GOVERNING EQUATIONS FOR MULTI-LAYERED TSUNAMI WAVES

Transcription

1 GOVERNING EQUATIONS FOR MULTILAYERED TSUNAMI WAVES Monzur Alam Imeaz Senior Lecurer, Faculy of Engineering & Indusrial Sciences, Swinburne Universiy of Tecnology, Haworn, Melbourne, VIC, AUSTRALIA Fumiiko Imamura Professor, Disaser Conrol Researc Cenre, Tooku Universiy, Sendai, JAPAN Jamal Naser Senior Lecurer, Faculy of Engineering & Indusrial Sciences, Swinburne Universiy of Tecnology, Haworn, Melbourne, VIC, AUSTRALIA ABSTRACT In order o develop generalized governing equaions for mulilayered long wave sysem, a ree layer sysem was considered I was anicipaed a equaions for op layer and boom layer will be independen on number of inermediae layers and equaion for inermediae layer will be generalized depending on number of inermediae layers However, from derived equaions, i is found a only op layer equaions are independen of number of inermediae layers; equaions for all oer layers are dependen on number, een and densiy of inermediae layers Momenum and coninuiy equaions for e op layer are eacly same as in e case of earlier developed governing equaions for wo layered sysem Coninuiy equaion for e boom layer is also eacly same as in e case of wolayered sysem Momenum equaion for e boom layer is dependen on een and densiy of op layer as well as all inermediae layers Coninuiy equaion for inermediae layer is affeced by levels of immediae boom layer Momenum equaion for e inermediae layer is affeced by een and densiy of upper layers Si governing equaions, wo for eac layer are derived from Euler equaions of moion and coninuiy, assuming long wave approimaion, negligible fricion and inerfacial miing Tis paper depics deails derivaions of e governing equaions Science of Tsunami Hazards, Vol 8, No, page 7 009

2 INTRODUCTION Mulilayered flow is relaed wi many environmenal penomena Termally driven ecange flows roug doorways o oceanic currens, sal waer inrusion in esuaries, spillage of e oil on e sea surface, spreading of dense conaminaed waer, sedimen laden discarges ino lakes, generaion of lee waves beind a mounain range and idal flows over sills of e ocean are eamples of mulilayered flow In ydraulics, is ype of flow is ofen ermed as graviy curren An eensive review on ydrodynamics of various graviy currens was provided by Simpson, JE 98 Tsunamis are generaed due o disurbances of free surfaces caused by no only seismic faul moion, bu also landslide and volcanic erupions Imamura and Imeaz, 995 Tsunami waves are also affeced by densiy differences along e dep of ocean Tere are some sudies on wolayered long waves or flows in e case of underwaer landslide generaed sunamis Hampon, 97; Parker, 98; Harbiz, 99; Jiang & Leblond, 99 Imamura & Imeaz 995 developed a linear numerical model on wolayered long wave flow, wic was successfully validaed by a rigorous analyical soluion Laer e linear model was eended o a nonlinear model and effecs of nonlineariy were invesigaed Imeaz & Imamura, 00 Madsen e al 00 developed a model of mulilayered flow based on Boussinesqype equaions, wic are suiable for sallow dep flow Lyne and Liu 004 developed anoer model of mulilayered flow using piecewise inegraion of Laplace equaion for eac individual layer and epanded e model for deep waer Tis paper provides deailed derivaion of mulilayered flow equaions based on NavierSokes equaion Fig y η H ρ η H ρ η H ρ Figure Scemaic diagram of ree layer profile Science of Tsunami Hazards, Vol 8, No, page 7 009

3 THEORETICAL CONSIDERATION Considering wodimensional case and a reelayered sysem, coninuiy equaions are as follows: For upper layer, u v y For inermediae layer, u v y For boom layer, u v y Momenum equaions in Xdirecion are as follows: For upper layer, u u u v u P = y For inermediae layer, u u u u v = y For boom layer, u u u v P u P = y Were, u = Uniform velociy over e dep along direcion v = Uniform velociy over e dep along ydirecion g = Acceleraion due o graviy P = Hydrosaic pressure of fluid ρ = Densiy of fluid represens for ime Subscrips, and denoes for upper layer, inermediae layer and boom layer respecively Science of Tsunami Hazards, Vol 8, No, page 7 009

4 BOUNDARY CONDITIONS Considering a funcion along any flow sreamline, wic is consan wi ime and differeniaing i wi ime, e following boundary condiions ave been derived A op surface ie a y = η, u = v A inerface beween op layer and inermediae layer ie a y = η, u = v u u = v u A inerface beween boom layer and inermediae layer ie a y = η, u = v u u u = v u u A boom ie a y =, u = v Were, η = Waer surface elevaion above sill waer level of layer η = Waer surface elevaion above sill waer level of layer η = Waer surface elevaion above sill waer level of layer = Sill waer dep of layer = Sill waer dep of layer = Sill waer dep of layer 4 GOVERNING EQUATIONS Afer inegraing coninuiy equaions along e dep, applying some subsiuion and boundary condiions, following governing equaions can be derived considering ydrosaic pressure disribuion Science of Tsunami Hazards, Vol 8, No, page

5 For upper layer Coninuiy equaion, M Momenum equaion, g D D M M 4 For inermediae layer Coninuiy equaion, M 5 Momenum equaion, g D D M M % &, 6 For lower layer Coninuiy equaion, M 7 Momenum equaion, g D D M M % &,, Were, η = Waer surface elevaion above sill waer level of layer η = Waer surface elevaion above sill waer level of layer η = Waer surface elevaion above sill waer level of layer D = η η, D = η η, D = η, α = ρ ρ, α = ρ ρ = Sill waer dep of layer = Sill waer dep of layer = Sill waer dep of layer dy u = M dy u M = dy, u = M, Science of Tsunami Hazards, Vol 8, No, page

6 Above derived equaions can be furer simplified neglecing nonlinear erms, assuming small ampliude waves ie η<< and D and no variaions of along direcion ie =0 Final simplified equaions are as follows: For upper layer Coninuiy equaion, M Momenum equaion, M g For inermediae layer Coninuiy equaion, M Momenum equaion, M & g %, For lower layer Coninuiy equaion, M Momenum equaion, M &,, g % From e derived equaions, i is found a momenum equaion for upper layer is no affeced by e properies of adjacen layer layer undernea However, coninuiy equaion of upper layer is affeced by surface elevaion of inermediae layer Coninuiy equaion for inermediae layer is affeced by e surface elevaion of boom layer Momenum equaion for inermediae layer is affeced by densiy and spaial cange of surface elevaion of upper layer Coninuiy equaion for boom layer is no affeced by eier uppermos layer or inermediae layer However, momenum equaion of boom layer is affeced by densiies and spaial canges in surface elevaions of all e layers above i Properies of all ese equaions are oulined in Table Science of Tsunami Hazards, Vol 8, No, page

7 TABLE Affec of a paricular layer for a paricular equaion from adjacen layers Layer Equaion Effec from adjacen layer Coninuiy Waer surface elevaion of inermediae layer Upper layer Momenum No affeced Coninuiy Waer surface elevaion of boom layer Inermediae layer Momenum Spaial cange of waer surface elevaion of upper layer Densiy of upper layer Coninuiy No affeced Boom layer Momenum Spaial cange of waer surface elevaion of upper layer Spaial cange of waer surface elevaion of inermediae layer Densiy of upper layer Densiy of inermediae layer Differeniaing coninuiy equaions wi and momenum equaions wi and en subracing eac from oer, ree differen equaions one for eac layer can be deduced as follows: For upper layer g 5 For inermediae layer g % &, For lower layer g % &,, Subsiuing Equaion 6 o Equaion 5 and rearranging, 0 = g g % 8 Science of Tsunami Hazards, Vol 8, No, page

8 Subsiuing Equaion 7 o Equaion 6 and rearranging, & g g = % [ ] 0 In Equaions 8 & 9, α =ρ ρ, β = and β = From Equaions 7, 8 & 9 wave celeriy for eac layer can be defined as: C C C = g = g = g 9 5 CHARACTERISTICS OF MULTILAYER EQUATIONS I is found a e developed governing equaions are complicaed aving influence from upper andor lower layer flow However, wave celeriies for ree differen layers are derived as sown above Wave celeriy of a paricular layer can be compared wi e wave celeriy of oer layer considering reasonable values of e associaed parameers Figure sows e relaionsip of normalized celeriy of upper layer wi β for differen α values Upper layer celeriy increases wi e increase of bo e β and α values aving a power relaionsip Cg Figure Relaionsip of normalized celeriy of upper layer wi β Science of Tsunami Hazards, Vol 8, No, page

9 Figure sows e relaionsip of normalized celeriy of inermediae layer wi β for differen α α values Inermediae layer celeriy increases wi e increase of bo e β and α α values aving a power relaionsip However, normalized celeriy of lower layer depends only on e raio of densiy of inermediae layer o e densiy of lower layer ie α = ρ ρ 50 C g DA 5 DA DA DA DA 4 DA 5 DA = Figure Relaionsip of normalized celeriy of inermediae layer wi β Celeriy decreases rapidly wi e increase of α values as sown in Figure Cg Figure 4 Relaionsip of normalized celeriy of lower layer wi α Science of Tsunami Hazards, Vol 8, No, page

10 Figure 5a sows e relaionsip of C C wi α α for differen values of α and Figure 5b sows e relaionsip of C C wi α for differen α α values I is sown a C C decreases significanly wi e increase of α α However, C C increases wi e increase of α 4 CC =099 4 DA = Figure 5a Relaionsip of C C wi α α CC 0 09 DA 05 DA DA DA DA 4 DA Figure 5b Relaionsip of C C wi α Science of Tsunami Hazards, Vol 8, No, page

11 Figure 6a sows e relaionsip of C C wi α α for differen values of α and Figure 6b sows e relaionsip of C C wi α for differen α α values I is sown a C C increases almos linearly wi e increase of α α However, for a paricular value of α α, C C increases almos eponenially wi e increase of α 40 C C Figure 6a Relaionsip of C C wi α α 40 DA = 5 CC DA DA DA Figure 6b Relaionsip of C C wi α Science of Tsunami Hazards, Vol 8, No, page 8 009

12 Figure 7a sows e relaionsip of C C wi α for differen values of α and Figure 7b sows e relaionsip of C C wi α for differen α I is sown a C C increases almos linearly wi e increase of α However, for a paricular value of α, C C increases almos eponenially wi e increase of α Figure 7a Relaionsip of C C wi α Figure 7b sows e relaionsip of C C wi α Science of Tsunami Hazards, Vol 8, No, page 8 009

13 Figure 8a sows e relaionsip of C C wi α for differen values of β and Figure 8b sows e relaionsip of C C wi α α for differen values of α I is sown a C C increases almos eponenially wi e increase of α However, for a paricular value of α, C C increases almos linearly wi e increase of α α Figure 8a sows e relaionsip of C C wi α Figure 8b sows e relaionsip of C C wi α α Science of Tsunami Hazards, Vol 8, No, page 8 009

14 6 CONCLUSION Governing equaions for ree layered sunami waves were developed ransforming NavierSoke equaions using proper boundary condiions and long wave approimaion Developed equaions for ree layers can be easily ransformed o equaions of any number of layers, considering e effecs of adjacen layers Evenually ree separae equaions one for eac layer were derived From ese derived equaions wave celeriy for eac layer wave was eraced and presened Caracerisics of wave celeriy for eac layer were discussed in relaion o relaive densiy andor relaive layer dep In regards o effec of adjacen layer on a paricular layer, i is found a e upper layer wave equaion is only affeced by waer surface elevaion of inermediae layer However, inermediae layer equaion is affeced by waer surface elevaion of boom layer, spaial cange of elevaion of upper layer and densiy of upper layer Boom layer equaion is affeced by spaial canges and densiies of all e layers above is layer A generalized sysem of governing equaions, wic will be able o calculae wave properies for any number of layers, can be developed from ese equaions A numerical finie difference model of developed equaions will be developed and simulaed For numerical simulaions, as Saggered Leap Frog sceme produced very good resuls for a wolayer long wave sysem, i epeced a same numerical sceme would be able o produce good resuls for a mulilayered sysem as well REFERENCES Hampon, M H 97 Te Role of Subaqueous Debris Flow in Generaing Turbidiy Currens, J Sedimenary Perology, Vol 4, No 4, pp77579 Harbiz, C 99 Numerical Simulaions of Slide Generaed Waer Waves, Sci of Tsunami Hazards, Vol 9, No, pp 5 Imamura, F & Imeaz, M A 995 Long Waves in TwoLayers: Governing Equaions and Numerical Model, Sci of Tsunami Hazards, Vol, No, pp 4 Imeaz, M A & Imamura, F 00 A Nonlinear Numerical Model for Sraified Tsunami Waves and is Applicaion, Sci of Tsunami Hazards, Vol 9, pp 5059 Jiang, L & Leblond, P H 99 Te Coupling of Submarine Slide and e Surface Waves wic i Generaes, J Geopys Res, Vol 97, No C8, pp7744 Lyne, P & Liu, PLF 004 Linear Analysis of e Mulilayer Model, Journal of Coasal Engineering, Vol 5, pp Madsen, PA, Bingam, HB & Liu, H 00 A New Boussinesq Meod for Fully Nonlinear Waves from Sallow o Deep Waer, J Fluid Mec, Vol 46, pp 0 Science of Tsunami Hazards, Vol 8, No, page

15 Parker, G 98 Condiions for Igniion of Caasropically Erosive Turbidiy Currens, Marine Geology, Vol 46, pp 077 Simpson, JE 98 Graviy Currens in e Laboraory, Amospere and Ocean, Ann Rev Fluid Mec, Vol 4, pp 4 NOTATIONS = Densiy of fluid M = Discarge per uni wid of flow = Waer surface elevaion above sill waer level D = Toal dep of layer = Raio of densiy of upper layer fluid o lower layer fluid = Disance along downsream direcion y = Disance perpendicular o direcion u = Uniform velociy over e dep along direcion v = Uniform velociy along ydirecion g = Acceleraion due o graviy P = Hydrosaic pressure of fluid = Raio of deps of lower layer o upper layer C = Wave celeriy Science of Tsunami Hazards, Vol 8, No, page