Jou nal o Compu a ional Physics 519 (2024) 113378
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Jou nal o Compu a ional Physics
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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
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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
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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
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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
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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:

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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]
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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
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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.
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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
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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. McDougall, Two nume ical models o landslide dynamic analysis, Compu . Geosci. 35 (5) (2009) 978–992.
[3] W. Wu, Compu a ional Ri e Dynamics, Ne Lib a y, Inc, CRC P ess, 2007.
[4] Z. Cao, Y. Li, Z. Yue, Mul iple ime scales o allu ial i e s ca ying suspended sedimen and hei implica ions o ma hema ical modeling, Ad . Wa e Resou .
30 (4) (2007) 715–729.
[5] J. Fe nández-Pa o, S. Ma ínez-A anda, P. Ga cía-Na a o, A 2D fini e olume simula ion ool o enable he assessmen o combined hyd ological and mo pho-
dynamical p ocesses in moun ain ca chmen s, Ad . Wa e Resou . 141 (2020) 103617.
[6] E. Biagioli, M. de’ Michieli Vi u i, F.D. Benede o, Modified shallow wa e model o iscous fluids and posi i i y p ese ing nume ical app oxima ion, Appl.
Ma h. Model. 94 (2021) 482–505.
[7] I. Eche e iba , S. Ma ínez-A anda, J. Fe nández-Pa o, R. Ga cía, P. B u au, P. Ga cía-Na a o, A model o compu ing he mally-d i en shallow flows, IOP
Con . Se . Ea h En i on. Sci. 1136 (1) (2023) 012037.
[8] A. Cos a, G. Macedonio, Nume ical simula ion o la a flows based on dep h-a e aged equa ions, Geophys. Res. Le . 32 (2005) L05304, h ps://doi .o g /10 .1029 /
2004GL021817.
[9] I. Eche e iba , S. Ma ínez-A anda, J. Fe nández-Pa o, P. Ga cía-Na a o, A GPU-based 2D iscous flow model wi h a iable densi y and hea exchange, Ad .
Eng. So w. 175 (2023) 103340.
Jou nal o Compu a ional Physics 519 (2024) 113378
20
J. O ega-Moya, S. Ma ínez-A anda, J. Fe nández-Pa o e al.
[10] M.D. Thanh, The Riemann p oblem o he shallow wa e equa ions wi h ho izon al empe a u e g adien s, Appl. Ma h. Compu . 325 (2018) 159–178, h ps://
doi .o g /10 .1016 /j .amc .2017 .12 .031, h ps://www .sciencedi ec .com /science /a icle /pii /S0096300317308901.
[11] M.R. Saleem, S. Zia, W. Ash a , I. Ali, S. Qama , The space– ime CESE scheme o shallow wa e equa ions inco po a ing a iable bo om opog aphy and
ho izon al empe a u e g adien s, Compu . Ma h. Appl. 75 (2018) 933–956, h ps://doi .o g /10 .1016 /j .camwa .2017 .10 .021, h ps://www .sciencedi ec .com /
science /a icle /pii /S089812211730679X.
[12] M. Che el, T. Pla z, E. Haube , D. Ba a oux, Y. La allée, D. Dingwell, La a flow heology: a compa ison o mo phological and pe ological me hods, Ea h
Plane . Sci. Le . 384 (2013) 109–120.
[13] G. Hulme, The in e p e a ion o la a flow mo phology, Geophys. J. In . 39 (2) (1974) 361–383.
[14] R.W. G iffi hs, The dynamics o la a flows, Annu. Re . Fluid Mech. 32 (1) (2000) 477–518.
[15] S. Ma ínez A anda, Efficien Simula ion Tools (EST) o sedimen anspo in geomo phological shallow flows, PhD Thesis, Uni e sidad de Za agoza, 2021.
[16] M. D agoni, S. Pond elli, A. Talla ico, Longi udinal de o ma ion o a la a flow: he influence o Bingham heology, J. Volcanol. Geo he m. Res. 52 (4) (1992)
247–254.
[17] M. Pas o , T. Blanc, B. Haddad, V. D empe ic, M.S. Mo les, P. Du o, M.M. S ickle, P. Mi a, J.A.F. Me odo, Dep h a e aged models o as landslide p opaga ion:
ma hema ical, heological and nume ical aspec s, A ch. Compu . Me hods Eng. 22 (1) (2015) 67–104.
[18] M. Jakob, O. Hung , Deb is-Flow Haza ds and Rela ed Phenomena, Sp inge P axis Books, Sp inge , Be lin Heidelbe g, 2005.
[19] S. Ma ínez-A anda, J. Mu illo, M. Mo ales-He nández, P. Ga cía-Na a o, No el disc e iza ion s a egies o he 2D non-New onian esis ance e m in geophysical
shallow flows, Eng. Geol. 302 (2022) 106625.
[20] M. Pas o , M. Quecedo, E. González, M.I. He e os, J.A.F. Me odo, P. Mi a, Simple app oxima ion o bo om ic ion o Bingham fluid dep h in eg a ed models,
J. Hyd aul. Eng. 130 (2) (2004) 149–155.
[21] D. Nae , D. Rickenmann, P. Ru schmann, B.W. McA dell, Compa ison o flow esis ance ela ions o deb is flows using a one-dimensional fini e elemen simula ion
model, Na . Haza ds Ea h Sys . Sci. 6(1) (2006) 155–165, h ps://doi .o g /10 .5194 /nhess -6 -155 -2006, h ps://nhess .cope nicus .o g /a icles /6 /155 /2006/.
[22] W. Wi mann, F. Sigmundsson, S. Dumon , Y. La allée, Pos -emplacemen cooling and con ac ion o la a flows: InSAR obse a ions and a he mal model o
la a fields a Hekla olcano, Iceland, J. Geophys. Res., Solid Ea h 122 (2017) 946–965, h ps://doi .o g /10 .1002 /2016JB013444.
[23] M.R. Pa ick, J. Dehn, K. Dean, Nume ical modeling o la a flow cooling applied o he 1997 Okmok e up ion: app oach and analysis, J. Geophys. Res. 109
(2004), h ps://doi .o g /10 .1029 /2003jb002537.
[24] P.Y. Bu gi, M. Caille , S. Hae eli, Field empe a u e measu emen s a E a’ale la a lake, E hiopia, Bull. Volcanol. 64 (2002) 472–485, h ps://doi .o g /10 .1007 /
s00445 -002 -0224 -3.
[25] L. Kesz helyi, A.J. Ha is, J. Dehn, Obse a ions o he effec o wind on he cooling o ac i e la a flows, Geophys. Res. Le . 30 (2003), h ps://doi .o g /10 .1029 /
2003GL017994.
[26] O. Hung , A model o he unou analysis o apid flow slides, deb is flows, and a alanches, Can. Geo ech. J. 32 (4) (1995) 610–623.
[27] M. Pas o , M. Quecedo, J.A. Fe nández Me odo, M.I. He o es, E. González, P. Mi a, Modelling ailings dams and mine was e dumps ailu es, Geo echnique 52 (8)
(2002) 579–591.
[28] S. Ma ínez-A anda, J. Fe nández-Pa o, P. Ga cía-Na a o, Mixing-phase model o shea -induced con ac i e/dila i e effec s in uns eady wa e -sedimen mix u e
flows, Ad . Wa e Resou . 188 (2024) 104710.
[29] R.M. I e son, D.L. Geo ge, A dep h-a e aged deb is-flow model ha includes he effec s o e ol ing dila ancy. I. Physical basis, P oc. R. Soc. A, Ma h. Phys. Eng.
Sci. 470 (2170) (2014) 20130819.
[30] M. Pas o , S. Tayyebi, M. S ickle, A. Yagüe, M. Molinos, P. Na as, D. Manzanal, A dep h in eg a ed, coupled, wo-phase model o deb is flow p opaga ion, Ac a
Geo ech. 16 (2021) 2409–2433.
[31] M. Quecedo, M. Pas o , M.I. He e os, J.A. Fe nández Me odo, Nume ical modelling o he p opaga ion o as landslides using he fini e elemen me hod, In . J.
Nume . Me hods Eng. 59 (6) (2004) 755–794.
[32] M. Pas o , B. Haddad, G. So bino, S. Cuomo, V. D empe ic, A dep h-in eg a ed, coupled sph model o flow-like landslides and ela ed phenomena, In . J. Nume .
Anal. Me hods Geomech. 33 (2) (2009) 143–172.
[33] E.N. da Cos a And ade, A heo y o he iscosi y o liquids. Pa II, Lond. Edinb. Dublin Philos. Mag. J. Sci. 17 (1934) 698–732, h ps://doi .o g /10 .1080 /
14786443409462427.
[34] C.V. Raman, A heo y o he iscosi y o liquids, Na u e 111 (1923) 532–533.
[35] I.A. Abdel aziq, T.H. Nie a , Rheology p ope ies o cas o oil: empe a u e and shea a e-dependence o cas o oil shea s ess, J. Ma e . Sci. Eng. 05 (2015),
h ps://doi .o g /10 .4172 /2169 -0022 .1000220.
[36] P. Kundu, I. Cohen, D. Dowling, Fluid Mechanics, Academic P ess, 2015, h ps://books .google .es /books ?id =uYe oQEACAAJ.
[37] J.B. Rawlings, Diffusion and Hea T ans e , Uni e si y Lec u e, 2014.
[38] R. Bi d, W. S ewa , E. Ligh oo , T anspo Phenomena, T anspo Phenomena, Wiley, 2006.
[39] S. Ma ínez-A anda, J. Mu illo, P. Ga cía-Na a o, A obus wo-dimensional model o highly sedimen -laden uns eady flows o a iable densi y o e mo able
beds, J. Hyd oin o m. 22 (5) (2020) 1138–1160.
[40] J. Mu illo, P. Ga cía-Na a o, Wa e Riemann desc ip ion o ic ion e ms in uns eady shallow flows: applica ion o wa e and mud/deb is floods, J. Compu .
Phys. 231 (2012) 1963–2001.
[41] S. Ma ínez-A anda, J. Mu illo, P. Ga cía-Na a o, A gpu-accele a ed efficien simula ion ool (es ) o 2d a iable-densi y mud/deb is flows o e non-uni o m
e odible beds, Eng. Geol. 296 (2021) 106462.
[42] H.R. Die e ich, E. Le , J. Chen, J.A. Richa dson, K.V. Cashman, Benchma king compu a ional fluid dynamics models o la a flow simula ion o haza d assess-
men , o ecas ing, and isk managemen , J. Appl. Volcanol. 6 (2017) 9, h ps://doi .o g /10 .1186 /s13617 -017 -0061 -x.