Shock and Bubble dynamics

 

The interest in Underwater Shock and Bubble Dynamics first arises in the early 1990’s when the Ministry of Defense (MINDEF) was keen to develop the capabilities to conduct WSSA (Whole Ship Shock Analysis) for those surface crafts procured by the Republic of Singapore Navy (RSN) from time to time. At that time, external consultants were engaged to study using the commercial software NASTRAN with USA (Underwater Shock Analysis) and DAA (Double Asymptotic Analysis) to analyze the performance of the surface craft. NASTRAN is essentially a linear structural code and one would not be able to carry out a very short range nonlinear analysis of an underwater explosion close to a structure.

 

In an underwater explosion, there are two important and fairly distinct phenomena separated by the typical time scales. At the very short time duration of O(ms), it is the underwater shock propagation, reflection and refraction at the structure and free surface. At the much larger time constant of O(seconds), it is the dynamics of bubble which dominates like the whipping effect and jet induced flow due to bubble collapse near the structure. The description of our research below is carried out along these two lines. These works are largely developed within NUS, and I am the main or corresponding author for several of the published papers.

 

(Underwater) Shock Dynamics in multi-medium flow

It is realized that there is not much work carried out in the literature for the case of a strong shock (like an underwater explosion) in a multi-medium flow where the properties are very different (like for air and water). We embarked on this work to develop a numerical method which is able to simulate accurately an underwater strong shock in the presence of the gaseous media separated by the free surface, and submerged/half-submerged structure. This is largely accomplished in our published work where the wave interface interaction is taken into account without knowing apriori the detailed refraction type via the use of the implicit characteristic method and ARS (Approximate  Riemann Solver) [Liu et al (2001), Comp. & Fluids, Vol. 30, pp 291-314; Liu et al (2001), Comp. & Fluids, Vol. 30, pp 315-337]. This is critical and the method is shown to work very well even for a very strong shock impacting on a material surface of very large density ratio, where most of the existing methods fail to provide satisfactorily results. The introduction of a submerged solid structure and the necessary treatment of the boundary conditions at the solid surface can be found in another work [Liu et al (2003), Int. J. Num. Meth. in Eng., Vol. 58, pp 609-630]; the result obtained for an underwater explosion is interesting and show the possible presence of a cavitation region next to the free surface giving rise to additional imposed torque on the loaded submerged structure.

 

Subsequent to our published works, we are keenly aware that the numerical method suggested can be rather complex and not so easily extended to 3D with complicated geometries. In recent times, it appeared in the literature a fairly powerful technique called the Ghost Fluid Method (GFM) which provides a simple means to treat a material interface and hence easy extension to multi-dimensions. While the original GFM [Fedkiw et al (1999), J. Comput. Phys. (JCP), Vol. 152, pp 457] or even the supposedly new GFM [Fedkiw (2002), JCP, Vol. 175, pp 200] works reasonably well in the presence of not-too-strong shock, it may and can lead to the incorrect physics when the shock strength becomes large. We carried out a thorough analysis to determine the cause and propose the modified GFM incorporating our developed APRS solver to overcome the problems/issues as reflected in Liu et al (2003) (JCP, Vol. 190, pp. 651-681). Two conditions are proposed necessary for any GFM-based scheme to be able to resolve the waves at the contact discontinuity correctly as expounded in Liu et al (2005) (JCP, 204, pp. 193-221). It is also apparent that the stability and accuracy of the GFM is dependent on the associated numerical scheme implemented. Consequent to our development of the modified GFM, we further put forward another powerful variant called the real-GFM for the simulation of compressible multi-medium flows [Liu et al (2006), SIAM J. Sci. Comput., Vol. 28, pp. 278-302]. The (natural) extension to strong shock in multi-medium flow next to compressible solid can be found in Liu et al (2007) (SIAM J. Sci. Comput., accepted for publication).

 

Our work in compressible multi-medium flow is extended to address shock-induced cavitation. In the literature, the cavitation region is commonly treated as constant or nearly constant (critical) pressure zone such that there is no flow or nor transmission of shock waves like the so-called cut-off cavitation model. We proposed a physics-based isentropic one-fluid cavitation model capable of simulating the creation/formation of the cavitating region and its subsequent collapse via a proper EOS (Equation of State) and which allows the proper transmission/refraction of shock. Our proposed model can be found in Liu et al (2004) (JCP, Vol. 201, pp. 80-108) and yet another simpler variant called the modified Schmidt model is put forward in Liu et al (2006) (Comp & Fluids, Vol. 35, pp. 1177-1192). Our full model is mathematically consistent and can be easily extended to higher dimensions in various configurations of near the free surface and compressible structures as demonstrated in Liu et al (2006) (Comm. in Comput. Phys. Vol. 1, pp. 898-919) and Xie et al (2006) (Applied Numerical Mathematics, APNUM, Vol. 57, pp. 721-733). Some preliminary works on employing the isentropic one-fluid cavitation model for the study of flow supercavitation where the cavitating region fully enveloped the object moving sufficiently fast in water for drag reduction are ongoing.

 

Further supporting evidences of the importance and significance of the work are:

(a) Invited to submit and present a paper on “Shock-structure-cavitation interaction” at the US/Singapore Workshop on Computational Mechanics and Simulation of Underwater Explosion Effects [organized jointly by ONR, NRL, DSTA and IHPC] held in Washington DC on 1-2, November 2000.

(b) The prestigious Defense Technology Prize for team effort in 1998 was awarded to the Underwater Shock Technology by MINDEF (Ministry of Defense, Singapore). BC is a team member of the Underwater Shock Technology Program in charge specifically of the shock and bubble dynamics. This is the first time a national award was given to a team outside of the Defense establishment.

(c) An exploratory project with ONR/NRL on both shock and bubble dynamics known as “Validation of the UNDEX (Underwater Explosion) Research” was initiated in 2001. We computed for different configurations of shock and bubble conditions for comparison to their results comprising both simulations and experiments. This inaugural and only project can be considered as part of the IEA (Information Exchange Agreement) signed between the US and Singapore Governments which will lead to further research collaboration. (Discussion is currently underway for a joint research effort on shock-cavitation.)

(d) At the 24th ISSW (International Symposium for Shock Waves) held at Beijing 2004 where our works on the shock in multi-medium flow and cavitations were presented, BC was subsequently approached by Dr. Christian Caron (email: Christian.caron@springer-sbm.com), the Executive Editor, Physics for Springer (www.springeronline.com) to write a book related to shock propagation in multi-medium flow and shock-cavitation. Dr. Caron mentioned that he has been closely following our works in this field

(e) Invited to present the work on is “Shock in an unsteady compressible multi-medium flow with cavitation” at the 1st TDSI (Temasek Defense System Institute at NUS) - NPS (Naval Postgraduate School) Conference, Monterey, California, on 4-5 November, 2004.

(f) Invited to present the work on “Simulation of homogeneous unsteady cavitation in multi-dimensional flow” at the International Conference on Scientific Computing, Nanjing China, 4-8 June 2005.

(g) Invited to give the Keynote lecture on “Numerical Modeling of Cavitation Inception in High-Speed Cavitating Flows: One-Fluid/Free Lagrange Model” at the International Conference on Enhancement and Promotion of Computational Methods Engineering Science and Mechanics (CMESM), Changchun China, 10-12 August 2006.

(h) Invited to participate at the Workshop Forum on “Maritime Security” organized by TDSI (Temasek Defence System Institute) – NPS (Naval Postgraduate School) – LLNL (Lawrence Livermore National Laboratories), Monterey, California, 20-22 June 2007.

(i) Invited to present the work on “The Pulse Detonation Engine for the UAV (Unmanned Air Vehicle): Simulation of 3D detonation” at the International Conference on Intelligent Unmanned Systems (ICIUS), Bali, Indonesia, 24-25 October 2007.

(j) There is continuous external funding and support for the work on topics related to underwater shock and bubble dynamics since the early 1990’s.

 

Future work:

(a) There is recent interest on the Pulse Detonation Engine (PDE) as expressed by MINDEF in conjunction with the development of unmanned air vehicle. PDE with the possibility of extended range of operation or power is a possibility for powering the new generation of unmanned air vehicle. MINDEF has provided an initial sum amounting to $500000 tenable over a period of two years for the PDE work. This research work is to be carried out with Temasek Laboratories which is administering the fund, and there is participation from University of Texas at Arlington mainly in the experiments. Our interest is on the numerical simulation of detonation as in the PDE. The project kicks off in July 2004.

Arising from the initial work on the PDE, in the pipeline is another research project on the liquid PDE in joint collaboration between Temasek Laboratories and University of Texas at Arlington with a grant of close to S$2 million. The work starts in early 2008.

(b) In a departure from Defense-related interest, we are starting to embark on the topic of

‘Shock in biological flow’. It is still an open question whether in the extracorporeal shock-wave lithotripsy (ESWL), renal calculi is destroyed solely by the shock wave (or due to the presence of shock-induced cavitation bubble with its associated jetting effect or both). This is because the time scale for the shock duration is very short and  called into question whether there is sufficient energy to break up the renal calculi or otherwise. A very preliminary work entitled “Numerical studies on shock cell interaction” was presented at the 24th International Shock Wave Symposium (ISSW) in July 2004. It is our intent to explore and engage in this new exciting field. There is possible collaboration with Professor Pei Zhong of Duke University under the auspice of the Duke-Singapore Medical Graduate School with local interest from medical doctors at NUH and SGH; some discussions are underway.

 

List of relevant publications

Liu, T G, B C Khoo* and K S Yeo, "The numerical simulations of explosion and implosion in air: Use of a modified Harten's TVD scheme". International Journal for Numerical Methods in Fluids, 31 (1999): 661‑680. (United Kingdom).

 

Liu, T G, B C Khoo* and K S Yeo, "The simulation of compressible multi‑medium flow. Part I: A New Methodology with Test Applictions to ID Gas‑Gas and Gas‑Water Cases". Computers & Fluids, 30, no. 3 (2001): 291‑314. (United States).

 

Liu, T G, B C Khoo* and K S Yeo, "The simulation of compressible multi‑medium flow, Part II: Applications to 2D underwater shock refraction". Computers & Fluids, 30, no. 3 (2001): 315‑337. (United States).

 

Liu, T G, B C Khoo* and K S Yeo, "Ghost fluid method for strong shock impacting on material interface". Journal of Computational Physics, 190 (2003): 651‑681. (United States).

 

Liu, T G, B C Khoo*, K S Yeo and C Wang, "Underwater shock‑free surface‑structure interaction". International Journal for Numerical Methods in Engineering, 58 (2003): 609‑630. (United Kingdom).

 

Hu, X. and B C Khoo*, "An interface interaction method for compressible multifluids". Journal of Computational Physics, 198 (2004): 35‑64. (United States).

 

Liu, T G, B C Khoo* and W F Xie, "Isentropic One‑Fluid Modelling of Unsteady Cavitating Flow". Journal of Computational Physics, 201 (2004): 80‑108. (United States).

 

Hu, X, B C Khoo*, D.L. Zhang and Z.L. Jiang, "The cellular structure of a two‑dimensional H2/O2/Ar detonation wave".  Combustion Theory and Modeling, 8 (2004): 339‑359. (United Kingdom).

 

Hu, X., D.L. Zhang, B C Khoo* and Z.L. Jiang*, "The Structure and evolution of a two‑dimensional H2/O2/Ar cellular detonation". Shock Waves, 14, no. 1‑2 (2005): 37‑44. (Germany).

 

Liu, T G, B C Khoo* and C W Wang, "The Ghost Fluid Method for Compressible Gas‑Water Simulation". Journal of Computational Physics, 204 (2005): 193‑221. (United States).

 

Liu, T G, W F Xie and B C Khoo*, "Ghost fluid method applied to compressible multi‑phase flows". Modern Physics Letters B, 19, no. 28‑29 (2005): 1475‑1478. (Singapore).

 

Qiu, J X, B C Khoo* and C W Shu, "A numerical study for the performance of the Runge‑Kutta discontinuous Galerkin method based on different numerical fluxes". Journal of Computational Physics, 212 (2005): 540‑565. (United States).

 

Wang, C W, T G Liu and B C Khoo*, "A real‑Ghost Fluid Method for the simulation of multi‑medium compressible flow". SIAM Journal on Scientific Computing, 28, no. 1 (2005): 278‑302. (United States).

 

Dou, H, B C Khoo* and K S Yeo, "Incipent separation in shock wave/boundary layer interactions as induced by sharp fin". Shock Waves, 15 (2006): 425‑436. (Germany).

 

Xie, W F, T G Liu and B C Khoo*, "Application of a one‑fluid model for large scale homogeneous unsteady cavitation: the modified Schmidt model". Computers & Fluids, 35, no.10 (2006): 1177‑1192. (United States).

 

Hu, X.Y., B C Khoo*, N.A. Adams and F.L. Huang, "A conservative interface method for compressible flows". Journal of Computational Physics, 219, no. 2 (2006): 553‑578. (United States).

 

Liu, T.G., B C Khoo* and W.F. Xie, "The Modified Ghost Fluid Method as Applied to Extreme Fluid‑Structure Interaction in the Presence of Cavitation". Communications in Computational Physics, 1, no. 5 (2006): 898‑919. (Hong Kong).

 

Qiu, J., T.G. Liu and B C Khoo*, "Runge‑Kutta Discontinuous Galerkin methods for compressible two‑medium flow simulations: one‑dimensional case". Journal of Computational Physics, 222, no. 1 (2007): 353‑373. (United States).

 

Xie, W.F., T.G. Liu and B C Khoo*, "The Simulation of Cavitating Flows Induced by Underwater Shock and Free Surface Interaction". Applied Numerical Mathematics, 57, no.5‑7 (2007): 721‑733. (Switzerland).

 

Liu, T. G. and B C Khoo*, "The accuracy of the modified Ghost Fluid Method for gas‑gas Riemann problem". Applied Numerical Mathematics, 57, no.5‑7 (2007): 721‑733. (Switzerland).

 

Xie, W.F., Y.L. Young, T.G. Liu and B C Khoo*, "Dynamic response of deformable structures subjected to shock load and cavitation reload". Computational Mechanics, (2006). (West Germany), accepted for publication.

 

Qiu, J.X., T.G. Liu and B C Khoo*, "Simulations of compressible two‑medium flow by Runge‑Kutta discontinuous Galerkin methods with the ghost fluid method". Communications in Computational Physics, (2007). (Hong Kong), accepted for publication.

 

Liu, T.G., W.F. Xie and B C Khoo*, "Then modified Ghost Fluid Method for coupling of fluid and structure with Hydro‑Elasto‑Plastic Equation of State". SIAM Journal on Scientific Computing, (2007). (United States), accepted for publication.

 

 

 

 

 

 

 

 

 

Dynamics of Bubble(s)

The motivation is to develop a software capable of simulating the dynamic behavior of multiple explosion bubbles in the presence of the free surface and submerged/half-submerged structure. To meet the objective and in view that viscous forces are not likely to contribute significantly to the dynamics of bubble-structure interaction, it was deliberated that the BEM (Boundary Element Method) be employed for such simulation. We started off with the axisymmetric bubble in the early 1990’s with two works published [Wang et al (1996), Comp. & Fluids, Vol. 25, pp 607-628; Wang et al (1996), Theoret. & Comp. Fluid Dyn., Vol. 8, pp 73-88]. At that time, the state-of-art in the literature was still limited to the 2D or axisymmetric configuration. Moving forward we introduced the 3D simulation of bubble dynamics resulting in the published work of two interacting bubbles in the presence of the free surface [Zhang et al (1998), JCP, Vol. 146, pp 105-123]. An important feature is the utilization of a nine-noded Lagrangian interpolation for the computation of the surface characteristics and material velocity, and the solid angles on the free surface are computed via direct approach. In a subsequent work, we extended the surgical-cut and vortex ring technique to general three-dimensional toroidal bubble problems with jet impact [Zhang et al (2001), JCP, Vol. 166, pp. 366-360]. The use of vortex ring has marked improvement over the vortex sheet technique where one may need to painstakingly track the evolution of the vortex sheet of a toroidal bubble. Improvement is made with the optimal distribution of the meshes representing the bubble surface in the form of the Elastic Mesh Technique (EMT) proposed where the mesh is advanced not by the material velocity but the optimal shift velocity obtained by minimizing the total elastic energy stored in every segment of the mesh at each time step [Wang et al (2003), Comp. & Fluids, Vol. 32, pp 1195-1212]. We further developed the criterion for bubbles merging during the evolution and compared to available experiments [Rungsiyaphornrat et al (2003), Comp. & Fluids, Vol. 32, pp 1049-1074]. The numerical tool developed for the bubble dynamics enable a comprehensive study of underwater explosion near a submerged structure [Klaseboer et al, 2005, J. Fluid Struct., Vol. 21, pp. 395-412].

In a marked departure from the Defense-initiated work on the bubble dynamics, we have began to apply the domain knowledge gained in this area for non-Defense work; this has resulted in two works on the collapsing bubble-induced micro-pump via jetting effect to be found in Khoo et al (2005) (Sen. & Act. A, Vol. 118, pp. 152-161) and Lew et al (2006) (Sen. & Act. A, to appear). The proposed micro-pump with no moving parts and yet with high impulse may have implication for commercialization. Other non-Defense related work is on the acoustic-induced bubble and its behavior near to simulated biomaterials for a range of bio-medical applications like UAL (Ultrasound-Assisted Lipoplasty), phacoemulsification, ultrasonic aspiration and others; discussion can be found in Fong et al (2006) (Ultrasound Med. & Bio., Vol. 32, pp. 925-942). The behaviour of non-equilibirum near to other elastic material or membrane surface can be found in Klaseboer and Khoo (2004) (J. Appl. Phys., Vol. 96, pp. 5808-5818), Klaseboer et al (2006) (Int. J. Multiphase Flow, Vol. 32, pp. 1110-1122) and Turangan et al (2006) (J. Appl. Phys., Vol. 100, 054910-1). The study of shock wave on bubble as present in shock lithotripsy treatment for the clinical removal of kidnet stones is greatly facilitated by our novel employment of pressure pulse in conjunction with BEM technique; the ensuing calculation depicting the correct flow physics is vastly superior and computationally efficient compared to the traditional approach [Klaseboer at al, Comp. Meth. Appl. Mech. Eng., Vol. 195, pp. 4287-4302].

 

The above-mentioned work on bubble dynamics is based on the direct formulation of the BEM equation. In parallel, we also embarked on the (alternative) indirect BEM approach. In the de-singularised version, there is a great speed up in the computation of the bubble evolution without compromise on the [Zhang et al (1999), Int. J. Num. Meth. in Fluids, Vol. 31, pp 1311-1320]. However, problems arise when two bubble surfaces come close together leading to the meeting of the de-singularised points. On the other hand, the de-singularised method would still be applicable for the free surface (provided it remains simply-connected which is always the situation) and can be used in conjunction with an improved indirect method as shown by Wang & Khoo (2004) (JCP, Vol. 194, pp. 451-480). In the latter, an expression for the self-induced velocity of a point on the boundary is obtained analytically and EMT is applied. The provision of both the indirect and direct BEM formulations for the bubble dynamics simulation presents the flexibility for linking to the nonlinear structural code PAM-shock by ESI International.

 

Consequent to our further development on the study of the dynamics of multiple bubbles and their interactions, we have successful embarked on a technique called the FMM (Fast Multipole Method) in combination with the pre-corrected FFT (Fast Fourier Transform) to formulate a new FFTM clustering (Fast Fourier Transform Multipole) suitable for our simulation of multiple bubbles via BEM approach [Bui et al (2006), JCP, Vol. 216, pp. 430-453]. Traditionally, FMM and pre-corrected FFT have been employed in electrostatic problems where the non-deformable geometries can be very complex. In our application, the bubble surfaces are expected to undergo large deformation and we have managed to compute over 25 full 3D bubbles in close interaction with each other using the FFTM clustering technique developed. The novel use of level-set technique for tracking the bubble interface(s) in conjunction with the BEM can be found in Tan el al (2007) (SIAM J. Sci. Comput., accepted for publication) carried out under the auspices of research collaboration with MIT provided by the SMA (Singapore-MIT Alliance). (SMA is a distance-learning graduate education in engineering between MIT, NUS and NTU and which started in 1999.) The level-set technique will overcome those related problems on numerical instability associated with front tracking technique especially when there is very large and rapid deformation of the surfaces as for the bubble and free surface.

 

Further supporting evidences of the importance and significance of the work are:

(a) Invited to submit and present a paper on “3D computation of bubbles near a free surface” at the US/Singapore Workshop on Computational Mechanics and Simulation of Underwater Explosion Effects [organized jointly by ONR, NRL, DSTA and IHPC] held in Washington DC on 1-2, November 2000.

(b) Based on a paper entitled, “A comparison of numerical simulation with experiment on bubble-structure interaction” presented and published in the 73rd Shock and Vibration Symposium held in Newport Rhode Island, November 18-22, 2002, an invitation was made by Professor W. D. Pilkey (Editor) and only to selected works to submit and publish in the journal Critical Technologies in Shock and Vibration initiated by SAVIAC.

(c) Collaboration with the French DGA on validation of the 3D bubble dynamic code. DGA provided all the necessary field data on underwater explosion bubble dynamics while we presented the simulations for exchange and comparison.  

(d) Collaboration with the ESI International (a multi-national corporation based in France) where the non-linear structural code PAM-shock has been linked up to our 3D bubble dynamic code to be licensed and marketed internationally on commercial terms. Already ESI has managed to license the bubble code to at least one large user in the US. According to ESI, there are more such licenses to be issued to the different US organizations because of their expressed deemed interest and other confirmed interest from the EU countries like Italy and France, and some encouraging enquiries from China who are interested on acquiring several licenses. (Such is a testament to the viability and up-to-date research content of the 3D bubble code developed over the past decade, and there are many more new capabilities in the pipeline to be released in an orderly manner.)

(e) Arising from our study of the collapse of a bubble near a solid surface with accompanying induced jet directed towards the said surface, a patent on ‘micro-pump based on bubble jet’ has been filed with the US patent office in 2002. It may be noted that the cost of the US patent application is fully underwritten by A*STAR.

(f) Invited to give a public technical seminar entitled, “What do micro-pumps and (ultrasonic) bio-medical treatments have in common? – Bubble Dynamics” at Hong Kong University of Science & Technology, Hong Kong, held on 5th May 2004.

(g) Invited to give a public technical seminar on ““Behavior of an oscillating bubble near an elastic membrane” at the Nanjing University of Aeronautics and Astronautics (NUAA), China, 6th June 2005.

(h) Invited to present the work on “Prediction of Free Surface Water Plume as a Barrier to Sea-skimming missiles” at the TDSI (Temasek Defense System Institute at NUS) - NPS (Naval Postgraduate School) organized International Maritime Protection Symposium Conference, Honolulu Hawaii, 12-14 December 2005.

(i) Invited to give a technical seminar on “Current development concerning bubble dynamics in Singapore” at the School of Chemical and Biological Sciences, Nanyang Technological University, 5 June 2006.

(j) Invited to give a public technical seminar on “What do micropumps and (ultrasound) biomedical treatment have in common?” at the North Carolina State University Special Summer Seminar, 14 July 2006.

(k) Invited to present the work on “One-fluid model for high speed cavitating flow” at Defence Technology and Systems Conference 2006 jointly organized by TDSI (Temasek Defence System Institute) – NPS (Naval Postgraduate School) – LLNL (Lawrence Livermore National Laboratories), Singapore, 6-8 December 2006.

(l) Invited to present the work on “Acoustic bubble simulations for biomedical applications using boundary element method (BEM)” at the 153rd Acoustic Society of America (ASA), Salt Lake City, Utah, 4-8 June 2007.

(m) Invited to present the work on “Ultrasound microbubble interaction” at the International Conference on Multiscale Modelling and Simulation (ICMMS 2008), IISc Bangalore, India, 2-4 Jan. 2008.

(o) There is continuous external funding and support for the work on topics related to underwater shock and bubble dynamics since the early 1990’s.

 

Future work:

(a) The linking of the bubble code to the PAM-shock not only represent the step towards commercialization of the combined code but also present the opportunities for further work into bubble-structure interaction as applied to the biological system. The collapse of a bubble entails a strong jet directed towards the structural surface and can be used for ‘non-invasive’ surgical procedure. Effort is underway to collaborate with the Dentistry Department (Professor Anil Kishen) on exploring the bubble dynamics to remove the biofilms commonly found in root canal procedure and others. Other efforts are directed at collaboration with the Eye Department at National University Hospital to explore the employment of non-equilibrium bubble for the clearing of the duct which blockage can lead to glaucoma.

(b) On the technical side, we will be embarking on the Stokes solver as employed under the BEM formulation with the intention of replacing the present Laplace solver for the present bubble dynamics simulation. If successful, we would be able to simulate for bubble in micron dimension which applicable to micro-fluidics. This will open for us a whole new avenue of research work.

 

List of relevant references

Wang, Q X, K S Yeo*, B C Khoo and K Y Lam, "Strong interaction between a buoyancy

bubble and a free surface". Theoretical and Computational Fluid Dynamics, 8 (1996): 73‑88. (United States).

 

Wang, Q X, K S Yeo*, B C Khoo and K Y Lam, "Non‑linear interaction between gas bubble

and free surface". Computers & Fluids, 25, no. 7 (1996): 607‑628. (United States).

 

Zhang, Y L, K S Yeo*, B C Khoo and W K Chong, "Three‑dimensional computation of bubbles

near a free surface". Journal of Computational Physics, 146 (1998): 105‑123. (United States).

 

Zhang, Y L, K S Yeo*, B C Khoo and W K Chong, "Simulation of three‑dimensional bubbles

using desingularised boundary integral method". International Journal for Numerical Methods in Fluids, 31 (1999): 1311‑1320. (United Kingdom).

 

Zhang, Y L, K S Yeo* and B C Khoo, "Three‑dimensional jet impact and toroidal bubbles". Journal of Computational Physics, 166 (2001): 336‑360. (United States).

 

Rungsiyaphornrat, S, E Klaseboer, B C Khoo* and K S Yeo, "The merging of two gaseous

bubbles with an application to underwater explosions". Computers & Fluids, 32, no. 8 (2003): 1049‑1074. (United States).

 

Wang, C, B C Khoo* and K S Yeo, "Elastic mesh technique for 3D BIM simulation with an application to underwater explosion bubble dynamics.". Computers & Fluids, 32, no. 9 (2003): 1195‑1212. (United States).

 

Wang, C and B C Khoo*, "An indirect boundary element method for three‑dimensional

explosion bubbles". Journal of Computational Physics, 194 (2004): 451‑480. (United States).

 

Klaseboer, E and B C Khoo*, "Boundary integral equations as applied to an oscillating bubble near fluid‑fluid interface". Computational Mechanics, 33, no. 2 (2004):129‑138. (West Germany).

 

Klaseboer, E. and B C Khoo*, "An oscillating bubble near an elastic material". Journal of

Applied Physics, 96, no. 10 (2004): 5808‑5818. (United States).

 

Khoo*, B C, E. Klaseboer and K.C. Hung, "A collapsing bubble‑induced micro‑pump using jetting effect". Sensors and Actuators A ‑ Physical, 118 (2005): 152‑161. (Netherlands).

 

Wang, Q X, K S Yeo*, B C Khoo and K Y Lam, "Vortex ring modelling of toroidal bubbles". Theoretical and Computational Fluid Dynamics, (2005): 1‑15. (United States).

 

Klaseboer, E, C Wang, C W Wang, B C Khoo*, K C Hung and P Boyce, "Experimental and numerical investigation of the dynamics of an underwater explosion bubble near a resilient/rigid structure". Journal of Fluid Mechanics, 537 (2005): 387‑413. (United Kingdom).

 

Soh, W K, B C Khoo* and W Y Yuen, "The entrainment of air by water jet impinging on a free surface". Experiments in Fluids, 39 (2005): 498‑506. (United Kingdom).

 

Ong, G P, B C Khoo*, C Turangan, E Klaseboer and S W Fong, "Behavior of oscillating bubbles near elastic membranes: an experimental and numerical study". Modern Physics Letters B, 19, no. 28‑29 (2005): 1579‑1582. (Singapore).

 

Klaseboer, E, B C Khoo* and K C Hung, "Dynamics of an oscillating bubbles near a floating structure". Journal of Fluids and Structures, 21 (2005): 395‑412. (United Kingdom).

 

Klaseboer, E. and B C Khoo*, "A modified Rayleigh‑Plesset model for a non spherically symmetric oscillating bubble with applications to Boundary Integral Methods". Engineering Analysis with Boundary Elements, 30 (2005): 59‑71. (Switzerland).

 

Bui, T.T., E.T. Ong, B C Khoo*, E. Klaseboer and K.C. Hung, "A Fast Algorithm for Modeling Multiple Bubbles Dynamics". Journal of Computational Physics, 216 (2006): 430‑453. (United States).

 

Fong, S.W., E. Klaseboer, C.K. Turangan, B C Khoo* and K.C. Hung, "Numerical analysis of a gas bubble near biomaterials in an ultrasound field". Ultrasound in Medicine and Biology, 32 (2006): 925‑942. (United States).

 

Klaseboer, E, C Turangan, S W Fong, T G Liu, K C Hung and B C Khoo*, "Simulations of pressure pulse ‑ bubble interaction using boundary element method". Computer Methods in Applied Mechanics and Engineering, 195, no. 33‑36 (2006): 4287‑4302. (Netherlands).

 

Klaseboer, E., C. Turangan and B C Khoo*, "Dynamic behaviour of a bubble near an elastic infinite interface". International Journal of Multiphase Flow, 32, no. 9 (2006): 1110‑1122. (United Kingdom).

 

Turangan, C., G.P. Ong, E. Klaseboer and B C Khoo*, "Experimental and numerical study of transient bubble‑elastic membrane interaction". Journal of Applied Physics, 100 (2006): 054910‑1 to 054910‑7. (United States).

 

Lew, K.S.F., E. Klaseboer and B C Khoo*, "A collapsing bubble‑induced micro‑pump: an experimental study". Sensors and Actuators A ‑ Physical, 133 (2007): 161‑172. (Netherlands).

 

Dao, M. H., K M Lim, B C Khoo* and K. Willcox, "Real‑Time Optimization Using Proper Orthogonal Decomposition: Free Surface Shape Prediction due to Underwater Bubble Dynamics". Computers & Fluids, 36 (2007): 499‑512. (United States).

 

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