1
NUMERICAL METHODOLOGY FOR THE CFD SIMULATION OF
DIAPHRAGM VOLUMETRIC PUMPS
Menéndez Blanco, Albe o ; Fe nández O o, Jesus Manuel  ; Meana-Fe nández, And és
Fluid Mechanics A ea, Depa men o Ene gy, Uni e si y o O iedo
C/ Wi edo Rica , s/n. Gijón, As u ias, 33204 Spain
 jesus o@unio i.es (co esponding au ho )
ABSTRACT
This pape p esen s he uns eady nume ical me hodology o he CFD simula ion o Ai -
Ope a ed Diaph agm Pumps. The model ep oduces he uns eady displacemen o he
diaph agm using dynamic mesh echniques and ully esol es he Fluid S uc u e
In e ac ion (FSI) esponsible o he mo ion o he check al es. The go e ning
pa ame e s ha e been modi ied wi h Use De ined Func ions (UDFs), using an implici
scheme o he g id mo ion ha gua an ees he s abili y and ealizabili y o he wo-
dimensional model adop ed.
The analysis o he ins an aneous deli e ed low a e, as a unc ion o he discha ged
ou le p essu e, has p o ided in e es ing and use ul in o ma ion o he u u e design o
new p o o ypes. The compa ison be ween nume ical esul s and he expe imen al
pe o mance cu es has con i med he accu acy o he model and he co ec mesh
selec ion in he small gaps and passages o he pump in e nal geome y.
The leakage lows, especially in he exhaus ing al e du ing he o wa d s oke, and he
ball apping esponsible o ins abili ies and high- equency noise du ing oscilla ions o
he al es has been accu a ely simula ed. A high-deli e ed p essu es, i has been
obse ed a cha ac e is ic ipple in he ins an aneous low a e du ing he decele a ion
o he diaph agm owa ds i s op-dead-cen e , associa ed o a pa ial e-opening o he
exhaus ing al e.
A close look o he dynamics o he balls has e ealed a s ong coupling be ween inle
and ou le check al es. In addi ion, despi e o he ema kable le el o accu acy (less
han 9% o de ia ion), he eci cula ing cells ound in he low ields inside he pump
sugges he con enience o he de elopmen o a ull-uns eady 3D model in he nea
u u e.
KEYWORDS: CFD Simula ion, Ai -Ope a ed Double Diaph agm Pump, Dynamic Mesh,
Nume ical Me hodology, Laye ing and emeshing echniques.
Re ised Ve sion 2.0, Sep embe , 17 h, 2018.
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HIGHLIGHTS
- CFD Modeling o an Ai -Ope a ed Double Diaph agm Pump o he i s ime.
- Dynamic meshes wi h implici schemes ully esol e he Fluid S uc u e In e ac ion.
- Expe imen al es s ha e alida ed he nume ical me hodology used in a 2D model.
- Typical phenomena like ball apping and low leakage a e accu a ely ep oduced.
- Flow pa e ns wi hin he pump ad ice o he need o a 3D model in u u e wo k.
1. INTRODUCTION
Diaph agm pumps a e posi i e displacemen pumps ha use ecip oca ing de o mable
diaph agms o gene a e a pulsa ing discha ge low. The d i ing mechanism can be
mechanical, using a ypical c ank gea mechanism o con e a o a ional mo ion in o an
al e na i e mo ion o a pis on a ached o he diaph agm, o luid-ope a ed, whe e a
seconda y hyd aulic luid d i es he mo ion om he opposing side o he diaph agm. In
he case o using comp essed ai as he d i ing luid, he pumps a e usually double-
ac ing, inco po a ing wo diaph agms and wo se s o check al es, wi h he ai low
being al e na ed be ween diaph agms ia a shu le al e. These pumps a e usually
known as Ai -Ope a ed Double Diaph agm (AODD) Pumps.
The wo king p inciple o hese AODD pumps is simple. The comp essed ai is shi ed
om one chambe o he o he by a pi o ing pneuma ic al e, while bo h diaph agms
a e mo ed simul aneously due o a cen al sha . This ecip oca ing mo ion c ea es and
collapses ca i ies o cing he liquid o ge ou o one chambe and go in o a discha ging
mani old, while he o he chambe is being illed wi h liquid coming om he inle
mani old a he same ime. In addi ion, a se o check al es, ope a ing due o he
p essu e di e ences in he chambe s, a e in oduced o gua an ee no eci cula ion.
AODD pumps a e highly eliable, due o he absence o in e nal pa s in ic ion, and hey
do no con ain sealing o lub ica ing oils, so leakage and con amina ion o he pumping
luid is a oided. The inclusion o in e media e elie al es and addi ional po s o ully-
load he hyd aulic chambe s help o enla ge he li espan o he pump. They a e used o
a wide ange o applica ions, going om pe ochemical o sani a y o ood and be e age
indus ies, whe e highly ab asi e o medium- o-high iscosi y liquids mus be
ans e ed. Slu ies and all so s o agg essi e luids, e en wi h high amoun o solid
con en s, can be also easily handled by hese pumps due o hei good suc ion
cha ac e is ics.
Manu ac u e s con igu e hese pumps o he speci ic needs o a cus ome ’s applica ion,
including selec ion o ma e ials and o e all pump capaci y, so he diaph agm has o be
compa ible wi h he pumped luid and sized o ma ch wi h he desi ed pump s oke and
deli e y low a e. Mos designs o AODD pumps a e based on a pe iphe al layou o he
low passages, wi h he pneuma ic mo o being placed in he cen e o he pump,
be ween bo h diaph agms, and ball al es ac ing as check al es [1,2,3]. The simplici y
and he compa ibili y o he design o di e en ma e ials and hea y luids explains i s
p edominan use in he indus y (Figu e 1, le ).
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Recen ly, a new con igu a ion wi h a cen al layou o he low passages has been
p oposed by SAMOA Indus ial S.A. [4,5]. The diaph agms a e placed in he cen al
posi ion o he pump, in combina ion wi h he check al es, loca ing he pneuma ic
mo o ou side o he co e and hus educing he o al leng h o he mani olds (Figu e 1,
igh ). This no el design p esen s lowe in e nal p essu e losses, highe compac ness
and supe io li espan o he diaph agms (due o he supp ession o non-axial o ces) wi h
espec o classical designs.
Figu e 1. Pe iphe al layou (le ) e sus Cen al layou ( igh ) o he low passages in an
AODD pump.
Due o he ad ances achie ed in he design o cen i ugal pumps du ing he las decades,
and hei low manu ac u ing cos s, olume ic pumps ha e played a mino ole in he
pumping machine y ield. Resea ch ac i i y has ollowed a simila end, wi h a wide
numbe o in es iga ions on cen i ugal pumps a ailable in he li e a u e, bu only a e y
educed signi ican esea ch on olume ic pumps (e en mo e ma ginal o diaph agm
pumps). Typically, he de elopmen o AODD pumps has been based on expe imen al
cha ac e iza ions o hei pe o mance [6,7], mainly ocused on he ma e ials
compa ibili y. Up o he au ho s’ knowledge, no scien i ic e e ences dealing wi h
diaph agm pumps can be ound in he open li e a u e.
In o de o p opose a nume ical model o he analysis o a diaph agm pump, o he ypes
o pumps wi h simila wo king p inciples can be e e enced. In pa icula , bo h
expe imen al and nume ical in es iga ions o axial pis on pumps wi h check al es can
be ound and ci ed as p elimina y s udies o AODD pumps. Expe imen al s udies ha e
analyzed he cha ac e is ics o he al e mo ion o a ecip oca ing pump [8] and e en
he de ec ion o ca i a ion phenomena using high-speed came as [9], while 1D
ma hema ical app oaches ha e been p oposed o model ecip oca ing pumps wi h sel -
ac ing al es nume ically [10,11], including p essu e pulsa ions [12], o e en al e na i e
pumps wi h mul iple ca i ies [13].
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In addi ion, CFD simula ions ha e ecen ly eme ged as a powe ul ool o analyze in
de ail he spa ial cha ac e is ics o he low pa e ns wi hin olume ic low pumps. The
pe o mance o plunge pumps as a unc ion o he c ank angle [14], he e alua ion o
i s inle s oke pe o mance [15] and e en ca i a ion incep ion [16,17] a e some
examples o he po en iali y o compu a ional modelling. The uns eady simula ion o ai -
ope a ed pis on pumps [18] and he nume ical analysis o axial pis on pumps [19,20] a e
also signi ican con ibu ions o he s udy o ecip oca ing pumps ia CFD.
Complemen a ily, he e a e also ele an examples o CFD modelling o he s udy o
he al e mo ion in check al es. The de elopmen o de o mable meshes and he
implemen a ion o de o ming g ids [21] ha e allowed he op imiza ion o check al e
designs [22] and he analysis o luid-s uc u e in e ac ions [23], e en unde incipien
ca i a ion [24].
In he p esen in es iga ion, a no el me hodology o he compu a ional analysis o
olume ic diaph agm pumps is p esen ed. In pa icula , an uns eady CFD modelling has
been de eloped using de o mable mesh echniques o simula e he low pa e ns wi hin
an AODD pump. A se o use de ined unc ions we e de eloped o model accu a ely he
dynamic mo emen o he pump diaph agm and he opening and closing o he check
al es. This displacemen is essen ial o model co ec ly he deli e ed low a e as a
unc ion o he p essu e di e ences in he chambe s. In addi ion, implici schemes o
he g id mo ion we e ound o be necessa y o s abili y and accu acy o he luid-
s uc u e in e ac ion (FSI) o he mo ing balls. Well-known phenomena like balls
apping, in e nal eci cula ions and olume ic losses ha e been success ully ep oduced
in he simula ions. Mo eo e , ypical delays o he opening and closing o he check
al es ha e been well- ep oduced by he model, and hei beha io as a unc ion o he
ai supply p essu e o he pump wo king condi ions has been also s udied. The
nume ical model has been complemen ed wi h a se o labo a o y es s o alida ion
pu poses. Uns eady p essu e signals in he diaph agm ca i ies ha e been measu ed o
di e en ope a ing condi ions o he pump and compa ed wi h he nume ical da a
p o ided by he CFD model.
2. STUDIED DIAPHRAGM PUMP (DP-200) AND OPERATING CYCLE
SAMOA Indus ial S.A. is a Eu opean manu ac u e o equipmen o luid ans e ,
disposing, dosing, eco e y and in en o y con ol. Since 1985, SAMOA G oup ha e been
comme cializing con en ional AODD pumps ha pump he luid a ound he ou side o
he pump. To o e come he issues associa ed wi h hese adi ional models, a adical
inside-ou concep was de eloped in 2000 o supp ess pe iphe al mani olds. The pump
was edesigned wi h a cen al low pa h con igu a ion and a new ai dis ibu ion sys em.
La e , he cen al low concep was eenginee ed in 2010 e ining he c i ical elemen s
and educing he numbe o in e nal componen s. The so-called Di ec lo® pump (DP) is
an ex emely simple machine, e y eliable wi h minimal main enance equi emen s
and ypical deli e ed low a es in he ange be ween 30 and 100 li e s pe minu e. In
2014, a new se ies o Di ec lo pumps has been designed o achie e deli e ed low a es
up o 200 l/min. The DP-200 se ies has equi ed a ca e ul de ini ion o he in e nal low
passages, he diaph agm size and he check al e geome ies. The pump has been inally
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comme cialized in Feb ua y 2017. Figu e 2 shows a pho og aph o he DP-200 pump, in
bo h me allic and plas ic e sions (le image). On he igh , an inne iew e eals he
di e en pa s o he cen al low concep .
Figu e 2. Me allic and plas ic p o o ypes o DP-200 pump (le ) and in e nal iew o he
basic elemen s ( igh ).
These pumps ea u e wo signi ican inno a ions: he lexible diaph agm suspension and
he ic ionless pi o ing ai al e. The diaph agm is a s uc u ed ci cula pla e ha
inco po a es a ibbed unde side o p o ide highe s eng h and li espan. The diaph agm
is made o EPDM (E hylene P opylene Diene Monome ) ubbe as basic ma e ial,
ein o ced wi h a ab ic and wi h a shee o PTFE (Poly Te a Fluo o E hylene) he mally-
adhe ed o he diaph agm side in con ac wi h he media. The diaph agms a e no igidly
ixed o he sha , educing a igue and con ibu ing o ex end hei se ice li e since
non-axial loads a e no ansmi ed. On he o he hand, a new ic ionless and ex emely
as ai al e was de eloped o educe he ai consump ion, educing he a igue wea
and allowing sho e s oke leng hs. Table 1 includes all he ele an geome ical da a
and summa izes he basic ope a ing pa ame e s o he pump.
Table 1. Geome ical da a and ope a ing pa ame e s
Diaph agm ex e nal diame e , De (mm) 200
Diaph agm in e nal diame e , Di (mm) 100
Diaph agm e ec i e diame e , Dd (mm) 150
Diaph agm s oke leng h, Ld (mm) 31.0
Diaph agm e ec i e a ea, Ad (cm2) 176.7
Deli e y pe s oke, Vd (cm3) 548
In e nal mani olds diame e , Dm (mm) 26.5
Ball diame e o check al es, D (mm) 31.75
Ball maximum displacemen , L (mm) 10
Check al e e ec i e a ea (max), A (cm2) 4.98
Ai -ope a ion p essu e, Pai (ba ) 0-8
P essu e a io (-) 1:1
Maximum ee deli e y (8 ba ), Q (l/min) 200
Maximum d i en eloci y, n (Hz) 3.0

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Figu e 3. Ske ch o he pneuma ic ope a ion (le ) and basic dimensions o he
diaph agm and check al es ( igh ).
The ope a ing cycle is a consequence o he al e na i e displacemen imposed in he
diaph agms by he pneuma ic al e. When one o he diaph agms mo es ou , a olume
in one chambe is c ea ed causing he p essu e o dec ease and suc ion luid in o he
pump. A he same ime, he o he chambe is collapsed and he luid in con ac wi h
he second diaph agm is discha ged ou o he pump. When diaph agms each hei
bo om and op dead cen e s (BDC, TDC) espec i ely, a pneuma ic end s oke senso is
igge ed and he di ec ional al e is in e ed. Due o he p essu e di e ences c ea ed,
he ball al es a e also ac i a ed a he beginning o e e y back-and- o h s oke, wi h
some delay due o ine ial o ces. Figu e 3 shows a ske ch o he ope a ing cycle,
including he basic dimensions o he wo king elemen s.
3. PERFORMANCE CURVES AND EXPERIMENTAL SENSORS
The expe imen al pe o mance cu es o he DP-200 AODD pump ha e been ob ained
by he manu ac u e in he p o o ypes labo a o y o he R&D depa men o SAMOA
Indus ial S.A. using wa e as he wo king luid. The es acili y is composed o a closed-
loop hyd aulic ci cui connec ing wo a mosphe ic anks wi h ee-su ace le el senso s,
ollowing he s anda ds ANSI/HI 10.6-2004 by he Hyd aulic Ins i u e [7] (see Figu e 4,
le ). The deli e ed low a e o he pump, ins alled be ween he anks, is measu ed
olume ically compa ing he ime equi ed o displace a gi en amoun o luid. The
amoun o wa e is calib a ed on-line using an elec onic balance. The p essu e ise gi en
by he pump is ob ained h ough he measu emen o he inle and ou le s a ic
p essu es o he pump wi h elec onic manome e s (Figu e 4, igh ).
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Figu e 4. Tes labo a o y in he R&D Depa men a SAMOA Indus ial S.A. (le ) and
de ail o he pump ins umen ed wi h he measu ing senso s ( igh ).
The pump is also ins umen ed wi h an induc i e senso , o ob ain he ins an aneous
posi ion o he diaph agm, and a p essu e ansduce o measu e he s a ic p essu e in
he hyd aulic chambe . In addi ion, he wo king poin o he pump can be modi ied
using a h o le al e in he p ima y line o he ci cui and he ai -supplied p essu e can
be also egula ed be ween 0 and 8 ba o di e en ai consump ions, which is measu ed
wi h a he mal lowme e . Figu e 5 shows a ske ch o he hyd aulic acili y and Table 2
gi es he accu acy and p ecision o he measu ing de ices.
Table 2. Accu acy and p ecision o he measu ing equipmen .
Measu ing de ice
Manu ac u e and model
Range
Unce ain y
Induc i e senso
PEPPERL-FUCHS NBN-8GM30-E2V1
0-3 mm
5% (hys e esis)
P essu e ansduce
TE Connec i i y MEAS U5244
-1 o 13 ba
± 1%
P essu e ansduce
ESI Genspec GS4002
-1 o 24 ba
± 1%
Elec onic manome e
Pa ke Se ice Junio SCJN-016-01
-1 o 16 ba
± 0.5%
The mal lowme e
Tes o 6442
12 o 3750 lpm
± 0.3%
Figu e 5. Ske ch o he hyd aulic ci cui o he expe imen al da abase (le ) and lis o
he basic elemen s ( igh ).
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To cha ac e ize he pe o mance cu es o he pump, h ee di e en ai -supplied
p essu es (2, 4 and 6 ba ) ha e been es ed o di e en discha ge p essu es in he
h o le al e. Hence, he expe imen al da abase is composed o 9 di e en wo king
condi ions (see able 3).
Table 3. Expe imen al da abase.
Tes
No.
Ai -supply
p essu e
(ba )
Discha ge
p essu e
(ba )
Volume
(l)
Time
(s)
Flow a e
(l/min)
D i ing
equency
(Hz)
No malized
E iciency
( – )
#1
2
0
200
107
112.2
1.85
0.00
#2
2
1
200
206
58.2
1.0
0.36
#3
4
0
200
83.9
144.6
2.45
0.00
#4
4
1.5
200
143
83.9
1.4
0.31
#5
4
3
200
504
23.8
0.45
0.47
#6
6
0
200
76
157.8
2.7
0.00
#7
6
1.5
200
104
115.4
1.8
0.24
#8
6
3.0
200
159
75.5
1.25
0.40
#9
6
4.5
200
282
42.6
0.75
0.50
The pe o mance cu es (p essu e ise and no malized e iciency as a unc ion o he
low a e o di e en ai -supplied p essu es) a e shown in Figu e 6. The e iciency has
been ob ained as he a io be ween he luid powe and he d i ing powe o he
pneuma ic supply. Acco ding o he manu ac u e , i has been de ined ha nominal
poin s co espond o wo king si ua ions whe e he discha ge gauge p essu e is
app oxima ely 1 ba ( ypical si ua ions o anspo o low- iscosi y liquids).
Addi ionally, using he heo y o ansmission o unce ain y, we ha e es ima ed he
ollowing p ecision o he igu es o me i o he olume ic pump: 0.12 ba in he
measu emen s o he p essu e ise gi en by he pump and maximum unce ain y o
0.7% in he measu emen o he low a e.
Figu e 6. Pe o mance cu es o he AODD pump: P essu e ise and no malized
e iciency as a unc ion o he low a e.
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4. NUMERICAL METHODOLOGY
The comme cial package ANSYS-FLUENT® 16 [25] has been used o esol e he se o
Reynolds-A e aged Na ie -S okes (RANS) equa ions wi h a ini e olume app oach. A
dynamic based echnique, using emeshing and laye ing unc ionali ies, ha e been
employed o he simula ion o he al e na i e de o ma ion o he diaph agm and he
induced mo ion o he ball al es.
4.1. Nume ical scheme
The Fini e Volume Me hod (FVM) is employed o esol e he luid go e ning equa ions
o incomp essible low using he RANS app oach:
• Con inui y equa ion: 𝜕𝑢𝑖
𝜕𝑥𝑖=0
(1)
• Momen um equa ion:
𝜌𝜕𝑢𝑖
𝜕𝑡 +𝜌𝜕(𝑢𝑖𝑢𝑗)
𝜕𝑥𝑗=−𝜕𝑝
𝜕𝑥𝑖+𝜇∇2𝑢𝑖+𝜕𝜏𝑖𝑗
𝜕𝑥𝑗
(2)
In his case, a second-o de upwind spa ial scheme has been employed o he
con ec ion e ms, wi h a G een-Gauss cell based me hod o he compu a ion o spa ial
g adien s in di usion e ms o he momen um equa ion. A seg ega ed sol e wi h he
SIMPLE p essu e- eloci y coupling was i s ly employed o p elimina y simula ions in a
s eady ashion. La e , i was swi ched o a PISO algo i hm o he dynamic simula ions
wi h he de o mable mesh.
In addi ion, due o he u bulen egime o he low inside he pump, a k-ε RNG
u bulence model wi h s anda d wall unc ions was selec ed due o i s obus ness and
e sa ili y o a wide ange o luid machine y applica ions [26]. Mo eo e , his op ion
includes he e ec o swi l in u bulence and also accoun s o low Reynolds numbe
e ec s. Thus, he Reynolds S ess Tenso in he momen um equa ion is modelled using
an Eddy Viscosi y Model, acco ding o:
𝜏𝑖𝑗 =−𝜌𝑢𝑖′𝑢𝑗′






=𝜇𝑡(𝜕𝑢𝑖
𝜕𝑥𝑗+𝜕𝑢𝑗
𝜕𝑥𝑖)
⏟
𝑆𝑖𝑗
−2
3𝜌𝑘𝛿𝑖𝑗
(3)
whe e 𝑘=1
2𝜌𝑢𝑘
′𝑢𝑘
′







is he u bulen kine ic ene gy and 𝜇𝑡=𝜌𝐶𝜇𝑘2
𝜀 is he u bulen
iscosi y, wi h 𝐶𝜇=0.0845. Addi ional anspo equa ions a e equi ed o he
u bulen kine ic ene gy and he u bulen dissipa ion a e, 𝜀=2𝜈𝑠𝑖𝑗
′𝑠𝑖𝑗
′






, being 𝑠𝑖𝑗
′=
1
2(𝜕𝑢𝑖′
𝜕𝑥𝑗+𝜕𝑢𝑗
′
𝜕𝑥𝑖). In he case o he RNG model, hese closu e equa ions a e o his case
(neglec ing buoyancy):
16
Complemen a ily, a hi d UDF based on he De ine-Execu e-A -End mac o is
implemen ed in he code o pe o m pos -p ocessing ou ines like collec ing ele an
da a and/o expo ing main a iables in o ou pu iles. Figu e 11 shows he necessa y
sequence ollowed o employ he laye ing mac o in an explici ashion. Mo e de ails wi h
comple e coding o he UDFs can be ound in [29].
Figu e 11. Sequence o he UDFs o he laye ing ou ines wi h he explici scheme.
4.6. Explici VS Implici schemes.
The explici scheme o he g id mo ion is desi able because allows he upda ing o he
mesh only once a he beginning o e e y ime s ep using he low a iables om he
p e ious ins an . Figu e 12, le , shows how he comple e explici ou ines o laye ing
and emeshing echniques a e inco po a ed in he compu a ional p ocedu e o he
ANSYS-FLUENT sol e . Howe e , signi ican es ic ions a ise o he g id mo ion in o de
no o iola e he mesh Cou an numbe [30], i.e. balls and diaph agms mo ions should
no exceed he size o he collapsing cells. Typically, his equi es he employmen o
educed ime s eps (below 5·10-5 s o he g id densi y o he s a ic model).
Ano he impo an es ic ion is associa ed o he e y small gaps used o he closu e
o he al es in he dynamic mesh. Since a comple e closu e is no possible (in o de o
a oid singula i ies, he mesh canno be comple ely anished), a minimum clea ance has
o be main ained o ealizabili y o he model. Hence, i has been ixed o 0.15 mm (a
0.5% o he ball diame e ) in he p esen simula ions, p ese ing only 2 cells in he al e
sea gap. In addi ion, due o he highly- e ined meshes in ha egion (y+= 4÷9), an
Enhanced Wall T ea men (EWT) was also employed o he u bulence modelling [31].
A hyb id mesh wi h app oxima ely 33000 cells was inally used o he whole domain.

17
This is an addi ional p oblem o he explici scheme because o he ex emely educed
ime s eps equi ed when he balls a e closing he check al es. P elimina y simula ions,
unde ee-discha ge condi ions, exhibi ed p oblems o con e gence due o uns able
calcula ions o he o ces ac ing on he balls and non- ealis ic eloci y es ima ions.
Addi ional a emp s educing he ime s eps om 10-5 o 10-6 and e en 10-7 did no
esol e he p oblems. A e a ew cycles, he ins abili ies a e ampli ied and he
simula ions blow-up due o he mesh co up ion p oduced by an ou -o -scale mo ion o
he balls ( igu e 12, igh ).
Figu e 12. Flowcha o he explici scheme (le ) and di e ging ins abili ies in he
compu a ion o he luid o ces exe ed on he balls ( igh ).
Mos o he simula ion pa ame e s we e also modi ied wi hou signi ican p og ess:
high-o de disc e iza ion schemes, o he u bulence models (S-A, k-ω), coupled sol e
and e en o he emeshing s a egies in he check al es. Hence i is concluded ha he e
is a high coupling be ween he solid-body mo ion o he balls and he luid- low
s uc u es (FSI) ha canno be sol ed wi h a con en ional explici scheme o he g id
mo ion.
To con i m his hypo hesis, a simpli ied case wi h a diaph agm and a single ball wi hou
al e-sea es ic ions was modelled. The ball was encapsula ed wi h a s uc u ed mesh
and he UDFs we e also simpli ied conside ing emeshing only wi h he explici scheme
18
(Figu e 13, le ). I was obse ed a clea end o nume ical ins abili y and a high
dependency on he ime s ep size. Figu e 13, igh , shows he e olu ion o he o ce
exe ed o e he ball o se e al cycles o he diaph agm mo ion, as he ime s ep is
p og essi ely educed. Only a na ow ange o ime s eps de i es in a con olled
e alua ion o he o ce. I he ime s ep is high, la ge oscilla ions a ise in he
compu a ions o he o ce; bu i he ime s ep is oo educed, new ins abili ies a e also
p oduced. Consequen ly, he ealizabili y o he model is se iously comp omised and
di icul o be de e mined in ap io is ic basis.
Figu e 13. Simpli ied model o he e alua ion o he explici scheme (le ) and
empo al e olu ion o he o ce in he ball as a unc ion o he ime s ep size ( igh ).
E ec i ely, he employmen o explici schemes in luid-solid in e ac ions is no
ecommended when he densi y a io o he luid wi h espec o he solid is a ound o
abo e uni y (he e, he balls a e made o PTFE wi h a ypical densi y o 2200 kg/m3). Small
p essu e luc ua ions may lead o signi ican oscilla ions in he body mo ion and
e en ually se o an uns able eedback ha b eaks up he simula ion [32], as p e iously
e idenced. The e o e, he nume ical me hodology was eo ien ed owa ds an implici
scheme, despi e o he la ge CPU imes equi ed. The signi ican inc emen o he
compu a ional cos s is based on he g id upda ing ha mus be comple ed now a e e y
i e a ion in he ime s ep. App oxima ely, 400 hou s (2-3 weeks) o CPU ime we e
necessa y o esol e one single cycle o he diaph agm in he wo-dimensional domain
o 33K cells using a 4-nodes In el Co e i7-5820K a 3.3 GHz and 64Gb RAM.
P e ious UDFs a e e-used, bu now upda ing bo h local and global a iables in e e y
i e a ion. In o de o imp o e he coupling be ween he eloci y and he o ces ac ing
on he balls, a second-o de backwa d app oach is also employed o calcula e he
eloci y o he balls:
19
𝑣𝑡= (4𝑣𝑡−∆𝑡 −𝑣𝑡−2∆𝑡 +2(𝐹
𝑚𝑏)∆𝑡) 3⁄
(8)
The con e gence o he implici scheme o he mesh is also enhanced in oducing
unde - elaxing ac o s du ing he i e a i e p ocess in a gi en ime s ep. The ollowing
equa ion is implemen ed [25] o con ol he ela i e changes in he g id eloci y
be ween consecu i e i e a ions:
𝑣𝑡(𝑛)= 𝑣𝑡∗(𝑛)𝜔𝑟 +(1−𝜔𝑟)𝑣𝑡(𝑛−1)
(9)
whe e 𝑛 indica es he cu en i e a ion, 𝑣𝑡(𝑛−1) s ands o he eloci y compu ed in he
p e ious i e a ion, 𝑣𝑡∗(𝑛) is he non- elaxed eloci y o he cu en i e a ion and 𝜔𝑟 is he
unde - elaxing ac o ( ha mus be ixed be ween 0 and 1). To speed up he whole
calcula ion, he unde - elaxing ac o is e-calcula ed in e e y i e a ion acco ding o he
ollowing ma hema ical exp ession:
𝜔𝑟=1−𝑟1𝑒−(𝑛2
𝑟2)
(10)
𝑟1 de ines he unde - elaxing ac o in he i s i e a ion, while 𝑟2 con ols he asymp o ic
e olu ion o he elaxing ac o owa ds uni y (i.e., when 𝜔𝑟 is no longe elaxing). In he
p esen simula ions, he alues 𝑟1=0.8 and 𝑟2=500 we e ound o p o ide a
easonable a io be ween economy and accu acy.
In addi ion, he mesh upda ing was inally no pe o med a e e y i e a ion. Only du ing
he i s numbe s o i e a ions, he upda e was main ained wi hou skipping. A e he
ini ial i e i e a ions, he upda e was pe o med e e y wo i e a ions and inally a e
e e y i e i e a ions. Figu e 14, igh , shows he asymp o ic e olu ion o he unde -
elaxing ac o (black solid line), and he i e a ing in e als o he mesh o be upda ed.
No e how he oscilla ions in he calcula ion o he o ce ( ed line) a e damped du ing he
ini ial i e a ions.
All hese ea u es ha e allowed o economize he compu a ional cos s and also o
inc ease he ime s ep sizes wi hou di e gence issues. The ins abili ies leading o non-
ealis ic mo ions and mesh di e gence du ing he ini ial i e a ions o he c i ical ime
s eps ( hose in he p oximi y o he al e sea ) a e now unde con ol. Figu e 14, le ,
shows he de ini i e lowcha o he dynamic calcula ions wi h UDFs o he implici
scheme.
20
Figu e 14. E olu ion o he unde - elaxing ac o s and g id upda ing in e als in a ime
s ep ( igh ) and inal lowcha o he implici scheme (le ).
4.7. Final uns eady esolu ion wi h FSI.
The inal nume ical model wi h implici scheme o he mesh upda e in he mo ing zones
was execu ed o all he ope a ing poin s cha ac e ized expe imen ally. To ep oduce
he di e en cases, he ou le bounda y condi ion (a p essu e d op p opo ional o he
kine ic ene gy) had o be ixed changing he alue o he loss coe icien . No e ha his
p ocedu e mimics he pa ial closu e o he h o le al e in he expe imen s. P ecisely,
he loss coe icien is es ima ed om he expe imen s, compu ing he a io be ween he
RMS alue o he s a ic p essu e measu ed wi h he elec onic manome e (see igu e
15, le ) and he mean bulk eloci y a he pump discha ge:
𝐾𝑣=𝑃𝑅𝑀𝑆
𝜌𝑣2=𝜋2𝐷𝑚
4𝑃𝑅𝑀𝑆
16𝜌𝑄2
(11)
The di e en loss coe icien s used o he nume ical simula ions a e p esen ed in Table
5 below ( ou h column). The ime s ep used in e e y case and he d i ing equency
imposed o he diaph agm (measu ed expe imen ally) is also gi en in he able. No e
21
ha , om a physical poin o iew, i is necessa y o p ese e a easonable Cou an
numbe (a ound uni y) in o de o ob ain a esolu ion wi h physical sense when
pe o ming he uns eady compu a ions. Conside ing ha he smalles cells in he al e
sea s a e in he o de o a ac ion o a millime e (10-4 m), wi h expec ed eloci ies in
he ange o 2.5 m/s, ime s eps we e ixed in he o de o 5·10-5 o 10-5 as a con enien
selec ion o accu acy and s abili y.
Table 5. Nume ical da abase. Ou le BC and ime s ep sizes.
Tes
No.
Supplied
p essu e
(ba )
Discha ge
p essu e,
PRMS (ba )
Loss
coe icien ,
K (-)
D i ing
equency (Hz)
Time s ep, Δ
(s)
#1
2
0
0
1.85
5·10-5
#2
2
1.16
75.1
1.0
5·10-5
#3
4
0
0
2.45
5·10-5
#4
4
1.99
62.1
1.4
5·10-5
#5
4
3.65
1411
0.45
5·10-5
#6
6
0
0
2.7
2.5·10-5
#7
6
1.97
32.5
1.8
2.5·10-5
#8
6
3.63
139.9
1.25
5·10-5
#9
6
4.14
500.8
0.75
5·10-5
In addi ion, i was necessa y o un he simula ions du ing wo ini ial cycles o elimina e
he ansien s a e in he pump. O cou se, his inc eased no ably he equi ed CPU ime,
bu i is una oidable o ob ain a ep esen a i e solu ion o he model. A ully-pe iodic
esponse was achie ed om he hi d cycle on, so moni o ing o he ele an a iables
and signi ican in e media e posi ions was hen ac i a ed.
5. NUMERICAL RESULTS
5.1. Compa ison o pe o mance cu es
A e he applica ion o he scaling ac o s de ined in eq. (6), he deli e ed low a e in
he wo-dimensional model, as a unc ion o he discha ge p essu e, is compa ed wi h
he expe imen al cu es in igu e 15, igh . All he low a e esul s ha e been duplica ed
aken in o accoun he exis ence o wo symme ic diaph agms in a pa allel
a angemen . E en wi h he 2D low assump ion, i is no iceable he o e all good
ag eemen , especially a high discha ged p essu es. A a mosphe ic ou le condi ions,
he nume ical model o e p edic s he deli e ed low a e because o he ele an
mechanical losses ( ic ion) a low p essu e egimes ha he simula ions a e no
modelling. Fo all he ai -supplied p essu es, he disc epancy be ween he expe imen al
and nume ical alues is enla ged as he ou pu p essu e dec eases. The highe
disc epancy is a ound a 9% in he case o he 4 ba cu e a ee-deli e y condi ions.
The plo in igu e 15, igh , also highligh s wo nominal poin s (a mode a e discha ge
p essu es) ep esen ing he si ua ions o high (6 ba ) and low (2 ba ) ai -supplied
p essu es. This is equi alen o high (1.8 Hz) o low (1.0 Hz) d i ing eloci ies. The

22
e olu ion o he low a e and he mo ion o he check al es o hese ope a ing poin s
will be discussed in de ail in he ollowing subsec ion.
Figu e 15. Le : S a ic p essu e a he pump discha ge ( ed line) and i s compu ed RMS
alue (black line) o Tes #5. Righ : Compa ison o CFD and expe imen al esul s.
5.2. Nominal ope a ion.
Following, he e olu ion o he p essu e in he diaph agm chambe and a he pump
ou le , as well as he low a e and he posi ion o he check al es, a e moni o ed du ing
a con e ged diaph agm cycle (deno ed as pe iod, T). Figu e 16, le , shows he esul s
o he high ai -supplied p essu e.
The p essu e in he chambe ises ab up ly wi h wo signi ican peaks a he beginning
o he diaph agm displacemen . The maximum p essu e coincides wi h he middle
posi ion o he o wa d s oke o he diaph agm (black line a =T/4). A ha momen ,
he exhaus ing al e is comple ely opened bu wi h he ball s a ing o d op. Be ween
T/4 and T/2, he deli e ed low a e expe iences a sudden bump, associa ed o a pa ial
e-opening o he exhaus al e. In he second hal o he cycle, he diaph agm is
suc ioning luid and he incoming low exhibi s a sinusoidal pa e n, in o al ag eemen
wi h he heo e ical esponse (dashed lines). No e ha he model is capable o
ep oduce he ine ial delays in he openings and closings o he al es, which a e
iden i ied as local decays o he low a es a he beginning o he back-and- o h
mo emen s.
Figu e 16, igh , shows a simila ep esen a ion bu o he case wi h low-p essu e. All
he e olu ions o he deli e ed low a e and he p essu e in he pump beha e mo e
heo e ically. Howe e , due o he lowe p essu e, he e a e much mo e oscilla ions o
he exhaus ing ball, which ep esen s a con inuous apping (high- equency noise) and
he o igin o ins abili ies in he closu e o he check al e. This ea u e is well-known by
he pump manu ac u e .
23
Figu e 16. Tempo al e olu ions o he p essu e, low a es and check al es du ing he
diaph agm cycle o high-p essu e (le ) and low-p essu e ( igh ) nominal condi ions.
To p o ide mo e insigh , Figu e 17 ep esen s he associa ed desc ip ion o he low
pa e ns a ou ele an in e media e posi ions o he diaph agm cycle o he high-
p essu e si ua ion. Only a T/4 he exhaus ing al e is pa ially opened; in he es o he
snapsho s, i is p ac ically closed. I is qui e signi ican he appea ance o se e al
eci cula ing cells o luid all h ough he in e nal passages, especially in he inal con ou
a he BDC o he diaph agm ( =T). This can be indica ing he need o a ull 3D geome y
o a mo e ealis ic desc ip ion o he in e nal low. This e idence is also associa ed wi h
he ac ha he low in he gap o he al es is de eloped wi h a wo high- eloci y je s
s uc u e a bo h sides o he ball. Ob iously, his is a clea es ic ion o he 2D
geome y, ha p obably may be di e en in a eal 3D model whe e he low can be
ci cum e en ially edis ibu ed a ound he ball. No e also he low sepa a ion in he
ou le mani old a =T/4 and he accu a e guidance o he suc ioning low when i is
in oduced in he diaph agm chambe ( =3T/4).
Figu e 17. Desc ip ion o he low pa e ns o he high-p essu e condi ions (6 ba ).
24
5.3. F ee-deli e y condi ions.
To conclude, ee-ou le condi ions a e also analyzed o high (6 ba ) and low (2 ba ) ai -
supplied p essu es. Because o he a mosphe ic condi ions a he pump discha ge, he
d i ing equencies o he diaph agm a e inc eased o 2.7 and 1.85 Hz espec i ely.
Figu e 18, le , shows he e olu ions o he main pa ame e s o he high p essu e case.
No e ha signi ican p essu es (up o 2 ba ) a e a ained in he diaph agm chambe
du ing he o wa d s oke. The mos ele an cha ac e is ic is he long delay o he
closu e o he exhaus ing al e, be ween =0 and T/8. Consequen ly, he e is a e y
impo an educ ion o he deli e ed low a e when he single diaph agm is pumping.
The ball expe iences a simila d op a e eaching i s uppe s oppe (be ween T/4 and
T/2). This is also mani es ed wi h a cha ac e is ic low a e bumping when he diaph agm
is decele a ing owa ds i s TDC ha e eals signi ican ins abili ies. Finally, in he second
hal o he cycle, i is also no iceable he same delay p oblem in he o he al e, wi h a
no able in e nal leakage o he suc ioning low a e be ween T/2 and 5T/8.
Fo he lowe p essu e case, Figu e 18 ( igh ), all he a iables exhibi a simila beha io .
Howe e , since he p essu e in he diaph agm chambe is lowe , he low a e bumping
a he end o he o wa d s oke is also less p onounced. As a consequence, due o a
smoo he ansi ion o he closu e o he exhaus ing al e, he leakage in he suc ion
al e is clea ly a enua ed.
Figu e 18. Tempo al e olu ions o he p essu e, low a es and check al es du ing he
diaph agm cycle o high-p essu e (le ) and low-p essu e ( igh ) ee-deli e y
condi ions.
A close look o he beha io o he al es is p esen ed in Figu e 19 wi h he
ep esen a ion o he o ces and eloci ies ob ained in he balls du ing he back-and-
o h mo ions o he diaph agm. The black solid line ep esen s he al e displacemen ,
he hin solid line is he ball eloci y and he dashed line is he o ce ac ing on he ball.
Plo s on op co espond o he exhaus ing check al e, whe eas plo s on he bo om
show he beha io o he al e in he suc ion egion. As be o e, le column shows he
esul s o he high p essu e and he igh column o he low p essu e.
25
The mos signi ican ea u e is he s ong coupling be ween he mo emen s o he balls.
No e, o ins ance, how du ing he closu e o he exhaus ing al e in he low p essu e
case, he e is an e iden slope change du ing i s displacemen , accele a ed in he inal
pa o he closu e. This e ec is clea ly p oduced by he o al closu e o he suc ion ball
when i eaches he al e sea .
Ano he example is clea ly obse ed in he high p essu e case. A e a pa ial e-opening
o he exhaus ing al e be ween T/4 and T/2, he al e is closed again, exhibi ing some
oscilla ions in i s e ical eloci y, bu also signi ican o ce luc ua ions a ise in he
suc ion al e (which is comple ely closed) in e u n. In addi ion, no e how nega i e
o ces ( o close he al es owa ds hei sea s) a e easily main ained, while posi i e
o ces a e no s able (due o g a i y) so he balls canno be main ained comple ely
opened o he ull s oke.
Figu e 19. Tempo al e olu ions o he eloci y and o ces on he check al es du ing
he diaph agm cycle o high-p essu e (le ) and low-p essu e ( igh ) ee-deli e y
condi ions.
6. CONCLUSIONS
An uns eady nume ical me hodology o he CFD simula ion o Ai -Ope a ed Double
Diaph agm (AODD) Pumps has been de eloped using dynamic meshes wi h laye ing and
emeshing echniques. In pa icula , a se o Use De ined Func ions (UDFs) we e
de eloped o model accu a ely he dynamic mo emen o he pump diaph agm,
implemen ed wi h a p esc ibed sinusoidal mo ion, and he opening and closu e o he
check al es, ully- esol ing he Fluid S uc u e In e ac ion (FSI) o e he balls.
In addi ion, an implici scheme o he g id mo ion was ound o be necessa y o he
s abili y and ealizabili y o he model adop ed. Explici schemes, less ime-consuming,
exhibi ed di e gence p oblems o he check balls app oaching he al e sea s e en wi h
ex emely educed ime s eps. A simpli ied model, wi hou geome ical es ic ions o
he ball s oke, also con i med his e idence. Mo eo e , o speed up he implici i e a i e
p ocess, an adap i e upda ing o he mesh mo ion, ein o ced a he ini ial i e a ions o
e e y ime s ep, was also in oduced.