous firing theconvergent focal point is the center of the sphere.We report here on simulations done during the designstages of the prototype. The rest of the paper is or-ganized as follows: Section2 describes results of thecavity/air-core formation simulations which were car-ried out for both water and lead. The results of the wa-ter simulations were then compared to in-house exper-imental data. Section3 deals with (i) pressure wavespropagation in the vessel, pistons and liquid lead, and(ii) air/vacuum cavity collapse.

2 AIR -CORE FORMATION

The results presented in this section were obtainedusing the open source CFD code OpenFOAM [1]. Theincompressible multiphase solver ‘interFoam’ was

used in these simulations. This solver implements theVolume Fraction (VOF) method for interface trackingand is suitable for the simulation of two immiscibleisothermal fluids. Full3D transient simulations werecarried on a computational mesh having approxi-mately2 million grid points.Our CFD computational model consists of a spherewith two drainage pipes open to the atmosphere atthe top and bottom of the sphere,i.e. north and southpoles. Liquid is pumped nearly tangentially into thesphere through a number of pumping nozzles locatedeither near the bottom of the sphere, Fig. 2a, or atthe equator, Fig. 2b. The injection flow rate wasapproximatelyq̇ = 0.007m3/s, which corresponds tothe nominal flow rate used in our experiments. Thesetup used in our water sphere experiments is shownin Fig. 3. Simulations were carried for two sets ofparameters: (i) diameter of the sphereDsphere= 1mand diameter of the drainageddrain = 0.1m, and(ii) Dsphere= 0.8m and ddrain = 0.065m. The for-mer geometry corresponds to the prototype underconstruction, and the latter to the in-house watersphere experiment. Simulations were carried out fordifferent sizes and number of the pumping nozzles.Between two and eight nozzles were used, and thenozzle size varied between2.54cm≤ dpump≤ 5.08cm.Simulations were performed for both water (ρwater =1000kg/m3, νwater = 1 × 10−6m2/s, σwater =0.07N/m) and lead (ρlead = 10000kg/m3, νlead =1.38× 10−7m2/s, σlead = 0.44N/m). It is worthnoting that a real reactor is designed to operate invacuum, such that the air-core will be replaced by avacuum-core. We expect however, that with respectto hydrodynamics, this will have little effect on theformation mechanism. Initially the sphere was filledwith air such that an entire process of filling followedby the air-core formation was simulated.A typical process of air-core vortex formation isshown in Fig. 4. Results are shown for the case ofequatorial injection with the seven pumping nozzleshaving a diameter ofdpump= 2.54cm. The total flowrate isq̇ = 0.007m3/s. For a given set of parametersa sphere is expected to be filled withinTf illing ≈ 75swhen the drainage hole is closed. During the fillingprocess, Fig. 4(a)-Fig. 4(c), the air originally residingin the sphere gets pushed out mostly through the upperdrainage hole. The air inside the sphere is rotated bythe swirling water and its rotation intensifies as thefilling process goes on and the volume of the air insidethe sphere goes down. Fig. 4(d) shows that once thesystem reaches its steady state a cylindrical air-corespanning the entire height of the sphere and stronglycommunicating with an outside atmosphere is formed.At this point it is important to reiterate, the main goal

of this part of the project is to demonstrate that weare able to provide a pumping system along with arange of parameters under which a stable air-coreis formed when the system reaches its steady state.With this in mind it is of interest to compare the flowcharacteristics at the latest stages of filling Fig. 4(c),with those when flow settles into an equilibrium stateFig. 4(d), since the shape of the volume occupiedby the gas is not too different. One can see thatnear the liquid-gas interface both the rotational andvertical velocities are significantly higher for the flowin equilibrium. There is also evidence of air flowinside the core due to its communication with thesurrounding atmosphere when the steady state hasbeen reached.It is important to note that during the filling of thesphere the system is open,i.e. there is no strongcoupling between the injected and drained fluid. An-gular momentum accumulates inside the sphere and isequal to the total angular momentum injected into thesphere minus angular momentum taken away by thedrained fluid, which is driven solely by gravity. Oncethe sphere is completely filled the system becomesa closed one,i.e. there is a strong coupling betweenthe input (injection) and output (drainage) of thesystem. Once in equilibrium, all angular momentumfurther added to the system by the injection needsto be taken out of the system by the draining fluid,so that angular momentum inside the sphere remainsconstant. This explains a sudden increase in therotational velocity at radial distances comparable withthe radius of the drainage once the system reachesequilibrium, as now the fluid has to leave the spherewith significantly higher angular momentum. Due tothe sudden increase of rotation, the pressure in thecenter of the sphere decreases. This is likely to be areason for the air-core formation,i.e. the air from thesurrounding atmosphere is sucked into the sphere dueto the developed pressure gradient which in turn oftenresults in the formation of a reasonably stable air-core.It is also worth noting that once pumping is turnedoff the system immediately switches back to the openstate which is accompanied by a sudden decrease inthe rotating velocity component. This sudden changein rotation is also clearly seen in our experiments bythe abrupt change in the cone angle of the fluid leavingthe drainage hole.The cylindrical air-core formation obtained in simu-lations is qualitatively similar to the one observed inour water sphere experiments: (i) when the injectionparameters are such that we were able to fill in thesphere completely, a nearly cylindrical air-core hasbeen always formed, (ii) formation process of thiscylindrical air-core is a sudden and violent event:

sometimes the existing gas region almost completelydisappears before a cylindrical air-core is formed bysucking the air from the surrounding atmosphere, (iii)rotational velocity of the water-air interface is muchhigher for the cylindrical air-core vortex formed oncethe sphere is completely filled, when compared to theslightly tapered air-core observed during the lateststages of the filling process.Once a cylindrical air-core was formed, the numericaldata was accumulated over the next10s to obtaintime-averaged structure of the flow field. Contoursof the mean vertical velocity and its profile close tothe bottom drainage hole (Z = −0.4m) are shown inFigs. 5(a) and (b), respectively. Near both drainageholes a vertical velocity field illustrates the phenomenaof bidirectional vortex,i.e. existence of the thin layerinside which a translatory component of the spiralmotion changes its direction,e.g. [3]. The strongestvelocity gradients across this layer occur in closeproximity of the drainage holes.Radial profiles of rotational velocity and pressurein the equatorial plane are shown in Figs. 6(a) and(b), respectively. Results with the bottom injectionare shown for both water and lead. As expected, theoverall vortex structure is a combination of viscouscore (forced vortex which is characterized by a nearlyconstant shear) in the center of the sphere and a freepotential vortex away from the center. The central partof the viscous core is occupied by an air-core, and itsdiameter is obtained from the field of volume fraction.The location of the gas-liquid interface can be alsoidentified by a small wiggle in the velocity profile.Shear stresses should be equal on the both sides of theliquid-gas interface therefore the velocity gradient isdiscontinuous across the interface between fluids withdifferent viscosity.The overall structure of the flow field in our sim-ulations is very similar to the one often occurringinside hydrocyclones, e.g. [4]. The mechanism of theair-core formation observed in hydrocyclones is ofpractical importance, as it significantly affects the effi-ciency of the hydrocyclone. This recently prompted anumber of studies on the subject with an ultimate goalof suppressing the air-core formation [6, 7, 8, 9, 10].The main goal of water sphere experiments andnumerical simulations is to get a feeling for theparameters required to recreate a similar gas cavityin the reactor prototype (argon-core within liquidlead). Comparing the results for water and leadone can see that the difference in Reynolds numberand surface tension have very little effect on thevelocity profile, but a radial pressure at the wall of thesphere is an order of magnitude higher for the lead.The radial pressure distribution can be estimated as

P(r) = ρ · R r0 v2

θ/r dr, therefore for the similar velocitydistribution the pressure scales with density. Thedensity of lead is10 times more than the density ofwater, therefore the increase of an order of magnitudein the pressure is exactly what we expect to get, pro-viding nearly the same velocity profiles. Comparisonbetween velocities measured in experiments withthose obtained in simulations showed a consistentsignificant overestimation of maximum velocity in thesimulations. This is not uncommon, analytical modelsand numerical simulations are tend to overestimatemaximum velocity, e.g. [5, 11].Fig. 7 shows rotational velocity and pressure forsimulations with equatorial injection. For the sameflow rate and injection geometry, a decrease in the sizeof pumping nozzles leads to increase of the injectionvelocity. This means that a decrease in the nozzlesize leads to an increase of the angular momentuminjected into the system, which in turns results inan increase of the maximum rotational velocity.Fig. 7 also shows that for higher equatorial injectionvelocities, velocity does not decay as1/r far from thecenter, but stays flat; this has also been observed inexperiments. Moreover, in the experiments with highequatorial injection velocities, the system was not ableto stay in an equilibrium for long: an initially stableair-core began to precess. The cylindrical air-core isgetting distorted and replaced by a rather violentlyspinning tapered vortex. Simulations for similar casesalso show development of instabilities leading to theprecessing of the air-core. One explanation for thisphenomenon is that with high injection velocities toomuch angular momentum is injected into the system.This leads to the high rotational velocity which in turnleads to increase in diameter of the vortex core. Withthe increase in the size of the air-core there is lessand less space available for liquid to drain out of thesphere. This carries on until air-core is finally pushedaway from the drainage hole to allow liquid easierescape from the sphere.

3 PRESSUREWAVES

Both CFD and FEA simulations were carried out tostudy pressure waves propagation inside the lead. Thepressure waves travel at the speed of sound in lead≈ 1800m/s; since this is so much greater than the rota-tional velocity≈ 10m/s the rotation of the fluid is nottaken into account for this part of the study. In thesesimulations, the cylindrical core of air in the CFD sim-ulations and vacuum in the FEA is artificially imposedat the centre. The pressure gradients developing in thelead as a result of fluid rotation are also not taken intoaccount. We expect this to have little effect on the re-sults as the increase in the pressure due to rotation is of

order of10atm, whereas the amplitude of the pressurewaves initiated by the impact of the pistons is of order10000atm.

The FEA analysis was done using commercial pack-age LS-Dyna [2]. The computational model used inFEA simulations is shown in Fig. 8. The model con-sists of steel pistons and a steel spherical vessel occu-pied by liquid lead with a cylindrical vacuum in theinterior. For steel parts a standard Lagrangian meshwas used, whereas the lead and the vacuum were mod-eled with Eulerian elements. The steel and lead werecoupled through fluid-structure interaction parameters.The outer piston was given an initial velocity of50m/s.It strikes the inner piston, creating a pressure wavewhich then travels through the inner piston to strikethe lead. Fig. 9 shows a typical cross-section pressuredistribution obtained in the simulations of an isolatedpiston. One can see that the pressure is greatest at thecenter and decreases towards the edges. This is due, inpart, to the geometry of the piston. The initial pressureacross the face is uniform, but the cylindrical sides ofthe piston expand and then contract back to regain theirequilibrium, i.e. zero stress, state. This contractionsends a wave back to the center of the piston, therebyconcentrating the pressure at the center of the piston.The pressure wave created by the impact of the pistonon lead is shown in Fig. 10. As the waves travel to-ward the center of the sphere, waves from the differentpistons merge creating a region of high pressure at thecenter of the sphere. A snapshot of the pressure profilejust before the vacuum cavity is impacted is shown inFig. 11. The interaction between the pressure in thelead and in the steel is also evident from the figure: thepressure imparted by the impact of the piston expandsin a hemispherical manner in the lead.In CFD simulations fluid-structure interaction betweensteel and lead was omitted completely, such that onlypressure waves through the lead were considered. Pis-ton impact was imposed through time-dependent pres-sure boundary condition applied at the surface of thesphere covered by the pistons. Impact pressure profilewas roughly taken from FEA simulations; pulse dura-tion is Tpulse= 1×10−4s and maximum amplitude isPmax= 1.4GPa. Currently the uniform pressure acrossthe entire piston face was prescribed. Pressure wavespropagation in the lead obtained in CFD simulations isshown in Fig. 12. The main differences between FEAand CFD simulations setups are: (i) In CFD, the steelsphere is treated as a rigid boundary, i.e. there is noenergy transfer between lead and steel. The pressurewaves are dissipated in CFD simulations only throughthe viscous mechanism. In FEA simulations both elas-tic and plastic response are modeled. This results in

higher amplitude of the pressure waves inside the leadand also a more regular wave pattern in the CFD sim-ulation; (ii) Uniform pressure distribution across faceof the piston in the CFD simulations. This leads to thehigher pressures predicted by CFD in particular nearthe poles of the sphere; (iii) Central cylindrical cav-ity is filled with air in CFD simulations as opposed tovacuum in FEA. This affects the later stages of the col-lapse, as air is going to push back on lead.The above differences between FEA and CFD simu-lations had a significant effect on the collapse of thecavity: in FEA simulations the cavity has always col-lapsed first in the middle of the sphere, whereas CFDsimulations consistently showed so called pinch-likecollapse, where the cavity first collapses close to thepoles and later in the middle. A physical mechanism ofthe cavity collapse is a very complex phenomenon thatdeserves a separate investigation, the results of whichwill be reported elsewhere. In a nutshell, the discrep-ancy between FEA and CFD results can be explainedby stronger pressure waves occurring near the poles ofthe sphere in CFD simulations as a result of differentthe initial setups.

ACKNOWLEDGMENTS

We would like to thank Dr Aaron Froese and Dr MerittReynolds for their highly valuable input during ournever ending discussions concerning vortex formationand pressure wave propagation mechanisms.

REFERENCES

[1] OpenFOAM. www.openfoam.org.

[2] LS-Dyna www.ls-dyna.com.

[3] J. Majdalani and S.W. Rienstra On the bidi-rectional vortex and other similarity solutionsin spherical coordinates. ZEITSCHRIFT FURANGEWANDTE MATHEMATIK UND PHYSIK(ZAMP), 58(2):289–308, 2007.

[4] R. Gupta and M.D. Kaulaskar and V. Kumar andR. Sripriya and B.C. Meikap and S. Chakraborty.Studies on the understanding mechanism of aircore vortex formation in a hydrocyclone.Chem-ical Engineering Journal, 144:153-166, 2008.

[5] B.A. Maicke and J. Majdalani A constant shearstress core flow model of the bidirectional vortexProc. R. Soc. A, 465:915-935, 2009.

[6] T. Dyakowski and R.A. Williams. Prediction ofair-core size and shape in a hydrocyclone.Int. J.Miner. Process., 43:1-14, 1995.

[7] T. Neesse and J. Dueck. Air core formation in thehydrocyclone.Minerals Engineering, 20:349-354,2007.

Figure 1: A schematic of the prototype which is cur-rently under construction in General Fusion.

[8] B. Wang and K. W. Chu and A. B. Yu. NumericalStudy of Particle - Fluid Flow in a Hydrocyclone.Ind. Eng. Chem. Res., 46:4695-4705, 2007.

[9] M. Narasimha and M. Brennan and P. N. Holtham.Large eddy simulation of hydrocyclone-predictionof air-core diameter and shape.Int. J. Miner. Pro-cess., 80:1-14, 2006.

[10] S. Kim and T. Khil and D. Kim and Y. Yoon.Effect of geometric parameters on the liquid filmthickness and air core formation in a swirl injector.Meas. Sci. Technol., 20:015403, 2008.

[11] A. Skerlavaj and A. Lipej and J. Ravnik and L.Skerget. Turbulence model comparison for a sur-face vortex simulation.25th IAHR Symposium onHydraulic Machinery and Systems,12,2010.

Figure 2: Computational model used in the air-corevortex generation simulations. (a) bottom injection;(b) equatorial injection.

Figure 3: Setup for the in-house water sphere exper-iment. With configuration shown water is injectedthrough the four pumping nozzles located near theequator. With current set of parameters a sphere ap-pears to be completely filled with water and a sta-ble cylindrical air-core spanning entire height of thesphere is clearly seen.

Figure 4: Air-core vortex formation process; parts (a)-(d) correspond to different times during the formationprocess. First column: streamlines; second column: volume fraction (red - water, blue - air); third column: swirlvelocity (corresponds to thex- velocity component for the cross-section presented); forth column: vertical velocity(z-velocity component). (Dsphere= 1m, ddrain = 0.1m, dpump= 2.54cm, q̇total = 0.007m3/s)

r

Z

-0.5 0 0.5

-0.5

0

0.5

-1.0 -0.8 -0.5 -0.3 0.0 0.3 0.5 0.8 1.0(a)

Y

Vz

-0.2 -0.1 0 0.1 0.2-2

0

2

r

interface(b)

Figure 5: (a) Mean vertical(Vz) velocity contours atX = 0 cross-section; (b) Vertical velocity profile atZ =−0.4m.

r [m]U

0 0.1 0.2 0.3 0.40

2

4

6

8

θ[m

/s]

air-core

drainage

~1/r

v=4.2m/sv=4.2m/sv=3.0m/s

water;lead;water;

(a)

r [m]

P

0.1 0.2 0.30

200

400

600

800

[kP

a](b)

Figure 6: Lead .vs water; (a) Average swirling veloc-ity profile atZ = 0; (b) Radial pressure distribution atZ = 0. Results are shown for the case of bottom in-jection with four pumping nozzles;̇q = 0.01m3/s forvin jection = 4.2m/s; q̇ = 0.00785m3/s for vin jection =3m/s, Dsphere= 0.8m.

r [m]

Rot

atio

nalV

eloc

ity[m

/s]

0 0.2 0.40

2

4

6

8

d=2.54 cm, water, q=0.07m3/s

d=3.81 cm, water, q=0.07m3/s

d=3.81 cm, water, q=0.07m3/s, shift

d=5.08 cm, water, q=0.07m3/s

d=2.54 cm, lead, q=0.05m3/s

(a)

r [m]

Pre

ssur

e[k

Pa]

0 0.2 0.40

20

40

60

80

d=2.54 cm, water, q=0.07m3/s

d=3.81 cm, water, q=0.07m3/s

d=3.81 cm, water, q=0.07m3/s, shift

d=5.08 cm, water, q=0.07m3/s

(b)

Figure 7: Typical average profiles atZ = 0 of (a)swirling velocity and (b) pressure. Results are shownfor the sphere withDsphere= 1m and seven pumpingnozzle at the equator. Results are shown for differentsizes of the pumping nozzles and different flow rates(as indicated in the legend). ’shift’ in the legend in-dicates that the pumping direction was slightly shiftedfrom being tangential. In this case the pumping nozzlewas shifted towards the center by6.35cm.

Figure 8: A view of the sphere used in the LS-Dynacalculations. The interior of the sphere, in red, is lead.It is surrounded by a30 cm casing of steel. Some ofthe 14 pistons are shown. The piston surface in contactwith the lead is curved to maintain a spherical shape.The vacuum mesh is not shown, thus the hole throughthe center of the sphere is the vortex created by spin-ning fluid.

Figure 9: A cross-sectional view of the pressure profilethrough both pistons. The delimiting edge of the pis-tons is approximately halfway through the view, wherethe pressure is discontinuous. (In this simulation of theisolated piston the computational mesh was finer thanthat used in full3D piston-sphere simulations.)

Figure 10: The pressure created from impacting steelpistons traveling into the lead. The pressure from dis-tant pistons is also seen.

Figure 11: An equatorial view of the pressure in thelead just before it impacts the vortex. The pressurewaves created from the individual pistons have mergedto create a uniform pressure wave at the center of thesphere. One can also see in this picture how the leadand steel interact. Near the south pole, pressure is leak-ing from the lead into the steel.

Figure 12: Propagation of the pressure waves insidethe lead obtained in CFD simulations; just after thepiston impact (top) and before hitting liquid-gas inter-face (bottom).