A vertically non-uniform temperature approach for the friction term computation in depth-averaged viscoplastic lava flows

Jou nal o Compu a ional Physics 519 (2024) 113378 A ailable online 30 Augus 2024 0021-9991/© 2024 The Au ho (s). Published by Else ie Inc. This is an open access a icle unde he CC BY-NC license (h p://c ea i ecommons.o g/licenses/by-nc/4.0/). Con en s lis s a ailable a ScienceDi ec Jou nal o Compu a ional Physics jou nal homepage: www.else ie .com/loca e/jcp A e ically non-uni o m empe a u e app oach o he ic ion e m compu a ion in dep h-a e aged iscoplas ic la a flows J. O ega-Moyaa,∗, S. Ma ínez-A andaa, J. Fe nández-Pa ob, P. Ga cía-Na a oa aFluid Dynamic Technologies -I3A, Uni e si y o Za agoza, Za agoza, Spain bEs ación Expe imen al Aula Dei, CSIC, Za agoza, Spain A R T I C L E I N F O A B S T R A C T Keywo ds: Viscoplas ic shallow flows Bingham model Liquefied la a flows Dep h-a e aged non-New onian models Tempe a u e exchange Fini e olume me hods Recen ly, dep h-a e aged shallow flow models ha e been adap ed o modelling liquefied la a flows, gene ally cha ac e ized by a ma ked empe a u e-dependen non-New onian heology. Modelling hese complex flows equi es o include he effec s o he dep h-a e aged empe a u e g adien s in he go e ning equa ions. The mos significan e m o co ec ly p edic he la a mobili y is he flow esis ance e m, which is widely es ima ed using he linea iscoplas ic Bingham model. This non-New onian model allows o ela e he bed shea s ess o he dep h- a e aged la a flow ea u es by means o a cubic equa ion wi h analy ical solu ion when assuming a uni o m empe a u e dis ibu ion along he e ical p ofile. Ne e heless, he la a empe a u e is non-uni o m along he e ical due o he hea ans e a he bo om and he ee su ace, and hence he classical cubic Bingham model is no alid anymo e. In his wo k, a dep h- a e aged shallow flow model is adap ed o ealis ic la a flows conside ing influence o he non- uni o m e ical empe a u e p ofile in he non-New onian esis ance. This equi es o modi y he heological iscoplas ic models o ensu ing he coupling be ween flow dynamics and empe a u e e olu ion. Th ee non-uni o m empe a u e e ical dis ibu ions a e conside ed: linea , piece-wise and diffusion p ofiles. Syn he ic es s a e used o show he influence o he empe a u e e ical p ofile on he nume ical esul s. Fu he mo e, labo a o y expe imen al da a a e used o alida e his no el iscoplas ic esis ance o mula ion and o show ha he calib a ion o i s pa ame e s is possible. 1. In oduc ion Dep h-a e aged shallow flow models a e widely used o he nume ical simula ion o hyd aulic/hyd ological ee su ace p ocesses, such as i e and coas al flows, unoff gene a ion in s eep slopes, as floods, sunami wa es, highly e osi e sedimen -laden flows, muddy slu ies and deb is flows [1–5]. Ne e heless, he dep h-a e aged app oach has also been applied o gene alized g a i y-d i en ee-su ace flows wi h complex beha iou , such as iscous and iscoplas ic flows [6]o oil spills o e land [7]. Recen ly, special a en ion has been dedica ed o modelling la a flows, which a e cha ac e ized by a empe a u e-dependen heology [8,9]. Usually, modelling hese complex flows equi es o include he effec s o he dep h-a e aged empe a u e g adien s in he go e ning equa ions [10,11]. Al hough he la a densi y sligh ly depends on he empe a u e, he main effec o he empe a u e a ia ion is he change o he heological p ope ies which con ol he flow mobili y. * Co esponding au ho . E-mail add ess: [email p o ec ed] (J. O ega-Moya). h ps://doi.o g/10.1016/j.jcp.2024.113378 Recei ed 4 Decembe 2023; Recei ed in e ised o m 30 July 2024; Accep ed 24 Augus 2024 Jou nal o Compu a ional Physics 519 (2024) 113378 2 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. Despi e he complexi y o cha ac e izing la a heology [12], he mos widesp ead model o add ess he mo phological bulk beha iou o la a flows is he linea iscoplas ic Bingham model [13,14]. This simplified cons i u i e model is de i ed om he gen- e alized He schel-Bulkley model o non-linea iscoplas ic fluids [15] conside ing ha he shea s ess 𝜏e ol es wi h he s ain a e, i.e. he a e o de o ma ion 𝛾, depending on wo empe a u e-dependen pa ame e s: he yield s eng h 𝜏𝑦 o he mo ion ini ializa ion and he Bingham iscosi y 𝜇accoun ing o he fluid consis ency. Usually, he dependence o hese heological pa ame e s on he em- pe a u e is add essed using exponen ial ela ions [16]. Since he flow s uc u e along he e ical coo dina e is los in dep h-a e aged models, he sui able applica ion o he Bingham cons i u i e model o es ima e he flow esis ance e m in ee-su ace iscoplas ic models equi es o exp ess he bounda y shea s ess a he bo om in e ace o he mo ing laye 𝜏𝑏as a unc ion o he bulk flow ea u es, i.e. he flow dep h ℎand he dep h-a e aged eloci y componen s 𝑢and 𝑣. The widesp ead solu ion consis s o assuming ha he ansien flow s uc u e is he same as ha o a s eady iscoplas ic flow unning o e an infini e uni o m slope. This assump ion is o en e e ed o as he infini e landslide [17,18], since i was ini ially o mula ed o landslides wi h complex heology, and leads o he classical cubic Bingham closu e o iscoplas ic ee-su ace flows [19–21]. The de i a ion o he cubic Bingham model was made unde he hypo hesis ha he yield s eng h emains cons an along he e ical di ec ion [19]and hence i also implies ha he empe a u e is cons an along he la a flow column [16]. Howe e , he empe a u e along he e ical coo dina e is likely o show a non-uni o m p ofile due o he hea ans e ( empe a u e exchange) a he bo om and ee su ace in e aces. The hea exchange e m a he uppe and bo om in e aces can be included in shallow flows and nume ous empi ical and analy ical models ha e been p oposed. The simples hea exchange model only includes na u al con ec ion and adia ion a he su ace and hea conduc ion a he bo om [22]. On he o he hand, he mos complex empe a u e ans e models ake in o accoun a ious mechanisms such as na u al and o ced con ec ion, adia ion, ain and la en hea o phase change [23]. In gene al, expe imen al e idence [24,25] indica es ha he main mechanism o hea ans e h oughou he ee su ace and he bo om in e aces o la a flows is con ec ion whe eas, inside he la a laye , empe a u e ans e occu s mainly by conduc ion. Conce ning he nume ical modelling o he la a flow mobili y using he iscoplas ic Bingham model, he main d awback o conside ing a non-uni o m empe a u e p ofile along he e ical coo dina e is ha he heological pa ame e s, i.e. yield s eng h and iscosi y, change along he flow column. Hence, he widesp ead cubic closu e ela ionship o he iscoplas ic flow esis ance is no alid anymo e. In he las decade, he mos ema kable effo s o include he effec s o he e ical a ia ion o he fluid d i en-p ope ies on he dep h-a e aged iscoplas ic dynamics ha e been de eloped o he po e-fluid p essu e in landslides and deb is flows. Diffe en app oaches ha e been de eloped o model he e ical dissipa ion o po e p essu e, om conside ing a ime- dependen linea non-hyd os a ic p essu e p ofile [26,27] o de i ing mo e complex quad a ic shape unc ions o he non-hyd os a ic componen based on he dila ion o he solid phase [28,29]. Fu he mo e, [17,30] p oposed a diffusion equa ion o po e p essu e ha was sol ed using a fini e diffe ence e ical mesh, allowing o model changes o bounda y condi ions a he bo om su ace. Al hough all hese wo ks we e ocused on he e ical a ia ion o he po e p essu e in sedimen -laden flows, he p oposed app oach can be ecalled o analyze he effec s o e ical empe a u e dis ibu ion on he dynamics o he mal-d i en iscoplas ic flows. The main goal o his wo k is o de i e a new dep h-a e aged shallow flow model [9] o ealis ic la a flows conside ing he influ- ence o he non-uni o m e ical empe a u e p ofile in he non-New onian esis ance a he bo om su ace. This equi es o modi y he heological iscoplas ic model o ensu ing he coupling be ween flow dynamics and empe a u e e olu ion. The dependence o he bo om shea s ess on he dep h-a e aged flow ea u es should be econs uc ed, achie ing a no el co ela ion be ween he flow mobili y and he empe a u e g adien s. The ou line o he ex is as ollows: fi s , in Sec ion 2, he dep h-a e aged go e ning equa ions a e ecalled wi h a pa icula emphasis on he gene a ion o a non-New onian esis ance model unde diffe en heological condi ions. The consequences o assuming a a iable empe a u e p ofile along he fluidized la a column a e highligh ed and a new esis ance model is p oposed in ha si ua ion. Then, he nume ical me hod used o sol e he go e ning equa ions is summa ized in Sec ion 4. Nume ical es cases a e included in Sec ion 5. Syn he ic es s a e used o show he influence o he assumed empe a- u e p ofile on he nume ical esul s, whe eas labo a o y expe imen al da a a e used o alida e ou o mula ion and o show ha a calib a ion o i s pa ame e s is possible. Finally, he main conclusions a e d awn in Sec ion 6. 2. Go e ning equa ions o iscoplas ic he mal-d i en su ace flows The gene al dep h-a e aged shallow flow equa ions can be de i ed by in eg a ing he Na ie -S okes sys em along he z-coo dina e om he bo om su ace 𝑧𝑏 o he ee su ace 𝑧𝑠=𝑧𝑏+ℎ, whe e ℎdeno es he fluid dep h. The mass-conse a ion law eads: 𝜕(𝜌ℎ) 𝜕𝑡 +𝜕 𝜕𝑥(𝜌ℎ𝑢)+ 𝜕 𝜕𝑦(𝜌ℎ𝑣)=0,(1) whe e 𝜌is he dep h-a e aged densi y whe eas 𝑢and 𝑣a e he componen s o he dep h-a e aged eloci ies along he 𝑥−and 𝑦−coo dina es, espec i ely. The momen um equa ions along he 𝑥−and 𝑦−di ec ions ake he o m: 𝜕(𝜌ℎ𝑢) 𝜕𝑡 +𝜕 𝜕𝑥 (𝜌ℎ𝑢2+1 2𝑔𝜌ℎ2)+𝜕 𝜕𝑦 (𝜌ℎ𝑢𝑣)=−𝑔𝜌ℎ 𝜕𝑧𝑏 𝜕𝑥 −𝜏𝑏𝑥 (2) 𝜕(𝜌ℎ𝑣) 𝜕𝑡 +𝜕 𝜕𝑥 (𝜌ℎ𝑢𝑣)+𝜕 𝜕𝑦 (𝜌ℎ𝑣2+1 2𝑔𝜌ℎ2)=−𝑔𝜌ℎ 𝜕𝑧𝑏 𝜕𝑦 −𝜏𝑏𝑦 (3) Jou nal o Compu a ional Physics 519 (2024) 113378 3 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. being 𝑔 he g a i y accele a ion and (𝜏𝑏𝑥, 𝜏𝑏𝑦) he componen s o he basal esis ance ec o 𝝉𝒃. No e ha he effec s o he densi y a ia ion along he flow dep h a e neglec ed in equa ions (2)-(3), in o de o simpli y he o mula ion o he con ec i e momen um fluxes and he hyd os a ic p essu e e ms [15]. Finally, he dep h-a e aged empe a u e equa ion is conside ed: 𝜕(𝜌ℎ𝑇) 𝜕𝑡 +𝜕(𝜌ℎ𝑇 𝑢) 𝜕𝑥 +𝜕(𝜌ℎ𝑇 𝑣) 𝜕𝑦 = 𝑄 𝑐𝑃 ,(4) whe e 𝑇is he dep h-a e aged empe a u e,  𝑄 he hea flux h oughou he uppe and bo om in e aces and 𝑐𝑃 he specific hea o he fluidized ma e ial. A he ee su ace, he hea ans e wi h he ai is mainly go e ned by con ec ion and adia ion exchange fluxes,  𝑄𝑐𝑜𝑛𝑣 and  𝑄𝑟𝑎𝑑 espec i ely. Fo he con ec ion mechanism, a simple law is assumed:  𝑄𝑐𝑜𝑛𝑣 =ℎ𝑐(𝑇𝑠−𝑇𝑎𝑖𝑟)(5) whe e ℎ𝑐is he con ec ion coefficien , 𝑇𝑎𝑖𝑟 deno es he ai empe a u e and 𝑇𝑠is he empe a u e o he flow a he ee su ace. Fo he adia ion mechanism, he S e an-Bol zmann Law is applied:  𝑄𝑟𝑎𝑑 =𝜖𝜎(𝑇4−𝑇4 𝑎𝑖𝑟)(6) whe e 𝜖is he su ace emissi i y and 𝜎=5.67 ⋅10−8 𝑊∕(𝑚2𝐾−4)is he S e an-Bol zmann cons an . Al hough he alue o 𝜖is s ill objec o deba e, expe imen al measu emen s in [24] sugges a alue o 0.74, whils o he au ho s p e e a alue nea 0.90-0.95 ([23]). When wo king wi h dep h-a e aged equa ions, he influence o he iscous s ess enso a ises in he momen um equa ion in he o m o he shea s ess ec o a he bo om su ace, i.e. he basal esis ance ec o 𝝉𝒃, which can be exp essed as: 𝝉𝒃=(𝜏𝑏𝑥,𝜏𝑏𝑦)=𝜏𝑏𝐧𝐮,(7) whe e 𝜏𝑏is he basal shea s ess and 𝐧𝐮=(𝑛𝑢𝑥, 𝑛𝑢𝑦) =(𝑢, 𝑣)∕√𝑢2+𝑣2is he flow uni ec o defined by he dep h-a e aged eloci y. I is essen ial o find a closu e ela ion o e alua e he shea esis ance a he bo om as a unc ion o he dep h-a e aged flow a iables. This ela ion will depend on he heological beha iou o he fluidized ma e ial. 2.1. Dep h-a e aged app oach o he iscoplas ic basal esis ance In o de o find a gene alized closu e ela ion o he basal shea s ess 𝜏𝑏, we conside he gene al exp ession o he iscous s ess enso : 𝝉=Φ(𝐼2𝐷)𝐃,(8) whe e 𝐃 ≡𝐷𝑖𝑗 =1 2(𝜕𝑗𝑢𝑖+𝜕𝑖𝑢𝑗)is he s ain a e enso and Φis a scala unc ion o 𝐼2𝐷=1 2T (𝐃2). In he case o he gene alized He schel-Bulkley iscoplas ic model, his unc ion can be exp essed [15]as: Φ(𝐼2𝐷)= 𝜏0 √𝐼2𝐷 +2𝐾Φ(4𝐼2𝐷) 𝑚−1 2,(9) whe e 𝜏0, 𝐾Φand 𝑚a e pa ame e s o be de e mined depending on he heological beha iou o he fluid. The infini e landslide me hod [17] assumes ha he flow s uc u e co esponds o ha o a simple shea s eady flow, whe e he eloci y along he flow column is almos ho izon al and only depends on he e ical posi ion: 𝐮 =(𝑢, 𝑣, 0) =(U(𝑧) 𝑛𝑢𝑥, U(𝑧) 𝑛𝑢𝑦, 0), being U(𝑧) =√𝑢2+𝑣2 he magni ude o he ho izon al eloci y, he iscoplas ic s ess enso can be exp essed as: 𝝉=⎛⎜⎜⎝ 00𝜏(𝑧)𝑛𝑢𝑥 00𝜏(𝑧)𝑛𝑢𝑦 𝜏(𝑧)𝑛𝑢𝑥 𝜏(𝑧)𝑛𝑢𝑦 0⎞⎟⎟⎠ ,(10) whe e 𝑧 ∈[0, ℎ]deno es he e ical coo dina e abo e he bo om su ace and 𝜏(𝑧)is he shea s ess along he flow column: 𝜏(𝑧)=Φ(𝐼2𝐷)1 2 𝑑U 𝑑𝑧 ,(11) 𝐼2𝐷=1 4(𝑑U 𝑑𝑧 )2 .(12) Then, a simplified linea shea s ess p ofile is assumed along he flow column as ollows: 𝜏(𝑧)=𝜏𝑏(1− 𝑧 ℎ),(13) whe e 𝜏𝑏=𝜏(𝑧 =0)is he alue o he esis ance a he bo om su ace. I is no ewo hy ha he linea shea s ess p ofile only ully holds in he case o e ically uni o m flow densi y. I he densi y weakly depends on ano he fluid d i en-p ope y, such as empe a u e, equa ion (13)should esul in a good app oxima ion o he s ess p ofile unde he infini e flow hypo hesis [17]. This Jou nal o Compu a ional Physics 519 (2024) 113378 4 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. assump ion is commonly used o de i e ela ionships o he basal shea s ess in dep h-a e aged models and will be implici in he ollowing iscoplas ic models. The shea s ess (13)is in eg a ed along he flow column o ob ain he eloci y p ofile U(𝑧 | ℎ, 𝜏𝑏), which depends on he basal shea s ess and he flow dep h. The eloci y p ofile is in eg a ed again o de i e an exp ession o he flow discha ge in ol ing he dep h-a e aged eloci y magni ude U(ℎ, 𝜏𝑏), allowing o compu e he basal shea s ess 𝜏𝑏 om he shallow flow ea u es as: U(ℎ,𝜏𝑏)= 1 ℎ ℎ ∫ 0 U(𝑧|ℎ,𝜏𝑏)𝑑𝑧 ⟹𝜏𝑏=(ℎ,U).(14) This p ocedu e is he me hod by which he mos common iscoplas ic esis ance ela ions o shallow flow equa ions ha e been de i ed [15], such as he classical Bingham model. I is wo h men ioning ha hese dep h-a e aged ic ion models should depend on he fluid d i en-p ope ies, i.e. he empe a u e 𝑇 o la a flows, no only h ough he a ia ion o he heological pa ame e s in (9)bu also h ough he densi y a ia ions. Howe e , he densi y dependence is assumed negligible due o he selec ion o a simplified linea shea s ess p ofile along he flow column. 3. Viscoplas ic esis ance models In his sec ion we in oduce he p ocedu e o de elop a unc ional ela ionship be ween he bed shea s ess and he dep h-a e aged flow ea u es o non-uni o m empe a u e dis ibu ions along he e ical coo dina e. Fi s , he basic case o uni o m empe a u e in he flow column is conside ed. Then, diffe en non-uni o m e ical empe a u e p ofiles a e add essed. 3.1. Classic Bingham model A common model o desc ibe iscoplas ic he mally-con olled fluids, such as la a, is he linea Bingham model. This is a pa icula case o he He schel-Bulkley model (9)using 𝜏0=𝜏𝑦, 𝐾Φ=𝜇and beha iou index 𝑚 =1, leading o: 𝜏(𝑧)=𝜏𝑦+𝜇𝑑U 𝑑𝑧 .(15) He e, 𝜏𝑦 ep esen s a yield s ess which is he minimum s ess necessa y o de o m he fluid and 𝜇is he dynamic iscosi y coefficien . Equa ion (15)is exp essed assuming ha 𝑑𝑈 𝑑𝑧 >0and hence 𝜏(𝑧) >𝜏 𝑦. The classical Bingham- ype models assume ha he yield s eng h and he iscosi y a e uni o m along he e ical coo dina e, i.e. ha he fluid empe a u e is homogeneous along he flow column. When 𝜏𝑏≤𝜏𝑦, hen U=0, o he wise, i 𝜏𝑏>𝜏 𝑦, his hypo hesis allows o exp ess he a ia ion o he eloci y along he e ical coo dina e 𝑧 ∈[0, ℎ]as: 𝑑U 𝑑𝑧 ={0i 𝑧0<𝑧<ℎ 1 𝜇[𝜏𝑏(1− 𝑧 ℎ)−𝜏𝑦]i 0<𝑧<𝑧 0 (16) whe e 𝑧0=ℎ (1− 𝜏𝑦 𝜏𝑏)deno es he shea laye dep h, i.e. he heigh whe e 𝜏(𝑧)becomes smalle han he yield s ess 𝜏𝑦. Hence, in he plug egion 𝑧 ∈[𝑧0, ℎ]abo e he shea laye , he s ain a e becomes null and he eloci y is uni o m. The e o e, he eloci y p ofile U(𝑧)is s aigh o wa d: U(𝑧)=⎧ ⎪ ⎨ ⎪ ⎩ 𝜏𝑏ℎ 2𝜇(1− 𝜏𝑦 𝜏𝑏)2i 𝑧0<𝑧<ℎ 𝜏𝑏−𝜏𝑦 𝜇𝑧−𝜏𝑏 2ℎ𝜇 𝑧2i 0<𝑧<𝑧 0 (17) and he flow discha ge can be exp essed as: ℎU= ℎ ∫ 0 U(𝑧)𝑑𝑧 =ℎ2 2𝜇(2 3𝜏𝑏−𝜏𝑦+ 𝜏3 𝑦 3𝜏2 𝑏),(18) om whe e we can finally find a cubic closu e ela ion o he basal esis ance 𝜏𝑏(ℎ, U): (𝜏𝑦 𝜏𝑏)3 −(3+ 6𝜇U 𝜏𝑦ℎ)𝜏𝑦 𝜏𝑏 +2=0 (19) which is alid o 𝜏𝑏≥𝜏𝑦and is usually ew i en as: U ℎ=⎧ ⎪ ⎨ ⎪ ⎩ 0i 𝜏𝑏<𝜏 𝑦 1 3(𝜏3 𝑦 2𝜇𝜏2 𝑏 −3𝜏𝑦 2𝜇+𝜏𝑏 𝜇)i 𝜏𝑏≥𝜏𝑦 (20) Jou nal o Compu a ional Physics 519 (2024) 113378 5 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. Fig. 1. Dependence o he fluid heological p ope ies on he empe a u e. I is wo h no ing ha he de i a ion o he classical Bingham cubic closu e (19)is made assuming ha he yield s eng h 𝜏𝑦and he iscosi y 𝜇a e uni o m along he flow column, as well as he flow densi y 𝜌. This equa ion will be used o calcula e he heo e ical uni o m empe a u e p ofile o compa a i e pu poses. Fu he mo e, some au ho s compu ed he solu ion o he bed shea s ess 𝜏𝑏 in he cubic equa ion (19)using i e a i e me hods, such as New on-Raphson, o deploying app oxima e quad a ic equa ions o he o iginal Bingham in o de o imp o e he obus ness o he bed esis ance es ima ion [20,31,32]. The Bingham ela ion (20) can be eo de ed in dimensionless e ms as: 𝜇U 𝜏𝑦ℎ=⎧ ⎪ ⎨ ⎪ ⎩ 0i 𝜏𝑏∕𝜏𝑦<1 1 3(𝜏2 𝑦 2𝜏2 𝑏 −3 2+𝜏𝑏 𝜏𝑦)i 𝜏𝑏∕𝜏𝑦≥1(21) which can be simplified o cases whe e 𝜏𝑏∕𝜏𝑦>3∕2 o he linea dimensionless exp ession: 𝜇U 𝜏𝑦ℎ=⎧ ⎪ ⎨ ⎪ ⎩ 0i 𝜏𝑏∕𝜏𝑦<3∕2 1 3(𝜏𝑏 𝜏𝑦−3 2)i 𝜏𝑏∕𝜏𝑦≥3∕2 (22) The exp ession (22)could be conside ed as a good app oxima ion o he classical cubic Bingham s ess cu e o alues 𝜏𝑏∕𝜏𝑦≥2, allowing an easie compu a ion o he flow esis ance in hese cases. Fo he pa icula pu e- iscous case, he yield s eng h is null (𝜏𝑦=0) and he simple New onian fluid 𝜏𝑏=3𝜇U∕ℎis ound. Fu he mo e, i he iscous componen is null, i.e. pu e-cohesi e fluids, he eloci y p ofile canno be ob ained using he infini e landslide hypo hesis unless an addi ional assump ion is made. In his sense, iscosi y egula izes he p oblem [17], allowing o de i e he e ical eloci y p ofile. 3.2. Va iable coefficien models In he case o a empe a u e-dependen Bingham fluid, he hypo hesis o uni o m heological p ope ies along he e ical coo - dina e is only sa isfied i he fluid empe a u e is also uni o m. Howe e , he hea exchange o he flow wi h he bo om laye and he uppe a mosphe e leads o a non-uni o m empe a u e p ofile in he flow column and hence he classical Bingham closu e is no alid anymo e. 3.2.1. Tempe a u e-dependen iscoplas ic coefficien s To find no el closu e ela ion o he basal shea esis ance, fi s , i is necessa y o o mula e he a iabili y o iscosi y and yield s ess wi h he fluid empe a u e. In his ex we will conside a ew models, bu he p ocedu e is easily ex ended o o he s. Fo he empe a u e-dependen iscosi y 𝜇(𝑇), he And ade model [33,34,9] eads: 𝜇(𝑇)=𝐴𝜇exp (𝐵𝜇∕𝑇),(23) being 𝐴𝜇[𝑃𝑎 ⋅𝑠]and 𝐵𝜇[◦𝐾] heological fi ing pa ame e s, whe eas he empe a u e-dependen yield s ess 𝜏𝑦(𝑇)is exp essed using he ollowing exponen ial ela ion [35,9]: 𝜏𝑦(𝑇)=𝐴𝜏+𝐵𝜏exp (𝐶𝜏𝑇),(24) whe e 𝐴𝜏[𝑃𝑎], 𝐵𝜏[𝑃𝑎], 𝐶𝜏[1∕◦𝐾]a e fi ing pa ame e s. Fig. 1depic s he a ia ion o he fluid iscosi y and yield s eng h on he empe a u e, depending on he heological pa ame e s conside ed. When he empe a u e p ofile 𝑇(𝑧)is non-uni o m, he shea s ess ela ion (15) ob ained is mo e complica ed, as i includes dependencies on he 𝑧−coo dina e in he heological pa ame e s: Jou nal o Compu a ional Physics 519 (2024) 113378 6 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. 𝜏(𝑇(𝑧),𝑧)=𝜏𝑦(𝑇(𝑧))+𝜇(𝑇(𝑧))𝑑U 𝑑𝑧 (25) In oducing he exp essions o 𝜏𝑦(𝑇)and 𝜇(𝑇), he s ain a e along he flow column is exp essed as: 𝑑U 𝑑𝑧 =⎧ ⎪ ⎨ ⎪ ⎩ 0i 𝜏(𝑧)<𝜏 𝑦(𝑇(𝑧)) 𝜏𝑏(1− 𝑧 ℎ)−[𝐴𝜏+𝐵𝜏exp (𝐶𝜏𝑇(𝑧))] 𝐴𝜇exp (𝐵𝜇∕𝑇(𝑧))i 𝜏(𝑧)≥𝜏𝑦(𝑇(𝑧))(26) The assump ion o empe a u e-dependen heological pa ame e s causes ha a clea diffe en ia ion o he shea laye and he plug zone along he flow column is no s aigh o wa d. The e o e, he e ical flow s uc u e depends on he local dynamics ea u es and he empe a u e p ofile. The in eg a ion o he s ain a e p ofile (26)is gene ally non-elemen a y and becomes impossible o comple e he p ocess as o iginally s a ed o he classical Bingham hypo hesis. E en conside ing ha he empe a u e p ofile 𝑇(𝑧)can be ully de e mined om he flow dep h ℎand he dep h-a e aged em- pe a u e 𝑇, p oposing a unc ional dependency o 𝜏𝑏=(𝑇, ℎ, U) may be a complex p ocess. The e o e, a dimensional analysis is pe o med in o de o simpli y he p oblem by applying he Vaschy-Buckingham ΠTheo em [36]. Fi s , we define he no malized a iables: 𝑧′=𝑧 ℎU′=U U 𝑇′=𝑇 𝑇ℎ (27) and we conside ha dimensionless empe a u e dis ibu ion 𝑇′(𝑧′)is known and ully cha ac e ized by a se o dimensionless pa- ame e s , which include he uni ocal shape- ela ion be ween he dep h-a e aged empe a u e 𝑇and he e e ence maximum empe a u e 𝑇ℎ. This e e ence empe a u e is conside ed cons an and mus be defined depending on he ype o e ical empe a- u e p ofile selec ed. The Vaschy-Buckingham ΠTheo em s a es ha , conside ing 𝑛physical pa ame e s in ol ed in a p oblem, hey can be combined in (𝑛 −𝑟)dimensionless a iables Π1, … Π𝑛−𝑟, whe e 𝑟is he ank o he dimensional ma ix o exponen s in he base o undamen al a iables in ol ed. In his case, he e exis 𝑛 =10dimensional pa ame e s {𝐴𝜇, 𝐵𝜇, 𝐴𝜏, 𝐵𝜏, 𝐶𝜏, 𝑇, 𝑇ℎ, U, ℎ, 𝜏𝑏}, while 𝑟 =4dimen- sions {leng h, ime, mass, empe a u e}a e in ol ed. The e o e, we can simpli y he dependencies o six dimensionless pa ame e s, Π1, Π2, ..., Π6, defined as: Π1=𝜏𝑏 𝐵𝜏 Π2=𝐴𝜇U 𝐵𝜏ℎΠ3=𝑇 𝑇ℎ (28) Π4=𝐴𝜏 𝐵𝜏 Π5=𝐶𝜏𝑇ℎΠ6=𝐵𝜇 𝑇ℎ (29) and he s ain a e along he e ical coo dina e (26) can be ew i en in dimensionless o m as: 𝑑U′ 𝑑𝑧′=Π1(1 − 𝑧′)−Π 4−exp(Π5𝑇′) Π2exp (Π6∕𝑇′)(30) Hence he no malized eloci y de i a i e (30) can be in eg a ed along he e ical coo dina e and he Π heo em ensu es he exis ence o he ela ionship: Π2=𝑓(Π1,Π4,Π5,Π6,)(31) whe e includes he dimensionless shape pa ame e Π3 o he no malized empe a u e e ical dis ibu ion 𝑇′(𝑧′). Howe e , once he heological pa ame e 𝐴𝜇, 𝐵𝜇, 𝐴𝜏, 𝐵𝜏, 𝐶𝜏o he fluid a e fixed and he e e ence maximum empe a u e 𝑇ℎis defined, he numbe s Π4, Π5, and Π6become cons an . Consequen ly, he dependency educes o an equa ion o he o m: Π2= 𝑓(Π1,)(32) and only he se {𝜏𝑏, ℎ, 𝑇}mus be specified. The e o e, se ing cons an dimensionless numbe s {Π4, Π5, Π6}, he dimensionless de i a i e (30) can be in eg a ed wice o find he a e aged eloci y U. I his in eg a ion is epea ed o a wide ange o {𝜏𝑏, ℎ, 𝑇}, he unc ional ela ionship  𝑓in (32) can be ound. Acco dingly, in his wo k, he ollowing p ocedu e o app oxima ing he bed shea s ess 𝜏𝑏is p oposed: 1. A se o disc e e alues o ℎ, 𝑇and 𝜏𝑏is conside ed, aking alues in some in e als o in e es : {𝑇1, 𝑇 2,…,𝑇 𝑁𝑇}⊂[𝑇min, 𝑇 max] {ℎ1,ℎ2,…,ℎ𝑁ℎ}⊂(0,ℎmax)] {𝜏1 𝑏,𝜏2 𝑏,…,𝜏𝑁𝜏 𝑏}⊂[𝜏𝑦,𝜏𝑏max] Jou nal o Compu a ional Physics 519 (2024) 113378 7 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. Fig. 2. No malized linea empe a u e p ofile wi h 𝑇ℎ= 300 Kand𝑇= 240 K. Table 1 Two diffe en se s o pa ame e s o desc ibe fluids wi h diffe en heolog- ical pa ame e . 𝐴𝜏(Pa) 𝐵𝜏(Pa) 𝐶𝜏(K−1)𝐴𝜇(Pa⋅s) 𝐵𝜇(K) Fluid 1 10 2000 -0.02 0.08 1200 Fluid 2 0 5.6⋅106-5.8⋅10−3 1.77 9500 2. The eloci y p ofile U(𝑧)is nume ically in eg a ed om (26) o each combina ion o pa ame e s. The e o e, 𝑁𝑇×𝑁ℎ×𝑁𝜏 diffe en nume ical eloci y p ofiles U(𝑧 | 𝑇, ℎ, 𝜏𝑏)a e compu ed. 3. Each eloci y p ofile is in eg a ed nume ically in o de o ob ain he dep h-a e aged eloci y U𝑖𝑗𝑘 =U(𝑇𝑖, ℎ𝑗, 𝜏𝑘 𝑏). The esul o his s ep is a se o uples {(𝑇𝑖, ℎ𝑗, 𝜏𝑘 𝑏, U𝑖𝑗𝑘)} wi h 𝑖 =1, 2, ..., 𝑁𝑇, 𝑗=1, 2, ..., 𝑁ℎand 𝑘 =1, 2, ..., 𝑁𝜏. 4. A unc ional dependency 𝜏𝑏=(𝑇, ℎ, U) is p oposed and fi ed o he disc e e alues p e iously compu ed. 3.2.2. Linea empe a u e model Fi s , he p oposed p ocedu e is illus a ed in he simples non-uni o m empe a u e p ofile co esponding o a linea empe a u e dis ibu ion along he flow column: 𝑇(𝑧)=𝑇ℎ−2(𝑇ℎ−𝑇)𝑧 ℎ(33) whe e he e e ence maximum empe a u e 𝑇ℎis cons an and se a he bed su ace 𝑧 =0(see Fig. 2). This is a special case o gene alized unc ional ela ionship (32)whe e he shape o he no malized empe a u e dis ibu ion 𝑇′(𝑧′)is ully desc ibed by he dimensionless pa ame e Π3=𝑇∕𝑇ℎ. Following he p ocedu e epo ed in he abo e sec ion, he eloci y p ofile in eg a ion is done o a wide se o alues o {𝜏𝑏, ℎ, 𝑇}. The esul s show ha , unde he condi ion ha 𝜏𝑏is conside ably la ge han 𝜏𝑦, i.e. 𝜏𝑏≫𝜏 𝑦, he dimensionless numbe Π2depends almos linea ly on Π1wi h a slope and axis-in e sec ion poin being unc ions o Π3. Hence, i can be exp essed as: Π2= 𝑓1(Π3)Π1+ 𝑓2(Π3).(34) As a final s ep, o simplici y, he fixed pa ame e s ha o m he dimensionless numbe s a e abso bed in o he unc ions  𝑓1and  𝑓2, esul ing in a dimensional ela ionship: U ℎ={0i 𝜏𝑏≤𝜏𝑏0 𝑓1(𝑇)𝜏𝑏+𝑓2(𝑇)i 𝜏𝑏>𝜏 𝑏0 (35) whe e 𝜏𝑏0=−𝑓2(𝑇) 𝑓1(𝑇) is an effec i e yield s ess. This linea dependence can be obse ed in he examples shown in Fig. 3 o wo diffe en pa ame e se s ep esen ing wo fluids wi h diffe en ia ed heology (see Table 1). To gene a e his g aph, 20 alues o bo h he basal shea s ess 𝜏𝑏and he flow dep h ℎwe e employed, along wi h 40 alues o dep h-a e aged empe a u e 𝑇, esul ing in almos 1.6 ⋅104in eg als ela ions. No all empe a u e alues a e ep esen ed, since he lines would no be dis inguishable, bu all alues o ℎa e plo ed. So each poin on he Fig. 3ac ually consis s o 20 o e lapping poin s co esponding o he alues o ℎ o a ce ain s ess and empe a u e. This confi ms he conclusion de i ed om he Π heo em: we can elimina e one pa ame e because Uand ℎa e no independen , i.e. only hei quo ien is ep esen a i e. I is wo h no ing he equi alence be ween he gene alized exp ession (35) o non-uni o m linea empe a u e dis ibu ion and he simplified ela ion o he uni o m empe a u e p ofile (22). Bo h o mulas s a e ha he ela ion be ween he bed shea s ess 𝜏𝑏and he dep h-a e aged eloci y Ucan be es ima ed by a linea ized app oxima ion (see Fig. 3). This simila i y pa ly explains Jou nal o Compu a ional Physics 519 (2024) 113378 8 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. Fig. 3. Linea dependency o U∕ℎwi h 𝜏𝑏 o he fluid heology in Table 1. Each line co esponds o a diffe en alue o a e aged empe a u e. Fig. 4. Fi ing o he empe a u e-dependen slope and axis-in e cep ion unc ions, 𝑓1(𝑇)and 𝑓2(𝑇), o he heological pa ame e s shown in Table 1. he s uc u e o he gene alized unc ional ela ionship (35). As pa icula case o (35), he New onian ela ion is he i ial choice, since i holds a pu e linea dependence o he shea s ess wi h he fluid iscosi y. Fo a classical Bingham fluid, he fi s e m on he igh -hand side in (20) becomes negligible compa ed o he second one o 𝜏𝑏≫𝜏 𝑦, and 𝜏𝑦and 𝜇a e unc ions o 𝑇, hence demons a ing he p oposed app oxima ed ela ionship. Thus, he p oblem has been educed o he linea fi ing o successi e lines a cons an empe a u e and he app oxima ion o wo unc ions o a single a iable, 𝑓1(𝑇)and 𝑓2(𝑇). I is no ewo hy ha he shape o hese unc ions a ies depending on he pa ame e alues used, bu a good fi can be easily ound. In pa icula , i has been obse ed ha unc ions ha wo k ela i ely well o a wide ange o diffe en pa ame e s a e: 𝑓1(𝑇)=𝐴1+𝐵1𝑇𝑛1(36) 𝑓2(𝑇)=𝐴2𝑇3+𝐵2𝑇2+𝐶2(37) whe e 𝐴1, 𝐵1, 𝑛1, 𝐴2, 𝐵2, and 𝐶2a e fi ing cons an s. The fi ing o 𝑓1(𝑇)and 𝑓2(𝑇) o he fluids conside ed in Table 1a e shown in Fig. 4. The significance o his esul is ha , i mo e complex empe a u e p ofiles a e es ed, he ela ionships ob ained he e hold app oxima ely, as i is shown in he nex sec ions. I is wo h no ing ha he ela ionships used in his wo k o he iscosi y and he yield s eng h, (23)and (24) espec i ely, can be eplaced by o he models wi h an a bi a y numbe o cons an pa ame e s. These addi ional pa ame e s will be abso bed in o cons an dimensionless numbe s in he same way ha occu ed wi h 𝐴𝜇, 𝐵𝜇, 𝐴𝜏, 𝐵𝜏and 𝐶𝜏. Howe e , o he dimensional analysis o be alid, he empe a u e p ofile needs o be unequi ocally desc ibed by he flow dep h ℎ, he dep h-a e age empe a u e 𝑇and an a bi a y numbe o cons an s wi h empe a u e uni s, which would ha e an analogous ea men o ha o 𝑇ℎ. This gi es some eedom o c ea e mo e ealis ic empe a u e dis ibu ions. Fu he mo e, he unc ions 𝑓1(𝑇)and 𝑓2(𝑇), in (36)and (37) espec i ely, do no ha e any physical backg ound and hey should be modified i equi ed by he model pa ame e s, he heology laws and he empe a u e p ofile chosen. Jou nal o Compu a ional Physics 519 (2024) 113378 9 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. Fig. 5. No malized piece-wise empe a u e p ofile wi h 𝑛=4,𝑇ℎ= 300 Kand𝑇= 270 K. Fig. 6. Compa ison be ween yield s ess and eloci y p ofiles in he case o uni o m empe a u e and a piece-wise p ofile (𝑛 =4, 𝑇ℎ= 300 K, 𝑇= 270 K) o an a bi a y heological configu a ion o pa ame e s (𝐴𝜏=10Pa, 𝐵𝜏= 2000 Pa, 𝐶𝜏=0.02 K−1 , 𝐴𝜇=0.08 Pa⋅s and 𝐵𝜇= 1200 K) wi h a basal s ess 𝜏𝑏=50Pa. 3.2.3. Piece-wise empe a u e model The na u al ex ension o he linea empe a u e p ofile is he piece-wise linea unc ion. A hea ans e will be assumed a he bo om and ee su aces so ha hey a e colde , wi h co esponding empe a u es 𝑇𝑏=𝑇𝑠, whe eas he maximum e e ence empe a u e 𝑇ℎis cons an and se a he cen e egion o he fluid column. Also, we also assumed ha he empe a u e g adien akes place in a laye o wid h 𝛿𝑇=ℎ∕𝑛. Hence, he empe a u e p ofile can be exp essed as: 𝑇(𝑧)=⎧ ⎪ ⎨ ⎪ ⎩ 𝑇𝑠+(𝑇ℎ−𝑇𝑠)ℎ−𝑧 𝛿𝑇i (ℎ−𝛿𝑇)<𝑧≤ℎ 𝑇ℎi 𝛿𝑇<𝑧≤(ℎ−𝛿𝑇) 𝑇𝑏+(𝑇ℎ−𝑇𝑏)𝑧 𝛿𝑇i 0<𝑧≤𝛿𝑇 (38) This model is simila o he one conside ed in [6]and we will ake 𝑛 =4assuming ha conduc ion as he unique hea ans e mechanism [8]. Bo h su ace empe a u es 𝑇𝑏=𝑇𝑠can be w i en in e ms o he dep h-a e aged empe a u e as 𝑇𝑏=𝑇𝑠=𝑛𝑇 − 𝑇ℎ(𝑛 −1), and hence he shape o he no malized empe a u e p ofile 𝑇′(𝑧′)is ully desc ibed by he dimensionless pa ame e Π3=𝑇∕𝑇ℎ. Fig. 5shows he g aphical ep esen a ion o he esul ing piece-wise p ofile. Fig. 6-(a) ep esen s he induced shea s ess 𝜏(𝑧)on he flow column and yield s eng h 𝜏𝑦(𝑧) o uni o m and piece-wise em- pe a u e dis ibu ions along he e ical di ec ion. An impo an diffe ence is obse ed be ween he uni o m empe a u e and he piece-wise empe a u e p ofile. While he o me shows an unique in e sec ion poin , sepa a ing he shea laye om he plug, he la e depic s wo in e sec ion poin s be ween he s ess p ofile 𝜏(𝑧)and he yield s ess p ofile 𝜏𝑦(𝑧). Hence, he eloci y unde he lowe in e sec ion poin is ze o, whe eas abo e he second in e sec ion poin he eloci y becomes cons an (plug zone), as i is plo ed in Fig. 6-(b). In e es ingly, such effec is simila o he o ma ion o wo c us s o solid ock abo e and below he liquid la a flows due he cooling p ocess. A his poin , we can epea he p ocess ollowed wi h he linea empe a u e p ofile (see Sec ion 3.2.2) in o de o fi he pa ame e s in he unc ional ela ionships (36)and (37). Then, he shea s ess 𝜏𝑏is compu ed om he model in (35). 3.2.4. Tempe a u e diffusion model o la a flows A mo e ealis ic empe a u e p ofile is now p oposed by using a simplified empe a u e diffusion model. I is no ewo hy ha he pu pose o his app oxima ion is no o p ecisely compu e he empe a u e dis ibu ion o fluid column bu o cap u e he main effec s o i s non-uni o mi y on he a e aged flow dynamics. The e o e, he p oposed model is a simplified app oach o he ealis ic desc ip ion Jou nal o Compu a ional Physics 519 (2024) 113378 16 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. Fig. 12. Tes 2. Basal s ess as a unc ion o empe a u e o U∕ℎ=1. Fig. 13. Tes 2 - F ee su ace ep esen a ion (𝑧𝑏+10ℎ) along he slope a =3450 s. Table 7 Tes 3 - Rheological pa ame e s used o each model. 𝐴𝜏(Pa) 𝐵𝜏(Pa) 𝐶𝜏(K−1)𝐴𝜇(Pa⋅s) 𝐵𝜇(K) Uni o m 0 35⋅106-0.01 0.05 9500 Linea 0 37⋅106-0.01 0.02 9500 Piece-wise 0 37⋅106-0.01 0.02 9500 Diffusion 0 37⋅106-0.01 0.02 9500 Fu he mo e, cases piece-wise L1 and piece-wise L2 a e almos indis inguishable, sugges ing ha conduc ion wi h he bed is negligible, which makes sense as i has a low he mal conduc i i y. 5.3. Tes 3: expe imen al case o la a flow o e 2D slope A benchma k p oposed in [42] o la a flow simula ion models is now used o alida ion pu poses. The es consis s o a 2D sloping plane wi h 𝑆0= 12, 25◦. O e his plane, a cons an flow o mel ed basal 𝑄𝑖𝑛 = 220 𝑚𝐿∕𝑠a 𝑇𝑖𝑛 = 1050 ◦𝐶is pou ed. Fo i s simula ion, he alues o emissi i y and con ec ion coefficien a e he ones sugges ed in he e e ence publica ion: 𝜖=0.95 and ℎ𝑐=2 W/(m2⋅K), oge he wi h a densi y unc ion 𝜌(𝑇) = 2350 −0.01(𝑇− 1073 K)kg/m3. The ic ion model is compa ed in he cases o uni o m empe a u e p ofile, aken as e e ence, and he linea , he piece-wise and he diffusion empe a u e p ofiles. Viscosi y and yield s ess pa ame e s a e chosen independen ly o uni o m empe a u e model and o he a iable coefficien s models, h ough a fi ing p ocess o he expe imen al da a. The selec ed heological pa ame e s a e p esen ed in Table 7. Fig. 16 displays he 2D map o dep h-a e aged empe a u e and flow dep h a ime 𝑡 =42 𝑠 o he piece-wise empe a u e p ofile. No e ha he cooling p ocess is as e o small dep h zones. Fig. 17-(a) depic s he e olu ion o he la a wa e along he longi udinal sec ion o he piece-wise empe a u e p ofile. Fu he - mo e, Fig. 17-(b) shows a compa ison be ween simula ed and obse ed wa e on loca ion o all he empe a u e dis ibu ions es ed. The heological pa ame e s we e se o ob ain he bes ag eemen be ween he compu ed wa e on loca ion and he obse ed da a. The coincidence be ween hem is accep able, al hough he e is a disc epancy due o a change o he flow on eloci y a 𝑡 =20 s app oxima ely. This change could be explained due o i egula i ies in he expe imen al pou ing. I is wo h no ing ha , in he o iginal wo k [42], all he conside ed models had he same disc epancy since he models we e se up o fi obse ed da a a e ha i egula i y occu ed. Jou nal o Compu a ional Physics 519 (2024) 113378 17 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. Fig. 14. Tes 2 - Flow dep h a = 20 s, = 200 s, = 1950 s and = 3450 s a e he damb eak. Fig. 15. Tes 2 - Dep h-a e aged empe a u e e olu ion along he damb eak a = 200 s and = 3450 s. Finally, Fig. 18-(a) shows he compa ison o he damb eak ee su ace a 𝑡 =42 𝑠compu ed wi h all he empe a u e dis ibu ions es ed. The e exis diffe ences be ween he piece-wise empe a u e p ofile and he o he empe a u e dis ibu ions, including he classical uni o m p ofile. These diffe ences a e clea ly e ealed when plo ing he empo al e olu ion o he flow dep h a 𝑥 =50 cm o all he empe a u e p ofile es ed in Fig. 18-(b). The expe imen al alues a e highe han he simula ion p edic ions, bu he piece-wise empe a u e model is able o ob ain sligh ly be e app oxima ion. 6. Conclusions The p esen wo k is de o ed o he o mula ion and nume ical esolu ion o a new dep h-a e aged iscoplas ic flow model wi h non-uni o m empe a u e a ia ion. A no el me hod has been de eloped and implemen ed o in oduce he effec s o a non-uni o m e ical empe a u e p ofile on he dep h-a e aged flow equa ions. The p oposed model has been disc e ized using a Fini e Volume Jou nal o Compu a ional Physics 519 (2024) 113378 18 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. Fig. 16. Tes 3 -2D maps o dep h-a e aged empe a u e and flow dep h o he piece-wise empe a u e p ofile case a =42 s. (Fo in e p e a ion o he colou s in he figu e(s), he eade is e e ed o he web e sion o his a icle.) Fig. 17. Tes 3 - Flow e olu ion along he cen al longi udinal sec ion. scheme ecen ly shown o be use ul o he efficien compu a ion o uns eady he mally-d i en shallow flows o e s eep opog aphy and wi h complex heology. Fu he mo e, he nume ical algo i hms a e implemen ed o mul i-CPU pa alleliza ion, dec easing hugely he compu a ion ime equi ed o pe o m high- esolu ion simula ions. This imp o es he pa ame e calib a ion p ocess, makes longe simula ions easible and acili a es he s udy o he sensi i i y wi h espec o he model pa ame e s. The no el non-uni o m empe a u e model o he iscoplas ic esis ance has been es ed in bo h 1D analy ical and 2D expe imen al cases. F om hese benchma k es s i is concluded ha he p oposed non-uni o m empe a u e p ofile modifica ion undoub edly has effec s ha can become e y ele an in he bulk flow dynamics. I has been obse ed ha , o low empe a u es, a non-uni o m empe a u e p ofile leads o lowe ic ion con ibu ion. This effec is explained because he inne laye s o he flow column a e ho e and hence mo e fluidized han unde he uni o m empe a u e hypo hesis. Fo high empe a u es, he beha iou depends on he pa ame e s and p ofile assumed, being possible o ha e he e e se beha iou . The eason o his phenomenon is ha he effec s associa ed wi h apid cooling o he uppe and lowe in e aces o he flow column can exceed he esis ance o a flow wi h uni o m high- empe a u e p ofile. The esul s also sugges ha , h ough a p ocess o calib a ing he pa ame e s indi idualized o each model, he esul s can be e y simila . This esul sugges s ha use o uni o m empe a u e models is subjec ed o a sui able calib a ion o he model pa ame e s, Jou nal o Compu a ional Physics 519 (2024) 113378 19 J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al. Fig. 18. Tes 3 - Compa ison o he obse ed and compu ed esul s along he cen al longi udinal sec ion o uni o m and linea empe a u e p ofile cases. bu i is necessa y o conduc u he es s o sus ain he hypo hesis. Finally, i has been de e mined ha he effec s o hea ans e by conduc ion a he bo om su ace o he flow laye a e negligible o he bulk flow dynamics, being sufficien o ake in o accoun adia ion and con ec ion hea ans e a he ee su ace. CRediT au ho ship con ibu ion s a emen J. O ega-Moya: W i ing – e iew & edi ing, W i ing – o iginal d a , Visualiza ion, So wa e, Me hodology, In es iga ion, Fo mal analysis, Concep ualiza ion. S. Ma ínez-A anda: W i ing – e iew & edi ing, W i ing – o iginal d a , So wa e, Concep ualiza ion, Me hodology. J. Fe nández-Pa o: W i ing – e iew & edi ing, W i ing – o iginal d a , So wa e, Concep ualiza ion. P. Ga cía- Na a o: W i ing – e iew & edi ing, W i ing – o iginal d a , Supe ision, Funding acquisi ion, Concep ualiza ion. Decla a ion o compe ing in e es The au ho s decla e ha hey ha e no known compe ing financial in e es s o pe sonal ela ionships ha could ha e appea ed o influence he wo k epo ed in his pape . Da a a ailabili y Da a will be made a ailable on eques . Acknowledgemen s This wo k was suppo ed by p ojec PID2022-137334NB-I00 unded by MCIN/AEI/10.13039/501100011033 and by ERDF/EU. The fi s au ho was unded by a “Deg ee Final Disse a ion” s uden g an om he A agón Ins i u e o Enginee ing Resea ch (I3A) - Uni e si y o Za agoza. This wo k has been pa ially unded by he Go e nmen o A agón, h ough he esea ch g an T32_23R Fluid Dynamics Technologies. Re e ences [1] S. Ma ínez-A anda, J. Fe nández-Pa o, I. Eche e iba , A. Na as-Mon illa, M. Mo ales-He nández, P. B u au, J. Mu illo, P. Ga cía-Na a o, Volume models and Efficien Simula ion Tools (EST) o Shallow flows, in: Ad ances in Fluid Mechanics: Modelling and Simula ions, Sp inge , Singapo e, 2022, pp. 67–137, Ch. Fini e. [2] O. Hung , S. 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