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Single-Phase Trapped Air Simulation in Water Flow
Seungtaik Oh
ETRI218 Gajeong-ro
Yusong305-700, Daejon, Korea
Il Kwon Jeong
ETRI218 Gajeong-ro
Yusong305-700, Daejon, Korea
ABSTRACTWe introduce a novel practical single-phase particle simulation for trapped air bubbles in a turbulent water flow.Our model for a trapped air bubble is a low-density rigid bodywith a spherical shape, and our bubble interacts withwater and other rigid bodies in a fully two-way manner. Our bubble is created at a trapped air pocket computedfrom the water volume. Stable and realistic bubble interactions are achieved using an impulse-based boundaryforce with non-positive coefficients of restitution. Subgrid-scale bubbles are also created to add more details usingprecomputed bubble data and an oscillating bubble mesh is used in rendering stage instead of a spherical shapefor a soft look of the bubble surface. Our method can be easilyimplemented by extending an existing rigid bodyinteraction of fluid solver, and it is fast compared to two-phase simulation because we do not simulate the air part.
KeywordsBubble, Fluid, SPH.
1 INTRODUCTIONThe existence of trapped air bubbles is one of the mostattractive features in water flow. Trapped air bub-bles mainly come from trapped air pockets capturedby turbulent water flow. Two-phase simulation is def-initely the most accurate way to capture trapped airpockets in water flow and to simulate the complex be-haviour of air bubbles, such as breaking and merging[Col03, Hon03, Hon05, Hon08, Son05, Zhe06]. How-ever, a two-phase simulation of water and air suffersfrom heavy computational cost and slow simulationtime since the air part should be simulated in the sameway as the water part.
In the present paper we propose a novel and practi-cal single-phase trapped air bubble simulation, within aparticle-based framework, Smoothed Particles Hydro-dynamics (SPH). Our main idea is to model a trappedair bubble as a low-density rigid body with a spheri-cal shape. In our approach, the air bubbles are naturallycoupled to the water and the other rigid bodies in a fullytwo-way manner since a trapped air bubble is simulatedas a rigid body. We follow the impulse-based approach[Oh09] for stable and realistic rigid body interaction in
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SPH, and in particular we apply nonpositive impulsesfor the interaction between bubbles. Nonpositive im-pulses produce a weak repulsion, and so it allows ap-proaching bubbles to overlap. A natural buoyant forcefor our bubble model is also induced by the impulse-based boundary force. Thanks to the single-phase fea-ture of our method, one can perform an air bubble simu-lation as fast as a normal single-phase water simulationwith a fairly small amount of additional memory.
Our approach provides some degree of freedom for userto enhance existing results. In the rendering stage, ourspherical bubble is replaced by an elliptically oscillat-ing shape whose axis are dynamically changed accord-ing to the bubble velocity. In addition, sub-grid scalebubbles can be recursively added in a post-processingstep. The creation and amount of sub-grid scale bub-bles are controlled by the gap between an air pocketand bubbles inside.
This paper is organized as follows. In the next sec-tion, the prior literature is surveyed. Section 3 explainsour new bubble model and related simulation issues.Section 4 explains how to create an oscillating bubblemesh and subgrid-scale bubbles. Section 5 presents anddiscusses the simulation results and conclusion is pre-sented in the last section.
2 PREVIOUS WORK
Two phase simulations have been studied for realisticair bubbles in grid-based Eulerian framework [Kan00,Hon03, Hon05, Son05, Zhe06]. Some hybrid methods
have also been presented to capture small scale air bub-bles [Hon08, Mih09].
A real-time method is suggested for bubbles and foamsin a shallow water simulation in [Thu07]. Millionsof dispersed bubbles were simulated using a stochasticsolver at a subgrid level in [Kim10].
In particle-based Lagrangian approaches, two-phasesimulations with a high density ratio were studied bymaking some modifications to the SPH equations in[Col03, Sol08]. A two-phase SPH fluid in a weaksense was simulated using a dynamic particle layerat water-bubble interface in [Mue05]. Small scalebubbles including foam were simulated within an SPHframework by [Cle07].
Trapped air bubbles were simulated as spheres in[Gre04]. The bubbles interact with water in an one-waymanner and several sophisticated bubble interactionsare applied. The bubble simulation is performed ina sub-simulation of a water simulation with a muchsmaller time step. Water might exist inside the bubblebecause the bubble cannot affect the water flow.Nonetheless, the idea that a trapped air bubble isrepresented by a sphere was practically simple and itbecame a motivation for our work. The underlyingidea of handling trapped air bubbles as spheres are thesame, but now a two-way coupling between the waterand the trapped air bubble increases the realism of ourmethod.
3 TRAPPED AIR BUBBLESLet us start with a brief introduction to SPH fluid simu-lation.
3.1 SPH Fluid SimulationThe fluid is updated by the two discrete SPH equationsderived from the momentum and continuity equations[Mon94]:
dua
dt= −∑
b
mb
(
pb
ρ2b
+pa
ρ2a+Πab
)
∇aWab + fa(1)
dρa
dt= ∑
b
mb (ua − ub)∇aWab, (2)
where the summations run over all neighboring parti-cles within the support of the kernelW , u is the veloc-ity, p is the pressure,ρ is the density,Π is the artificialviscosity, andf is the body force such as gravity. Thepressure is given by an equation of state
p = p0((ρ/ρ0)γ−1), (3)
whereγ = 7 for water. With this formulation, we obtaina weakly compressible fluid and the density variation isdetermined by the ratio of the maximum velocity to thespeed of sound,cs, more precisely:dρ/ρ ≈ (umax/cs)
2.For an incompressible fluid such as water, we setcs =10umax and obtain a 1% density variation. All simula-tions in this paper useumax = 10 m/s.
3.2 Bubble ModelOur method simulates a trapped air bubble by a low-density rigid body. Every trapped air bubble is repre-sented by a spherical rigid body, and its shape is un-changed during the simulation. For a shape change ofbubble, the spherical bubbles are replaced with dynami-cally changing shapes at rendering stage, which is com-putationally efficient compared to applying soft bodysimulation for each bubble.
The stability of a weakly compressible SPH is guaran-teed only under the condition that the density ratio isless than ten to one in the fluid particles evolving in thesimulation [Col03][Sol08], and the same is true for ourrigid body interaction, such as water with bubbles. Forthis reason, we would make use of a relaxed densityfor our bubble model and set the density of our bubbleto one-tenth of that of water, although the real densityratio is one-one thousandth. A density difference be-tween water and a boundary object generates a naturalbuoyancy on the boundary object in our rigid body in-teraction scheme.
3.3 Bubble DynamicsOnce a trapped air bubble is modeled as in the previousmanner, the interactions of the bubbles with the waterand other rigid bodies are almost straightforward sincethe bubble is just a rigid body.
3.3.1 Impulse-based Boundary Force
Monaghan suggested aboundary force scheme, whichis a common method of solving rigid body interac-tions in SPH [Mon94][Mon03]. An improved boundaryforce based on impulse is suggested for stable and re-alistic complex rigid body interactions such as a hugestacking problem [Oh09]. Given an impulseJ anda time step∆t, the complete impulse-based boundaryforceFI for a fluid particle and a rigid body is
FI = J/∆t +Ff , (4)
where the fluid forceFf consists of the pressure of thefluid particle and the friction between the fluid particleand the rigid body.
A benefit in using the impulse-based boundary force isthat the density difference between the fluid and thebubble generates a natural buoyancy for our bubblemodel. The impulse-based boundary force has a waterpressure termFf in Equation (4), which make a forcedifference between on the top and the bottom of thebubble. Fig. 1 shows three rigid bodies with differ-ent densities experiencing the correct buoyancy fromthe impulse-based boundary force. At the beginning,the objects have zero relative velocity with respect tothe water, that is,J = 0, but it experiences a non-zerobuoyancy boundary force from the fluid pressure.
(a) (b)Figure 1: Buoyancy test of three objects. From the leftthe density ratios of objects to water are 10 (cube), 1(sphere), and 0.1 (cylinder). The buoyant force is nat-urally generated by the impulse-based boundary forcewithout artificial buoyancy.
An auxiliary buoyant force is added to increase the ris-ing velocity, as in [Mue05, Mih09]. The Stokes veloc-ity we are going to use is the terminal velocity of an airbubble maintaining its spherical shape while rising inwater, which is given by
vs =29
gR2
ν, (5)
whereg is the gravity,R is the radius of the bubble,andν is the kinematic viscosity of water (see [Bat67],p. 234). The auxiliary buoyant force controls the risingvelocity of bubbles in the vertical direction.
A drag force can also be simulated by the fluid forceterm, more precisely, the friction. The frictional forceis given by a damping model and its magnitude dependson the fluid viscosity and the object’s friction coeffi-cient:
Fdrag =−(µ + k)∆v, (6)
whereµ is the fluid viscosity,k is the object’s frictioncoefficient, and∆v is the relative viscosity between thefluid and the object.
3.3.2 Nonpositive Impulse for Bubble Interac-tion
The bubble interaction can be also attained by theimpulse-based boundary force in a unified way sinceour bubble model is rigid. For the interaction betweenbubbles, we use a modified impulse with nonpositivecoefficient of restitution(COR) denoted byε. Gen-erally, only ε ≥ 0 is considered for non-penetrationin interaction. However, in principle, a nonpositiveCOR is also admissible and it is observed thatε = −1generates no interaction force and, for−1 < ε < 0, aweak repulsive force occurs. Based on this, we controlCOR by settingε ≤ 0 for proper bubble interactionssuch as merging and splitting. Attraction between bub-bles can be realized by settingε1 < 0 for approachingbubbles. For overlapping bubbles, the overlapping canbe accelerated or decelerated by setting anotherε2 withε1 < ε2 ≤ 0.
procedures1 compute neighboring info
- construct search grid for current particle positions- find trapped air pocket and create bubbles
2 calculate particle interactions- compute bubble interactions- delete bubbles
3 update particles position and velocity- update bubble motion
4 go back to 1Table 1: Procedures of trapped air simulation
3.4 Bubble Creation and DeletionFor the creation of a bubble, we follow an approachbased on the flood fill algorithm, similar to that in[Gre04]. The existing neighbor-searching grid is usedfor the grid structure for the flood fill algorithm to min-imize the overhead. We choose an atmosphere cell withno particles first and then all atmosphere cells can befound by the flood fill algorithm. The empty cells notmarked as atmosphere become candidates for air pock-ets. For a randomly chosen air pocket component, abubble is created by computing the center and the opti-mal size of bubble fit to the air pocket component.
A bubble is deleted when one of the following condi-tion is satisfied:(1)Bubble age is too high. (2) A bubblehas been floating on the free surface too long. (3) Twobubbles overlap too much (4) A floating bubble experi-ences a strong impact by water.
The overall procedure for the trapped air simulation inthe SPH formulation is shown in Table 1.
4 BUBBLE MESHING AND SUB-GRIDSCALE BUBBLES
4.1 Oscillating Bubble MeshIf a bubble is rendered as a sphere, the bubble surfacelooks hard and unrealistic. In order to overcome thisdefect, the spherical bubble is replaced by an oscillat-ing bubble mesh during the rendering stage. The oscil-lating bubble mesh has an elliptic shape whose axis isdynamically rotating according to the bubble velocitydirection.
An arbitrarily oriented ellipsoid with centerc is givenby (x− c)T A(x− c) = 1, whereA is a positive definitesymmetric matrix. In fact,A = RT ΛR for a rotation ma-trix R and a diagonal matrixΛ. For our bubble mesh,R is chosen to be a rotation matrix mapping the direc-tion of the bubble’s velocityv to thez-axis, that is, therotation axis isez × v and the rotation angle is∡(e3,v).LettingΛ = diag(α−2,β−2,γ−2), we set
α(t) = r+ δ (t) (7)
β (t) = α(t) (8)
γ(t) =3V
4π α(t)β (t), (9)
(a) (b)
(c) (d)Figure 2: Free surface mesh and its application to bub-ble visualization. The upper row shows a comparisonbetween the full surface mesh (a) and the free surfacemesh (b) in a general water simulation. In the lowerrow, the full surface mesh (c) and the free surface mesh(d) results in bubble visualization are given. In (c), onecan see some artifacts caused by the full surface mesh(grey) at the water–bubble interface.
wherer is the radius of the bubble,δ (t) is an oscillat-ing function with an attenuation in time, andV is thebubble’s volume. The oscillating functionδ (t) is givenby
δ (t) =
{
kr(t/T −1)2sin(2nπt/T) : 0< t ≤ T
0 : t > T(10)
wherek ≪ 1 is the magnitude,T is the maximum oscil-lation time, andn is the frequency. From the the defi-nition of γ(t), we see that the volume of our oscillatingbubble is preserved. This oscillating bubble mesh cre-ates the look of an elastic bubble without the need forsoft body modelling of the bubble in the simulation.
A free surface extraction for SPH water particles isneeded for effective rendering of bubble mesh. In oursituation, a conventional water mesh extraction fromthe water particles causes a rendering problem in theproximity of bubbles since two interfaces, water and thebubble mesh, co-exist around a bubble (Fig. 2(c)). Toremove this problem, we modify our Marching Cubesalgorithm by skipping grid cells close to rigid bound-aries or bubbles so that the water mesh has a free sur-face part only. The free surface mesh has no polygonsnear a boundary object (Fig. 2(b)). Since our bubbleis simulated as a boundary object, the free surface onlyextraction is also available for our bubble. A free sur-face mesh is useful in a general situation since the sizeof the mesh data is significantly reduced and a clearrendering around the interface of a boundary object ispossible(Fig. 2(d)). Once a free surface only mesh is
Figure 3: Comparison for subgrid-scale bubbles: with-out subgrid-scale bubbles (a) and with subgrid-scalebubbles (orange)(b).
obtained, it is straightforward to substitute a sphericalbubble shape with the oscillating bubble shape.
4.2 Sub-grid Scale Bubble GenerationOur trapped air bubble is slightly smaller than the corre-sponding trapped air pocket consisting of cubic cells inthe neighbor searching grid. Sub-scale bubbles whosesize is less than the search grid size are created at thegap between the bubble and the pocket. The exact com-putation of the gap size, however, is expensive and im-practical. Roughly, the gap size is inversely related tothe area fraction in water, which is computed as the ra-tio of the number of the surface particles of a bubble in-sider the water to the number of all surface particles ofthe bubble. With the area fraction in water, we can de-termine the numbers and positions of the subgrid-scalebubbles around a newly created trapped air bubble. Thecreation sites exist outside of the trapped air bubble andthe sites are randomly chosen from the mesh verticesof three concentric spheres whose center is that of theexisting trapped air bubble.
The subgrid-scale bubbles represented by particles arepassively advected by the water flow and the secondorder Runge–Kutta method is used for the time inte-gration of its motion. The vertical component of thebubble velocity is also controlled by the Stokes veloc-ity andumax in the same way as before. Sub-grid scalebubble generation can be done recursively as decreas-ing the size so that one can obtain a sufficient num-ber of subgrid-scale bubbles. A comparison result forsubgrid-scale bubbles is shown in Fig. 3.
This process is a post-process, performed after the sim-ulation, and so artists have some control to make theirown subgrid-scale bubbles as needed.
5 RESULTS AND DISCUSSIONSome examples are presented to show the effectivenessof our method. All the simulations were performed ona machine with two quad core CPUs at 2.33 GHz. Inall simulations, free surface meshes were used for ren-dering the water and a trapped air bubble was renderedusing an oscillating bubble mesh. The frames per sec-ond rate (FPS) was set to 30 and so a frame represents1/30 sec.
(a) (b)
Figure 4: Pouring water.
The first example is water pouring (Fig. 4). Water ispoured by an inflow into an initial volume of water with474k particles. The maximum number of water par-ticles reaches 923k at the end of simulation. Totally1140 trapped air bubbles are created and in order to adddetails 3050 subgrid-scale bubbles are generated by apost-process. It takes 5–12 minutes for a single frame,and the overhead for the trapped air bubbles increasesthe simulation time by 1.9% only for the main simula-tion, and only by 5.8% including the subgrid-scale bub-ble generation.
The second example is a diving bunny simulation (Fig.5). As mentioned earlier, a trapped air bubble is con-trolled so that it is not created in the interior of thebunny. In total, 61 trapped air bubbles are createdand 1716 subgrid-scale bubbles are added by the post-processing. The simulation has 474k water particlesand it takes about five minutes to simulate one frameof the water and trapped air bubbles. In this case, theoverhead for the main trapped air bubbles is negligible,and, including the subgrid-scale bubble generation, theoverhead is estimated to be 8.1%. An artificial buoy-ant force based on Stokes law was applied to increasethe rising velocity of the subgrid-scale bubbles. Note,too, that since we had no water mesh around the bunnyfrom the free surface mesh extraction, the bunny is ableto be clearly rendered, which is the main benefit of freesurface mesh extraction.
Our current method has the following limitations. Thedensity ratio of the water to our trapped air bubble isset to ten to one, which is one hundred times smallerthan the real density ratio. We expect that more realis-tic results can be achieved if we can increase the densityratio while preserving the time step and stability. Someadvanced techniques could improve the reality of theoscillating bubble mesh by considering the nearby dy-namics of water, such as the pressure and vorticity. Al-though the oscillating bubble mesh is used in renderinginstead of a spherical bubble, we still have a problem
in simulating a natural bubble’s merging and splitting.It should also be mentioned that our SPH solver per-formance is not good enough because no accelerationmethod has been applied as yet, so we expect that someGPU techniques can improve the performance signifi-cantly.
6 CONCLUSIONWe have presented a new practical method to simulatewater and trapped air bubbles in a single-phase SPHframework. Our trapped air bubble is modeled as alight weight spherical rigid body and its interaction iscomputed through a stable and realistic impulse-basedboundary force. The trapped air bubbles interact withthe water in a fully two-way manner, which increasesthe realism of the water simulation. Sub-grid scale bub-bles are also created to add details using the existingtrapped air bubble data and oscillating bubble meshesare rendered instead of spherical shapes to provide asoft look to the bubble surfaces. Finally, our method isalmost as fast as single-phase water simulation becausewe don’t simulate the air, and it has been shown that wehave a trapped air simulation with an overhead less than10% of that for water-only simulations.
7 ACKNOWLEDGMENTSWe also give many thanks to our colleagues in ETRI fortheir continuous support. This work was supported bythe ‘Cross-Ministry Giga KOREA Project’ of the Min-istry of Science, ICT and Future Planning, Republic ofKorea[GK130100, Development of Interactive and Re-alistic Massive Giga-Content Technology].
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