A NUMERICAL STUDY OF THE EFFECT OF CONTACT ANGLE ON THE DYNAMICS OF A SINGLE BUBBLE DURING POOL BOILING

Proceedings of IMECE ASME Inernaional Mechanical Engineering Congress & Exposiion November 17-,, New Orleans, Louisiana IMECE-33876 A NUMERICAL STUDY OF THE EFFECT OF CONTACT ANGLE ON THE DYNAMICS OF A SINGLE BUBBLE DURING POOL BOILING H.S.Abarajih and V.K.Dhir! Universiy of California, Los Angeles, Mechanical and Aerospace Engineering Deparmen, Los Angeles, CA 995, U.S.A, email: vdhir@seas.ucla.edu ABSTRACT The effec of conac angle on he growh and deparure of a single bubble on a horizonal heaed surface during pool boiling under normal graviy condiions has been invesigaed using numerical simulaions. The conac angle is varied by changing he Hamaker consan ha defines he long-range forces. A finie difference scheme is used o solve he equaions governing mass, momenum and energy in he vapor and liquid phases. The vapor-liquid inerface is capured by he Level Se mehod, which is modified o include he influence of phase change a he liquid-vapor inerface. The conac angle is varied from 1 o 9 and is effec on he bubble deparure diameer and he bubble growh period are sudied. Boh waer and PF56 are used as es liquids. The conac angle is kep consan hroughou he bubble growh and deparure process. The effec of conac angle on he parameers like hermal boundary layer hickness, wall hea flux and hea flux from he microlayer under various condiions of superheas and subcoolings is also sudied. INTRODUCTION Boiling, being he mos efficien mode of hea ransfer is employed in various energy conversion sysems and componen cooling devices. In order o have a good undersanding of he process, a number of analyical, experimenal and numerical sudies have been carried ou in he pas hrough he modeling of bubble dynamics including he growh and deparure of he vapor bubbles. In his work complee numerical simulaions of bubble dynamics in pool boiling are carried ou o qualify he effec of conac angle. Friz (1935) was he firs o develop an equaion for he bubble deparure diameer involving conac angle by balancing buoyancy wih surface ension forces acing on a saic bubble. His empirical expression for bubble deparure diameer involving conac angle is given as D =.8 ϕ σ /[ g( ρ ρ )] (1) d l v where ϕ is he conac angle in degrees. Sainszewski (1959) suggesed a correlaion including he effec of bubble growing velociy as D =.71ϕ dd σ /[ g( ρ ρ )](1 +.435 * ) () d l v d dd where is given in inches per second jus prior o d deparure. Lee and Nydahl (1989) calculaed he bubble growh rae by solving he flow and emperaure fields numerically from he momenum and energy equaions. They used he formulaion of Cooper and Llyod (1969) for micro layer hickness. However hey assumed a hemispherical bubble and wedge shaped microlayer and hus hey neiher accouned for nor evaluaed he radial variaion of he microlayer hickness during he growh of he bubble. Zeng e al. (1993) used a force balance approach o predic he bubble diameer a deparure. They included he surface ension, inerial force, buoyancy and he lif force creaed by he wake of he previous depared bubble. Bu here was empiricism involved in compuing he inerial and drag forces. The sudy assumed a power law profile for growh rae and he coefficiens were deermined from he experimens. Mei e al. (1995) derived resuls for he bubble growh and deparure ime assuming a wedge shaped microlayer. They also assumed ha he hea ransfer o he bubble was only hrough he microlayer, which is no correc. The sudy did no consider he hydrodynamics of he liquid moion induced by he growing bubble and inroduced empiricism hrough he assumed shape of he growing bubble. Welch (1998) has used a finie volume! auhor for correspondance 1 Copyrigh by ASME

mehod and an inerface racking mehod o model bubble dynamics. The conducion in he solid wall was also aken ino accoun. However, he microlayer was no modeled expliciely. Lay and Dhir (1995) carried ou complee analysis of he microlayer including disjoining pressure erm, vapor recoil pressure and inerfacial hea ransfer resisance o deermine he shape of he microlayer for various conac angles. They used balance beween forces due o curvaure of inerface, disjoining pressure, hydrosaic head and liquid drag, which in urn deermined he shape of he vapor-liquid inerface. Qiu and Dhir (1) performed experimens for he deerminaion of he bubble deparure diameer and ime period of growh wih waer as well as PF56 as es liquids. They observed smaller deparure diameers and growh periods in he case of PF56 han hose for waer and hey aribued his o he difference in he conac angles of he wo liquids. Smaller conac angle causes smaller deparure diameers and shorer growh periods. Son e al. (1999) numerically simulaed he bubble growh during he nucleae boiling by using he Level Se mehod. This mehod has been applied o adiabaic incompressible wo-phase flow by Sussman e al. (1994) and o film boiling near criical pressures by Son and Dhir (1998). The compuaional domain was divided ino wo regions viz. micro and macro regions. The inerface shape, posiion and velociy and emperaure fields were obained from he macro region by solving he conservaion equaions. The micro region equaions, which include he disjoining pressure in he hin liquid film, were solved by employing he lubricaion heory. The soluions of he micro region and macro region were mached a he ouer edge of he micro layer. The purpose of he presen sudy is o evaluae and analyze he effec of conac angle on he bubble growh and hea ransfer associaed wih i. The work of Son e al. (1999) is exended here o find he variaions of bubble deparure diameers, growh period and hea ransfer associaed wih i for various conac angles. NOMENCLATURE A = dispersion consan, J c = specific hea a consan pressure, kj/(kg K) p D = Lif-off diameer of he bubble, m g e = graviaional acceleraion a earh level, m/s g = graviaional acceleraion a any level, m/s H = sep funcion h = grid spacing for he macro region h ev = evaporaive hea ransfer coefficien, W/(m K) C pl* T w Ja* = Jackob number, h fg = laen hea of evaporaion, J/Kg h fg K = inerfacial curvaure, 1/m l = characerisic lengh, m M = molecular weigh m = evaporaive mass rae vecor a inerface, kg/(m s) p = pressure, Pa q = hea flux, W/m R = radius of compuaional domain, m R = universal gas consan, - R = radius of dry region beneah a bubble, m R = radial locaion of he inerface a y=h/, m 1 r = radial coordinae, m T = emperaure, K = ime, s = characerisic ime, l / u, s u = velociy in r direcion, m/s u = inerfacial velociy vecor, m/s in u = characerisic velociy, m = evaporaive mass rae from micro layer, kg/s micro V c = volume of a conrol volume in he micro region, m 3 v = velociy in y direcion, m/s Z = heigh of compuaional domain, m z = verical coordinae normal o he heaing wall, m α = hermal diffusiviy, m /s β = coefficien of hermal expansion, 1/K δ = liquid hin film hickness, m δ T = hermal layer hickness, m ε ( ) = apparen conac angle, deg δ φ = smoohed dela funcion, - ϕ φ = level se funcion θ = dimensionless emperaure, (T-T s )/(T wall - T s ) = hermal conduciviy, W/mK κ µ = viscosiy, Pa s ν = kinemaic viscosiy, m /s ρ = densiy, kg/m 3 σ = surface ension, N/m Γ = mass flow rae in he micro layer, kg/s = heaing wall superhea, K T s Subscrips lv, = liquid and vapor phase rz,, = / r, / y, / swall, = sauraion, wall in = inerface = infinie MATHEMATICAL DEVELOPMENT OF THE MODEL ASSUMPTIONS The assumpions made in he model are: 1) The process is wo dimensional and axisymmeric. ) The flows are laminar. 3) The wall emperaure remains consan hroughou he process. 4) Pure waer and PF56 a amospheric pressure are used as he es fluids. Copyrigh by ASME

5) The hermodynamic properies of he individual phases are assumed o be insensiive o he small changes in emperaure and pressure. The assumpion of consan propery is reasonable as he compuaions are performed for low wall superhea range. 6) Variaions of conac angle during advancing and receding phases of he inerface are no included. ANALYSIS The model of Son e al. (1999) is exended o sudy single bubble growh in nucleae boiling for various conac angles. The compuaional domain is divided in o wo regions viz. micro region and macro region as shown in Fig.1. The micro region is a hin film ha lies underneah he bubble whereas he macro region consiss of he bubble and i s surrounding. Boh he regions are coupled and are solved for simulaneously. The calculaed shapes of he inerface in he micro region and macro region are mached a he ouer edge of he micro layer. According o he lubricaion heory, he momenum equaion in he micro region is wrien as, pl r u = µ z where pl is he pressure in he liquid. Hea conduced hrough he hin film mus mach ha due o evaporaion from he vapor-liquid inerface. By using modified Clausius-Clapeyron equaion, he energy conservaion equaion for he micro region yields, k ( T T ) ( p p ) T l wall in l v v = h T T ev + in v δ ρ h l fg The evaporaive hea ransfer coefficien is obained from kineic heory as, h ev 1/ ρ h v fg (5) M = π RT T v (6) v and T = T ( p ) v s v (4) The pressure of he vapor and liquid phases a he inerface are relaed by, where A q p = p σ K + (7) l v δ A 3 ρ h v fg is he dispersion consan or Hamaker consan. The second erm on he righ-hand side of equaion (7) accouns for he capillary pressure caused by he curvaure of he inerface, he hird erm is for he disjoining pressure, and he las erm originaes from he recoil pressure. The curvaure of he inerface is defined as, MICRO REGION The equaion of mass conservaion in micro region is wrien as, q 1 δ = ρ l. rudz h r r (3) fg where q is he conducive hea flux from he inerface, defined kl( Twall Tin ) as wih δ as he hickness of he hin film. δ Lubricaion heory and one dimensional hea ransfer in he hin film have been assumed in a manner similar o ha in he earlier works by Sephan and Hammer (1994) and Lay and Dhir (1995). 1 δ δ K = r / 1+ r r r r The combinaion of he mass conservaion, equaion (3), momenum conservaion equaion.(4), energy conservaion, equaion (5) and pressure balance equaion, (7) along wih equaion (8) for he curvaure for he micro-region yields a se of hree nonlinear firs order ordinary differenial equaions (7),(8) and (9) δ δ (1 + δ ) (1 + δ ) = + r r σ 3/ r r r r ρ h q A q + l fg T T in v 3 T h δ ρ h v ev v fg (8) (9) 3 Copyrigh by ASME

T qδ 3Th µ Γ in = + r κ + h δ κ + h δ ρ h rδ [ ] Γ rq = r h r v ev ( ) l ev l ev l fg fg (1) (11) The above hree differenial equaions (9)-(11) can be simulaneously inegraed by using a Runge-Kua mehod, when iniial condiions a r = R are given. In presen case, he radial locaion R1 he inerface shape obained from micro and macro soluions are mached. As much his is he end poin for he inegraion of he above se of equaions. The radius of dry region beneah a bubble, R, is relaed o R 1 from he definiion of he apparen conac angle, anϕ =.5 h/( R1 R). The boundary condiions for film hickness a he end poins are: δ = δ, δ =, Γ = a r = R r δ = h/, δ = a rr r = R 1 (1) where, δ is he inerline film hickness a he ip of microlayer, which is calculaed by combining Eqs. (3) and (4) and requiring ha Tin = Twall a r = R and h is he spacing of he wo dimensional grid for he macro-region. For a given T a r =, a unique shape of he vapor-liquid inerface is in, obained. R MACRO REGION For numerically analyzing he macro region, he level se formulaion developed by Son e al. (1999) for nucleae boiling of pure liquid is used. The inerface separaing he wo phases is capured by φ which is defined as a signed disance from he inerface. The negaive sign is chosen for he vapor phase and he posiive sign for he liquid phase. The disconinuous pressure drop across vapor and liquid caused by surface ension force is smoohed ino a numerically coninuous funcion wih a δ - funcion formulaion (refer o Sussman e al., 1994, for deail). The coninuiy, momenum and energy, conservaion equaions for he vapor and liquid in he macro region are wrien as, ρ + = ( ρu ) (13) T ρ ( u + u u) = p+ µ u + µ u + ρg ρβ ( T T ) g( ) σk H (14) p T ρc T u T T H s ( + ) = κ for > (15) T = T ( p ) for H = (16) s v The fluid densiy, viscosiy and hermal conduciviy of waer are defined in erms of he sep funcion H as, ρ = ρv + ( ρl ρv) H (17) 1 1 1 1 µ = µ + ( µ µ ) (18) v l v H 1 1 κ κ l H = (19) where, H is he Heviside funcion, which is smoohed over hree grid spaces as described below, 1 if φ 1.5h H = if φ 1.5h φ πφ.5 + + sin /( π) if φ 1.5h 3h 3h The mass conservaion equaion (13) can be rewrien as, u = ρ + ρ ρ m u = ρ ρ ( u ) / () (1) The erm on righ hand side of equaion (19) is he volume expansion due o liquid-vapor phase change. From he condiions of he mass coninuiy and energy balance a he vapor-liquid inerface, he following equaions are obained, m = ρ( u u) = ρ ( u u ) in l in l () = ρ ( u u ) v in v m= κ T h (3) / fg where m is he waer evaporaion rae vecor, and u in is inerface velociy. If he inerface is assumed o advec in he same way as he level se funcion, he advecion equaion for densiy a he inerface can be wrien as, ρ + u in ρ = (4) Using equaions (18), () and (1), he coninuiy equaion, (19) for macro region is rewrien as, (5) The vapor produced as a resul of evaporaion from he micro region is added o he vapor space hrough he cells adjacen o he heaed wall, and is expressed as, 4 Copyrigh by ASME

1 dv m mic = δε ( φ) Vc d Vcρ v mic (6) where, Vc is he volume of a conrol volume, m mic is he evaporaion rae from he micro-layer and is expressed as, κ ( T T ) R1 l w in mic = rdr (7) R hfgδ m The volume expansion conribued by micro layer is smoohed a he vapor-liquid inerface by he smoohed dela funcion δ ( φ ) = H / φ (8) ε In level se formulaion, he level se funcion φ is used o keep rack of he vapor-liquid inerface locaion as he se of poins where φ =, and i is advanced by he inerfacial velociy while solving he following equaion, φ = φ u in (9) To keep he values of φ close o ha of a signed disance funcion φ = 1, φ is reiniialized afer every ime sep, φ φ = φ + h (1 φ ) (3) where, φ is a soluion of equaion (7). The boundary condiions for velociy, emperaure, concenraion, and level se funcion for he governing equaions, (11)-(14) are: u = v =, T = T, wall φ = cosφ a z = z u = v =, T = T, z z s φ = a z = Z z u = v = T = φ = a r =, R r r r (31) SOLUTION The governing equaions are numerically inegraed by following he procedure of Son e al. (1999). The compuaional domain is chosen o be ( R/ l, Z / l ) = (1,4), so ha he bubble growh process is no affeced by he boundaries of he compuaional domain. The iniial velociy is assumed o be zero everywhere in he domain. The iniial fluid emperaure profile is aken o be linear in he naural convecion hermal boundary layer and he hermal boundary layer hickness, δ T, is evaluaed using he correlaion for he urbulen naural convecion on a horizonal plae as, 1/3 δt = 7.14( να l l / gβt T) The calculaions are carried ou over several cycles of bubble growh and deparure unil no cycle-o-cycle change in he bubble growh paern or in he emperaure profile is observed.. The mesh size for all calculaions is chosen as 98 98. I represens he bes rade-off in calculaion accuracy and compuing ime, has been shown by Son e al. (1999). The procedure given by Son e al. (1999) o mach he soluions for he micro and macro regions is adoped here o vary he conac angle. 1) The value of A, he Hamaker consan is guessed for a given conac angle. ) The macrolayer equaions are solved o deermine he value of R 1 (radial locaion of he vapor-liquid inerface a δ = h /.) 3) The microlayer equaions are solved wih he guessed value of, he Hamaker consan o deermine he A value of R (radial locaion of he vapor-liquid inerface a δ =. ) 4) The apparen conac angle is calculaed using equaion anϕ =.5 h/( R R ) and repea seps 1-4 for a differen value of 1 A, he Hamaker consan, if he values of he given and he calculaed apparen conac angles are differen. For he numerical calculaions, he governing equaions for micro and macro regions are non-dimensionalized by defining he characerisic lengh,, he characerisic velociy, u, and he characerisic ime, l as l = σ /[ g( ρ ρ )]; u = gl ; l v = l / u ( 3) 5 Copyrigh by ASME

RESULTS AND DISCUSSIONS BUBBLE DEPARTURE DIAMETER AND GROWTH PERIOD Fig. (a) shows he variaion of bubble deparure diameer wih conac angle for various wall superheas in sauraed waer. The bubble diameer increases slighly nonlinearly wih conac angle for a given wall superhea. This is due o he increase in he base area in conac wih he wall, which increases he conribuion of downward force due o surface ension. The force due o surface ension increases wih he increase in he conac angle which in urn increases he vapor volume required for bubble deparure. higher conac angles. The bubble shapes for various conac angles are given in Fig. 3 a ime, which is half of he bubble growh period. For a conac angle of 9, he bubble is hemispherical and approaches nearly a spherical shape for a conac angle of 1. Fig. (a). The Variaion of Equivalen Bubble Diameer wih Conac angle for various wall superheas, T sub = C a 1 am pressure for waer. Fig. 3 Comparison of Bubble shapes for various Conac angle a T w =8 K and T sub = K a 1 am pressure a a ime insan of = g / (i.e., he half growh) for waer. Fig. (b). The Variaion of Time period of Growh of bubble wih Conac angle for various wall superheas, T sub = C a 1 am pressure for waer. The growh period of he bubble also increases nonlinearly wih he increase in he conac angle. Fig. (b) shows he variaion of bubble growh period wih conac angle for various wall superheas in sauraed waer. This is due o he increased conribuion of he surface ension force in he case of DISPERSION CONSTANT The value of, he Hamaker consan or he dispersion consan is found ou by ieraion so as o mach he bubble shape a he ouer edge of he microlayer wih ha of he macrolayer. Fig. 4 shows he variaion of he dispersion consan, wih conac angle for waer and PF56. The A dispersion consan A A goes from negaive o posiive value a around 18 indicaing he change o aracive naure beween he liquid and wall. The value of he dispersion consan doesn vary much beween waer and PF56 for he same conac angle and a wall superhea of 8 C. A 6 Copyrigh by ASME

ha of experimenal values of Qiu e al (1). Again agreemen beween he shapes is reasonable. Fig. 4. The Variaion of Hamaker s Consan, A wih Conac angle for T w = 8 C, T sub = C a 1 am pressure for waer and PF56. HEAT TRANSFER RATES Fig. 5(a) shows he hea ransfer rae corresponds o evaporaion from he microlayer and in o he bubble for various conac angles and a wall superhea of 8 C in sauraed waer. The hea ransfer raes increase wih he increase in conac angle because of he increase in bubble base area as shown earlier in he fig. 3. Fig. 5(b) shows he hea ransfer rae corresponding o evaporaion from he macrolayer surrounding he bubble. The macrolayer conribuion also increases subsanially wih he increase in conac angle because of a subsanial increase in he bubble diameer and bubble growh period wih conac angle. The oal hea ransfer raes i.e., he sum of he hea ransfer raes form microlayer and macrolayer was found mach well wih he hea ransfer rae obained from vapor volume growh rae Fig. 5 (a) The Variaion of Hea Transfer Rae wih Time from Microlayer for various Conac angles a T w = 8 C, T sub = C a 1 am pressure for waer. Q = ρ h oal v fg dv d where V is volume of he bubble a any ime insan which provides a validaion for he numerical code. EXPERIMENTAL VALIDATION Fig. 6 shows he comparison of he ime dependence of equivalen bubble diameer for PF56 wih conac angle, ϕ =1, obained numerically wih he daa of Qiu e al (1) for a wall superhea of 19 C and a liquid subcooling of.6 C. The equivalen bubble diameer is he diameer of he sphere having he same volume as he bubble. The numerical resuls mach well wih he experimenal daa. The value of he -1 dispersion consan, A calculaed for PF56 is abou *1 J, which indicaes he aracive naure beween he wall and he liquid and weing naure of he liquid resuling in he lower bubble deparure diameers and smaller growh periods. Prediced bubble base diameer is also found o be in agreemen wih ha obained in he experimens. Fig. 7 shows qualiiaive comparison of he bubble shapes generaed numerically wih (33) Fig. 5 (b) The Variaion of Hea Transfer Rae from Macrolayer wih Time for various Conac angles a T w = 8 C, T sub = C a 1 am pressure for waer. COMPARISON OF WATER AND PF56 Fig. 8 (a) shows he comparison of prediced bubble deparure diameers wih conac angle a a wall superhea of 8 C in sauraed waer and PF56 whereas Fig. 8(b) shows such a comparison when he bubble diameers are nondimensionalized wih l. The Non-dimensional bubble deparure diameer of PF56 is higher han ha of waer a all conac angles whereas he acual deparure diameers of waer are always greaer han ha of PF56. This is due o he large value of Jackob number, Ja* for PF56 (Ja*=.16) in comparison o ha for waer (Ja*=.15). This mainly reflecs on he increase in liquid ineria as a resul of faser growh of he bubble. 7 Copyrigh by ASME

Fig. 6 The Variaion of Equivalen Bubble Diameer wih Time for T w = 19 C, T sub =.6 C a 1 am pressure for PF56 wih conac angle, ϕ =1. Fig. 8(a). The Variaion of Bubble Deparure Diameer wih Conac angle for waer and PF56 a T w = 8 C, T sub = C a 1 am pressure Fig. 8(b) The Variaion of Non-dimensional Bubble Deparure Diameer wih Conac Angle a T w = 8 C, T sub = C a 1 am pressure for PF56 and waer. Fig. 7 Comparison of numerical and experimenal growhdeparure cycles for PF56 a earh normal graviy and amospheric pressure, T sub =. 6 C, T w =19. C. 8 Copyrigh by ASME

CONCLUSIONS 1) The bubble deparure diameer increases wih he increase in he conac angle. The conac angle is relaed o he magniude of he Hamaker consan, which is found o change wih he surface weabiliy. ) The dispersion consan, goes from negaive o posiive A value a around 18 indicaing he change in he repulsive o aracive naure beween he wall and he liquid. 3) The magniude of dispersion consan does no differ much beween waer and PF56 for he same conac angle for same superhea. 4) The Non-dimensional deparure diameers of PF56 are greaer han hose for waer due o he higher values of he Jackob number. ACKNOWLEDGMENTS This work received suppor from NASA under he Micrograviy Fluid Physics program. REFERENCES 1. Friz, W., 1935, Maximum Volume of Vapor Bubbles, Physik Zeischr., Vol.36, pp. 379-384.. Sainszewski, B. E., 1959, Nucleae Boiling Bubble Growh and Deparure, Technical Repor, No. 16, Division of Sponsored Research, Massachuses Insiue of Technology, Cambridge, MA. 3. Lee, R.C and Nyadhl, J.E., Numerical Calculaion of Bubble Growh in Nucleae Boiling from incepion o deparure, Journal of Hea Transfer, Vol. 111, pp.474-479. 4. Zeng, L.Z., Klausner, J.F. and Mei, R., 1993, A unified Model for he predicion of Bubble Deachmen Diameers in A Boiling Sysems-1.Pool Boiling, Inernaional Journal of Hea and Mass Transfer, Vol. 36, pp 61-7. 5. Mei, R., Chen, W. and Klausner, J. F., 1995, Vapor Bubble Growh in Heerogeneus Boiling-1.Growh Rae and Thermal Fileds, Inernaional Journal of Hea and Mass Transfer, Vol. 38 pp 91-934. 6. Qiu, D., and Dhir, V.K., 1, Ineracing Effecs of Graviy, Wall superhea, Liquid Subcooling and Fluid Properies on Dynamics of a Single Bubble. In press. Submied o Journal of Hea Transfer. 7. Welch, S. W. J., 1998, Direc Simulaion of Vapor Bubble Growh, Inernaional Journal of Hea and Mass Transfer, Vol. 41,pp. 1655-1666. 8. Son, G., Dhir, V.K., and Ramanujapu, N., 1999, "Dynamics and Hea Transfer Associaed Wih a Single Bubble During Nucleae Boiling on a Horizonal Surface", Journal of Hea Transfer, Vol.11, pp.63-63. 9. Son, G., Dhir, V.K.,1998, "Numerical Analysis of Film Boiling Near Criical Pressure wih Level Se Mehod", Journal of Hea Transfer, Vol.1, pp.183-19. 1. Lay, J. H., and Dhir, V. K., 1995, Numerical Calculaion of Bubble Growh in Nucleae Boiling of Sauraed Liquids, Journal of Hea Transfer, Vol. 117, pp.394-41. 11. Sephan, P., and Hammer, J., 1994, A New Model for Nucleae Boiling Hea Transfer, Hea and Mass Transfer, Vol.3, pp. 119-15. 1. Sussman, M., Smereka, P and Osher, S., 1994, A Level Se Approach for Compuings Sluions o Incompressible Twophase flow, Journal of Compuaional Physics, Vol. 114, pp. 146-159. 9 Copyrigh by ASME