Coupling of a fluids BEM code with a DYNAFLOW, Inc., 7210 … · 2014. 5. 13. · Coupling of a fluids BEM code with a structural FEM code for fluid-structure interaction simulation

Coupling of a fluids BEM code with a

structural FEM code for fluid-structure

interaction simulation

G.L. Chahine, K.M. Kalumuck, R. Duraiswami

DYNAFLOW, Inc., 7210 Pindell School Road,

MD f 07JP,

ABSTRACT

We have developed computer programs (2DynaFS, 3DynaFS) to study poten-tial nonlinear free surface flows in axisymmetric and three-dimensional geome-tries, and have applied them to the study of the dynamics of bubbles and theirinteraction with nearby bodies. In previous research, the neighboring structureswere modeled as rigid boundaries with no motion or only simplified motion. How-ever, the structural response can be important, and could significantly affect thebubble dynamics. Reliable prediction of loads on materials requires fully cou-pled fluid- structure modeling. In this paper we report our progress in couplingthese BEM based fluid codes to existing finite element structural codes (NIKE2D,NIKE3D), and the modifications required for the modeling of both the rigid bodymotion of the body and its deformation.

INTRODUCTION

Study of the interaction of bubbles and structures is of importance at severalscales. The growth and collapse of micron- sized bubbles near boundaries such aspropeller blades is thought to hold the key to understanding the deleterious effectsof cavitation on such structures. The interaction of much larger bubbles withunderwater and off-shore structures has important naval and marine applications.In both these problems, the final stage of bubble collapse is associated with theformation of a high-speed liquid jet directed towards the solid structure. Localforces associated with the collapse of such bubbles can be very high and cansignificantly affect the structure.

Earlier studies have concentrated on the problem of studying the bubble dy-namics in isolation, or near idealized infinite rigid walls [1-2] or compliant walls[3]. While these studies are very useful in pointing out qualitative behavior ofthe bubble-dynamics near boundaries, to simulate the real situation closely, themodeling of the structural response should also be performed closely to includethe influence of the flexibility of the structure, as well as structural motion onthe bubble dynamics. The structure response can strongly modify the bubbledynamics and affect the resulting pressures and forces on the structure.

In this study we present results from our efforts to create an integrated bubble-

Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

582 Boundary Elements

structure simulation program. The structural portion of the program has beendeveloped from the NIKE suite of finite element programs developed by LawrenceLivermore National Laboratories. Two programs have been developed - onebased on our BEM axisymmetric code 2DynaFS and on NIKE2D , and the otherbased our 3D BEM code 3DynaFS and NIKE3D. The results presented hereconcentrate on a model problem of the interaction of a relatively large bubble ata distance from a spherical shell in an infinite medium of liquid.

FLUID MODELING

One of the numerical methods that has proven to be very efficient in solving thetype of free boundary problem associated with bubble dynamics is the BoundaryElement Method. Among others, Blake et al [1,2], Guerri et al [4], and Wilkerson[5] used this method in the solution of axisymmetric problems of bubble growthand collapse near boundaries. This method was extended to three-dimensionalbubble dynamics problems by Chahine et al. [6-8]. For large (but subsonic)bubble wall velocities one can neglect viscosity and compressibility effects on thebubble dynamics. These assumptions, classical in bubble dynamics studies, resultin a potential flow (velocity potential,

Boundary Elements 583

Boundary Element Formulation

In order to enable the simulation of bubble behavior in complex geometry andflow configurations including the full non-linear boundary conditions, a three-dimensional Boundary Element Method was developed by Chahine et al [6-8].This method was chosen because of its computational efficiency. By consideringonly the boundaries of the fluid domain it reduces the dimension of the problemby one. This method is based on Green's equation which provides $ anywhere inthe domain of the fluid (field points P) if the velocity potential, $, and its normalderivatives are known on the fluid boundaries (points Af ), and if $ satisfies theLaplace equation:

where a?r = ft is the solid angle under which P sees the fluid.

a = 4, if P is a point in the fluid,a = 2, if Pis a point on a smooth surface, and

a < 4, if Pis a point at a sharp corner of the surface.

If the field point is selected to be on the surface of the bubble or its image, thena closed set of equations can be obtained and used at each time step to solve forvalues of d$/dn (or $ ) assuming that all values of $ (or d$/dn) are known atthe preceding step.

To solve Equation (4) numerically, it is necessary to discretize the geometryinto panels, perform the integration over each panel, and sum up the contribu-tions to complete the integration over the entire bubble surface. The initiallyspherical bubble is discretized into a geodesic shape using flat, triangular pan-els [6]. Equation (4) then becomes a set of N equations (N is the number ofdiscretization nodes) of index i of the type:

N

Zŵ )-̂ * (5)

where A,-j and B{j are elements of matrices which are the discrete equivalent ofthe integrals given in Equation (4).

To evaluate the integrals in (4) over any particular panel, a linear variation ofthe potential and its normal derivative over the panel is assumed. In this manner,both $ and d$/dn are continuous over the bubble surface, and are expressed asa function of the values at the three nodes which delimit a particular panel. Thetwo integrals in (4) are then evaluated analytically. In order to compute thecurvature of the bubble surface - to evaluate the dynamic boundary condition atthe bubble surface - a three-dimensional local bubble surface fit, f(x,y,z) = 0,is first computed. The unit normal and the curvature at a node can then beexpressed as:

and C = V.n. (6)



To obtain the total fluid velocity at any point on the surface of the bubble,the tangential velocity, V& , must be computed at each node in addition to thenormal velocity, Vn = d$/dn n . This is also done using a local surface fit tothe velocity potential, $/ = h(x,y,z). The following gives a good approximationfor the tangential velocity

Vt = nx(V$, xn). (7)

The basic procedure can then be summarized as follows. With the probleminitialized and the velocity potential known over the surface of the bubble, anupdated value ofd$/dn can be obtained by performing the integrations in (4) andsolving the corresponding matrix equation (5). D$/Dt, the material derivativeof the potential is then computed using Bernoulli's equation.

Using an appropriate time step the values of $ on the boundaries can be updatedusing $ at the previous time step and D$/Dt. New coordinate positions of thenodes are then obtained using the displacement:

/d$dM = l—n + Vt.

where n and e^ are the unit normal and tangential vectors. This time steppingprocedure is repeated throughout the bubble growth and collapse, resulting in ashape history of the bubble.

To numerically integrate the above equations we use a simple Euler steppingscheme. The choice of the time step is made using an adaptive scheme based onthe variations in $ or the maximum value of the velocity on the boundaries.

Specialization to axisymmetric problems

In axisymmetric problems, the physical variables are independent of the angularcoordinate. Thus the angular coordinate only enters the formulation through theargument of the Green's function, Q, in Equation (4) The integration of thesedependent quantities can be explicitly carried out. Let C represent the traceof the geometry under consideration in a meridional plane. Let r, 0, z be thecylindrical coordinates of point M, running point on the boundary, and withoutloss of generality we select the coordinates of P to be (R, 0, Z). The integralequation (4) can then be written

r d / F** \ f 36 r**(R, 0,Z) = /


The equation for the potential may then be written as:

(12)To convert the above integral formulation to a boundary element code we

discretize the axisymmetric geometry with N straight line panels. We assumethat over each bubble panel the potential $ is distributed linearly, while d$/dn isassumed to be constant, while over each solid body panel the converse is the case.The integral equation is then collocated at the centers of the panels. Integrationover each panel is performed using Gauss schemes. In the case the panel underconsideration does not contain the collocation point a 12-point Gauss- Legendrescheme is used. In the case that the collocation point is on the panel underconsideration, the elliptic integral K(m) has a logarithmic singularity at thecollocation point. In this case a 10-point Gauss scheme which accounts for thesingularity is used. The matrix equation obtained is then solved using a standardLU decomposition technique.

During the growth and collapse of the bubble, the original panels changelengths. This can cause the bubble shape to become severely distorted and affectthe quality of the approximation and numerical calculations. To prevent this thefollowing regridding scheme is used. At the end of each time step the lengthof each of the panels is determined. A cubic spline interpolation of the bubblegeometry and of the variables is performed using the cumulative length of thepanels as parameter. New panels of equal lengths are generated, with new valuesof the variables obtained via interpolations using cubic spline representations.This prevents the panels from becoming distorted, and was seen to improve theaccuracy of the numerical simulation.

Validation of BEM numerical codes

The use of the BEM to study axisymmetric bubble dynamics has been vali-dated by the various authors quoted earlier. This has included both comparisonswith a quasi-analytical solution for spherical bubbles - Rayleigh-Plesset Equa-tion - and experimental validation for the relatively simple cases of spherical andaxisymmetric bubble collapse near flat solid walls. Comparison of the results ofthe 3D code against previously published and confirmed results in the literaturefor the relatively simple cases have been very favorable. For spherical bubbles,comparison with the Rayleigh-Plesset "exact" solution revealed that numericalerrors for a "coarse" discretization of a 102-node bubble was about 2 percent ofthe achieved maximum radius, but was very small, 0.03 percent, of the bubbleperiod. The error on the maximum radius was less than 0.14 percent for a dis-cretized bubble of 162 nodes (320 panels), and dropped to 0.05 percent for 252nodes (500 panels). Comparisons were also made with studies of axisymmetricbubble collapse available in the literature [1-3], and have shown, for the coarsediscretization, differences with these studies on the bubble period of the orderof 1 percent. Finally, comparison with actual test results of the complex three-dimensional behavior of a large bubble collapse in a gravity field near a cylindershows very satisfactory results, [6-8].



FLUID-STRUCTURE CODES COUPLING

The axisymmetric and three-dimensional finite element structural dynamics codesNIKE2D and NIKE3D have been coupled to our axisymmetric and three-dimensionalBEM bubble dynamics codes 2DynaFS and SDynaFS to allow modeling of theeffects of the interaction of structural flexibility, deformation, and motion on bub-ble dynamics. In addition, a rigid body motion capability was incorporated toour fluid code. Both the axisymmetric and 3D codes employ a complete couplingof the fluid and the structure at the wetted surfaces of the structure. Calcula-tions are performed at each time step in both the fluid and in the structure. Thealgorithms are structured such that the fluid code is the "main" routine whilethe structural code functions as a set of subroutines driven by the "main" fluidroutines.

The structure and fluid boundaries can either have the same discretizationor be discretized separately. When a separate discretization is adopted a fluidboundary "net" can be overlaid on the wetted structural surface and a correspon-dence is created between the nodes in the fluid discretization and those in thestructure.

At each time step, pressures calculated by the fluid model are passed tothe structural code and used to generate loads along the wetted surface. Thestructural code then solves the structural equations at this time step with thisnewly computed load and knowing the state of the structure at the previousstep. This results in displacement and velocity of each node of the structure, andtherefore a new position and 'shape' for the structure. The normal velocity ateach structural node is set equal to the normal component of the gradient of thepotential at that point as in Equation (2) which gives a boundary condition forthe fluid model at the structure boundary.

NIKE2D and NIKE3D are powerful general purpose implicit finite element struc-tural modeling codes developed by Lawrence Livermore National Laboratories[9,10].) For the current work, analysis is performed in the time domain. Bothcodes utilize the Newmark beta method for time integration of the equationsof motion. The codes can handle both geometric (large strain) and materialnonlinearities. In the work reported here, linear elastic material models using afour-noded axisymmetric element were selected. In the three dimensional model-ing, triangular shell elements have been employed. These selections enable a oneto one correspondence between the nodes in the fluid model and those for thestructure thus requiring no interpolation. Implicit structural codes were selectedto allow stability to be maintained while allowing the time step to be deter-mined by the fluid code, thus enabling adaptive time stepping during the bubbleoscillations.

Full coupling between 2DynaFS and NIKE2D or between SDynaFS andNIKE3D is achieved through the following outlined procedure:

• Read input data and initialize fluid and solid routines.

• Begin time stepping.

• (*) Solve boundary element equations in the fluid.



• Compute pressure at each node of the fluid net overlaid on the wettedstructure by use of the unsteady Bernoulli equation.

• Transform pressures calculated on the fluid net to pressures at the wettedstructural nodes, if necessary.

• Pass pressures to structural code as loads at each node.

• Solve the structural equations with this new load, knowing the state ofthe structure at the previous time step, to obtain the new displacement,velocity and acceleration of each node of the structure.

• Obtain structural stresses and strains (if desired).

• Pass new velocities and displacements to the fluid code as new boundaryconditions for the next time step.

• Set the velocity normal to the structure at each fluid node along the wettedsurface equal to the normal component of the gradient of the potential,d$/dn, at that point.

* Update the position of the wetted surface.

• Increment time and return to (*).

RESULTS AND DISCUSSION

Axisymmetric Calculations

The results of a series of calculations that highlight the effects of structural mo-tion on bubble dynamics are presented here. In these calculations characteristiclength, pressure and time scales are given by #max> the maximum radius the bub-ble would achieve in an infinite medium; PQ, the ambient pressure at the locationof the initial center of the bubble; and T, the Rayleigh bubble time - the naturalperiod of a bubble in an infinite medium and in the absence of gravity given by

(13)

For these calculations, an arbitrary bubble of size sufficient to make gravityimportant was selected together with a simple spherical structure. We thus con-sider a bubble of initial radius RQ = 0.177 m, and initial pressure Pg = 6.83MPa growing and collapsing in a gravity field at a depth of 189 Rm*x above aspherical structure of radius 4 Rm&\- In that case Pgo/Po = 7.4 and Ro/Rm&x =0.4. The sphere is hollow with a thickness of 0.075 #max- The gas constant usedis k = 1.25. This results in a value of T = 0.0146 s. The material model is lin-ear elastic with three parameters to be specified - Young's modulus, the Poissonratio, and the material density. Here, the structure is taken to be composed of ahomogeneous linear elastic material with Young's Modulus E = 10,3007̂ , Pois-son's ratio y = 0.3, with a total mass M = 225 times the mass of water displacedby the bubble at its maximum size. The interior of the sphere is pressurized toPQ to ensure initial equilibrium.



In performing these calculations, it was found that, for this set of parameters,the coupled calculation exhibits unstable oscillations in the absence of damp-ing. This situation is common in structural dynamics [11]. By the introductionof damping, the calculations were stabilized. Rayleigh damping which can beexpressed as

(14)

was employed. &. is the fraction of critical damping of mode r of frequency /,.,and otm and a, are the mass and stiffness damping coefficients (given in units ofs~* and 5, respectively.) The former produces damping that varies inversely withfrequency while the stiffness damping varies directly with frequency. Details ofthe implementation of this damping in NIKE2D can be found in [10]. Briefly, forthe local structure motion the finite element equations can be expressed as

MU+CU +KU = P (15)

where ° indicates time derivative, and M, K, and C are the mass, stiffness anddamping matrices while U and P are the nodal displacement and load vectors.The damping employed is incorporated by setting

C =


Figure 3 presents a comparison of the locations of the bubble node nearestthe structure and farthest from the structure as functions of time. These curvesshow the influence of the structure flexibility and motion on the bubble dynamics.Varying the damping at different frequencies changes the response of the structureand, in turn, modifies the bubble dynamics. These effects are most pronouncedon the least damped case ( a, = 10""*s, am = 0 s~*) compared to the rigid fixedcase. The bubble period is lengthened for the flexible case. The bubble periodis actually shortened by rigid body motion as compared to the fixed case. Inaddition the curves indicate a dramatic difference in this motion at the end of abubble cycle of oscillation. As can be seen from Figure 4 which presents bubblecross-sections at different times, very significant modifications of the bubble shapeare seen at the end of the collapse and beginning of the rebound phase. In therigid body case, the bubble completes its collapse without the formation of asignificant jet, then during rebound a very thin jet is formed and correspondsto similar results near solid walls recently published in [12]. Simultaneously,the bubble growth is such that the bubble practically touches the structure atthe time the very thin jet impacts it. For the flexible and movable structure,the qualitative behavior is similar. However, in the flexible wall case the jetis wider, occurs earlier, and is faster, so that it penetrates the other side of thebubble before it achieves a volume large enough to touch the structure. When thestructure is only allowed to undergo motion with no deformation, a very differentbehavior is observed in which the re-entering jet development is prevented or atleast very significantly delayed.

Figures 5-7 present a comparison of the position of the structural node nearestthe bubble, farthest from the bubble and at mid distance, as a function of time forthe various cases. In the rigid body motion case, the point closest to the bubbleon the structure is seen to constantly be pushed away from the bubble. To thecontrary, the flexible body case shows initial repulsion from the bubble followedby attraction then repulsion again. This follows more closely the variations ofthe pressure field shown at the closest node in Figure 2.

An understanding of the observed behavior can be obtained by comparingthe displacements at the structure node nearest the bubble (Fig. 5) with thatlocated along the side of the sphere (90 degrees away, Fig. 6), and with thefarthest away point (Fig. 7), In the light damping case, the primary frequencyexhibited in the displacement history of the node nearest the bubble is thatof the bubble - approximately 30 Hz. A second higher frequency deformationof smaller amplitude can also be observed. This is the frequency seen in thepressure histories - approximately 110 Hz due to the structure. Inspection ofthe displacement of the structure at nodes along the side and furthest from thebubble also clearly show the 110 Hz frequency. This suggests that the structureis responding at this 110 Hz frequency. However, the dynamics of the structurenear the bubble are being dominated by the bubble and the dominant frequencyis that of the bubble, or, more precisely, the coupled bubble-structure system.Figure 8 presents the deformed structure shapes at three instants for the mostlightly damped case: during the growth, near maximum bubble size, and nearthe end of the collapse. In this figure, the deformations have been exaggerated by



a factor of 25 for clarity. The influence of the loading by the bubble is apparent.During the early growth and the collapse, the portion of the structure nearest thebubble (top) is pushed away from the bubble. Between these times, this portionof the structure is drawn toward the bubble.

Comparison of the simultaneous pressure histories generated on the structureat the node nearest the bubble (Fig. 2) with those located furthest from thebubble (180 degrees location) apart and along the side (90 degree location) -practically flat behavior- show a net force on the structure directed away fromthe bubble during the initial growth and final collapse stages, while a smallernet force on the structure is directed towards the bubble during the intermediatetime.

Our coupled code is capable of calculations at bubble distances from thewall less than 1 Rm&x- An example is given in Figures 9 and 10 for the sameconditions as above but with a distance of 0.75 Rm^x' Here, the (a, = lO""̂ ,arn=0 j-i) damping case is compared to that of a fixed rigid structure and amovable rigid structure. No bubble rebound is observed in these cases. Littledifference is observed between the fixed and the rigidly moving cases. However,the deformable case again shows an increase in bubble period and an increase inpressure generated on the structure.

Three-Dimensional Calculations with Deformation

Initial work with NIKE3D was performed with simple structures discretized usingtriangular (3-noded) shell elements which enable a one to one relationship be-tween the finite element node points and the node points in the fluid model atthe structural boundary with no interpolation required. As a preliminary exam-ple the code has been run for the conditions at a distance of 0.75 Rmax from thestructure for a 288-panel bubble and a "cylindrical" body structure representedby 66 nodes and 128 triangular shell elements.

Figure 11 presents a cross sectional view of these preliminary calculationsshowing bubble and structure behavior. Since structure motion is not easy todistinguish on this figure, Figure 12 presents the motion and deformations ofpoints on the structure with time together with the location of bubble node atthe jet.

Three-Dimensional Rigid Body Motion Model

The capabilities of 3DynaFS have also been expanded to account for the effectsof submerged solid body motion in response to the hydrodynamic forces generatedby the bubble dynamics. A fully coupled general approach that allows for sixdegrees of freedom - translation and rotation about the x, y, and z directions- has been taken. Equations of both linear and angular momentum are solved.Coordinate systems that are both fixed in space and moving with the body areemployed [13]. The current state of the submerged body is tracked by knowingthe position of its center of gravity and its angular orientation relative to a fixedreference frame. Transformation matrices between the coordinate systems fixedin space and affixed to the body are employed. At each time step, the bubbledynamics routines provide the pressure along the submerged body while the rigidbody dynamics routines provide the instantaneous position and velocity of the



body surface as a boundary condition to the bubble dynamics calculations.Calculations have been performed for the case of a bubble generated imme-

diately under the center of the submerged body. In these calculations, one planeof symmetry was employed. Figures 13 and 14 present the results for an 85-nodebubble (85 nodes and 144 panels in the half bubble) beneath a submerged cylin-der at a distance of 1.5 times the maximum bubble radius.. The submerged halfbody contained 89 nodes and 156 panels. The body mass was set equal to themass of displaced water to make the body neutrally buoyant. The resulting bodymotion is thus due to the hydrodynamic forces exerted by the bubble flow field.

Figure 14a shows the bubble contours and body cross sections at the x = 0plane. The thick lines on this cross section are due to body motion. In thiscase, the body was found to move downward (toward the bubble) throughout thecalculation. Figure 14b presents the same information for a base case in whichthe body was not allowed to move. The effect of the body motion on the bubbledynamics can be seen in Figure 13 which presents plots of the positions of thebubble north and south poles as a function of time for these two cases. As canbe seen, the collapse time of the bubble when the body is allowed to move isdecreased by approximately 8 percent of the base case and its center migratesdownward.

CONCLUSIONS

Preliminary results were presented from Boundary Element Method free sur-face hydrodynamics codes (2DynaFS, SDynaFS) coupled to Finite ElementMethod structural dynamics codes (NIKE2D, NIKE3D) to study the effects onbubble dynamics in a gravity field of a deformable and interacting nearby struc-ture. The results indicate significant modification to the bubble behavior and itsinfluence on the structure due to allowing motion and deformation of the struc-ture. In particular, bubble period modification accompanied by a modification ofthe reentrant jet formation and of the pressures generated along the solid bodyare predicted. Results also indicate the ability to simulate close standoff bubbledynamics-structure interaction for cases in which the bubble is deformed by thepresence of the structure. This work represents early results of an ongoing studyto develop and utilize these simulation techniques.

ACKNOWLEDGMENTS

We would like to thank Mr. Thomas Spelce, Laurence Livermore National Labo-ratories, Livermore, CA for providing us with the source code for the NIKE suiteof FEM structural codes

REFERENCES

1. Blake, J. R., Taib, B.B. and Doherty, G., "Transient Cavities Near Bound-aries. Part I. Rigid Boundary," Journal of Fluid Mechanics, vol. 170, pp.479-497, 1986.

2. Blake, J. R, and Gibson, D. C., "Cavitation Bubbles Near Boundaries, "Ann. Rev. Fl. Mech., Vol. 19, pp. 99-123, 1987.



3. Ligneul, P., "Influence des Conditions aux Frontieres sur le Phenomene decavitation", Doctorat d'Etat es-Sciences, Univerity Pierre and Marie Curie,Paris, 1989.

4. Guerri, L., Lucca, G., and Prosperetti, A., "A Numerical Method for theDynamics of Non-Spherical Cavitation Bubbles," Proc. 2nd Int. Coll. onDrops and Bubbles, JPL Publication 82-7, Monterey CA, November, 1981.

5. Wilkerson, S., "Boundary Integral Technique for Explosion Bubble CollapseAnalysis," ASME Energy Sources Technology Conference and Exhibition,Houston Tx., Jan.1989.

6. G.L. Chahine and T.O. Perdue, "Simulation of the Three-Dimensional Be-havior of an Unsteady Large Bubble Near a Structure," in "Drops and Bub-bles" edited by T.G. Wang, A.I.P. Conference Proceedings, 197, 169-187,1989.

7. Chahine, G.L., "A Numerical Model for Three-Dimensional Bubble Dynam-ics in Complex Configurations, "22nd. American Towing Tank Conference,St. Johns, Newfoundland, Canada, August, 1989.

8. G.L. Chahine, "Dynamics of the Interaction of Non-Spherical Cavities,"in "Mathematical Approaches in Hydrodynamics" ed. T. Miloh, SIAM,Philadelphia, 1991.

9. Maker, B.N., Ferenz, R.M. and Hallquist, J.O. "NIKE3D: a Nonlinear, Im-plici Three-Dimensional Finite Element Code for Solid and Structural Me-chanics, User's Manual", Laurence Liver more National Lab Report UCRL-MA-105268, November, 1990.

10. Englemann, B., and Hallquist, J.O. "NIKE2D: a Nonlinear, Implicit, Two-Dimensional Finite Element Code for Solid Mechanics, User's Manual"Laurence Livermore National Lab Report UCRL-MA-105413, April, 1991.

11. Hayrettin Kardestunger, "Finite Element Handbook", Me. Graw Hill Inter-national Editions, Engineering Studies, 1989.

12. Best, J.P., and Kucera, A., "A numerical Investigation of Non-sphericalRebounding Bubbles", J. Fluid Mech., vol. 245, pp 137-154, 1992.

13. Goldstein, H., "Classical Mechanics", Addison-Wesley, New York, 1980.



freq. (Hz)FIGURE I. FRACTION OF CRITICAL DAMPING. &AS A FUNCTION OF FREQUENCY FOR VARIOUSMASS AND STIFFNESS DAMPINGCOEFFICIENTS.

FIGURE 3. LOCATION HISTORIES OF TOP ANDBOTTOM POINTS ON THE BUBBLE FOR FIXED.RIGIDLY MOVING. AND DEFORMINGSTRUCTURE.

0.02 0.01 0.04t(s)

FIGURE 1 PRESSURE HISTORIES CALCULATEDON STRUCTURE NODE NEAREST BUBBLE FORFIXED. RIGIDLY MOVING. AND DEFORMINGSTRUCTURE.

**%

%

00 0.01 0.02

FIGURE 5. COMPARISON OF DISPLACEMENTHISTORIES FOR STRUCTURE NODE NEARESTTHE BUBBLE.

FIGURP. 1. COMPARISON OF CALCULATED BUBBLE COLLAPSECONTOURS (LEFT TO RIGHT: FIXED, RIGIDLY MOVING, AND DE-FORMING STRUCTURE).



M**dl* mo»m«

/ v \I \

,..,.̂^('/ ' ' / >-,'-> /\y i / I i *.\u*

v' f» •/

FIGURE 6. COMPARISON OF DISPLACEMENTHISTORIES FOR SIDE STRUCTURE NODE(LOCATED 90 DEGREES AWAY FROM THEBUBBLE).

1=0.329 T

0.01 0.02 0.03 004 (t (s)

FIGURE 7 COMPARISON OF DISPLACEMENTHISTORIES FOR STRUCTURE NODE FURTHESTFROM BUBBLE (180 DEGREES AWAY).

1=2.456 T

FIGURE 3. COMPARISON OF DEFORMED STRUCTURE SHAPES ATVARIOUS TIMES. DASHED LINE: DEFORMED SHAPE, DEFORMA-TIONS EXAGGERATED 25 TIMES: SOLID LINE: UNDEFORMED SHAPE.a* = 0, at. = 10"**.

FIGURE 9. COMPARISON OF CALCULATED DUDDLE COLLAPSECONTOURS FOR A STANDOFF OF 0.75 R̂ ..(LEFT TO RIGHT: FIXED,RIGIDLY MOVING, AND DEFORMING STRUCTURE).



FIGURE 10. LOCATION HISTORIES OF TOP ANDBOTTOM POINTS ON THE BUBBLE FOR FIXED.RIGIDLY MOVING. AND DEFORMINGSTRUCTURE FOR A STANDOFF OF 0.75 R̂ .

FIGURE 11. BUBBLE COLLAPSE CONTOURSBENEATH A LONG FINITE DEFORMABLE BODYSHOWN PARTIALLY IN CROSS - SECTIONCALCULATED WITH 3DYNAFS/NIKE3D.

FIGURE 12. CALCULATED DISPLACEMENTS OF 4STRUCTURAL NODES AND BUBBLE JETPOSITION FOR CASE OF FIGURE 11.

a. Movable body

FIGURE 13. COMPARISON OF LOCATIONS OF TOPAND BOTTOM BUBBLE NODES FOR CASES OF ABUBBLE BENEATH A RIGIDLY MOVING BODYAND AN IMMOVABLE BODY.

b. Immovable bodyi

FIGURE II. CLOSE UP OF BUBBLE GROWTH AND COLLAPSE CON-TOURS BENEATH RIGIDLY MOVING AND IMMOVABLE BODIES.


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Coupling of a fluids BEM code with a DYNAFLOW, Inc., 7210 … · 2014. 5. 13. · Coupling of a fluids BEM code with a structural FEM code for fluid-structure interaction simulation