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FLOW OVER BROAD-CRESTED WEIR WITH INFLOW BY
APPROACH SHAFT – NUMERICAL MODEL
Jakub Majo 1, Ma in O ánus 2 and Zbyněk Zacho al 3
1. AQUATIS a.s., Bo anická 834/56, B no, Czech Republic;
jakub.majo @aqua is.cz
2. Slo ak Uni e si y o Technology in B a isla a, Facul y o Ci il Enginee ing,
Depa men o Hyd aulic Enginee ing, B a isla a, Slo akia;
ma in.o anus@s uba.sk
3. B no Uni e si y o Technology, Facul y o Ci il Enginee ing, Ins i u e o Wa e
s uc u es, B no, Czech Republic; zacho al.z@ ce. u b .cz
ABSTRACT
In he case o low o e ec angula b oad-c es ed wei , whe e he in low is ealized by
app oach sha , occu s in luence o wa e su ace le el by app oach low eloci y. The pape
desc ibes nume ical model o low including wei , app oach and ou le sha . Simula ions o low
we e c ea ed by 2D and 3D model wi h using h ee me hods o u bulen modelling. In his pape a
wa e su ace le el o each model se up is e alua ed and hen i is compa ed wi h measu ed
alues.
KEYWORDS
App oach sha , B oad-c es ed wei , F ee su ace le el, Nume ical model
INTRODUCTION
In p ac ice he e a e also used b oad-c es ed wei s wi h app oach sha o de e mine he
discha ge, e en ual o egula e he wa e su ace le el (labo a o ies, pond inle s uc u es,
was ewa e ea men plan s, wei s o e en ion basins in sewe sys ems, e c.) (Chyba! Nenalezen
zd oj odkazů.).
In he p o essional li e a u e, he low o e he men ioned wei wi h he app oach sha is
desc ibed in he publica ions Chyba! Nenalezen zd oj odkazů., Chyba! Nenalezen zd oj
odkazů. and Chyba! Nenalezen zd oj odkazů., which a e based on ex ensi e expe imen al
esea ch. F om he expe imen al esea ch is known he wa e su ace p o ile in he longi udinal
plane o symme y o he app oach sha and wei , as well as he p essu e heigh on he app oach
sha walls o he ull ange o geome ic dimension a ios used in p ac ice [2], [4]. The wa e
su ace le el in he app oach sha and he eloci y ield in he wake a ea a he wei c es a e
known only o speci ic geome ic and low condi ions [3], [4].
As he au ho s know, a p esen only models o o e low o e b oad-c es ed wei wi h
e ical in low a e pe o med [5], [6], [7], [8] and [9], bu none o he au ho s deal wi h nume ical
modelling o o e low o e b oad-c es ed wei wi h app oach sha .
The aim o he esea ch was c ea ing a sui able nume ical model o he mos eliable
desc ip ion o he low o e ec angula b oad-c es ed wei wi h in low by app oach sha . Model
was alida ed on he basis o measu ed wa e su ace le el p o ile. The esul s o he simula ions
could be used o supplemen he measu ed da a om expe imen al esea ch.
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Fig. 1 – Scheme o low o e b oad-c es ed wei s wi h app oach sha
WEIR GEOMETRY
The wei geome y, app oach sha and ou le sha we e c ea ed in he so wa e FLOW-3D
e sion 11.0.4 [10], in which calcula ions we e also pe o med. Two concep s o model cons uc ion
we e used o e alua e hei ad an ages and disad an ages. The 2D model used he concep o
ixed blocks, which de ined he space o low by wo blocks o he ne wo k (Figu e 2, le ). The
wid h o he ixed blocks was 2 m, he wid h o he blocks o he compu e ne wo k was he same
as he wid h o he wei . The e we e h ee solid blocks. The i s block ( ed in Figu e 2) o med he
opposi e wall o he inle sha wi h espec o he wei . The second block (blue in Figu e 2) o med
a b oad-c es ed wei , an adjacen wall o he inle sha and an adjacen wall o he ou le sha . The
hi d block (yellow in Figu e 2) o med he opposi e wall o he ou le sha . Thei mu ual loca ion
was chosen so ha he nodes o he ne wo k we e in one case a he bounda ies o he
compu a ional a ea and in he o he case we e no . The 3D model used he concep o i e blocks
o a ne wo k o de ine he space h ough which wa e can low (Figu e 2).
The wei had a leng h in he di ec ion o low L = 0.650 m. The side walls o he wei and he
sha we e ele a ed 0.500 m abo e he c es o he wei . The wei , he app oach sha and he
ou le sha we e same wid h b = 0.500 m. The leng h o he app oach sha was l = 0.300 m and i s
heigh was 4 m. In he case o a 3D model due o he educ ion o he numbe o cells, he heigh
was only 2 m. The leng h and heigh o he ou le sha we e he same as o he app oach sha .
The o e low was ee wi h a ully ae a ed nappe in he ou le sha .
SIMULATIONS
To desc ibe he wa e su ace le el a simula ion using a 2D and 3D model o he head
h = 0.253 m (measu ed 0.05 m om he sha wall) a discha ge 0.090 m3/s) was pe o med,
he e o e o a l/h a io o app oxima ely 1.2. Simula ions we e pe o med wi h RANS (Reynolds-
a e aged Na ie -S okes) u bulence models k-ω, k-ε and LES (La ge eddy simula ion). In o ma ion
on indi idual u bulence models can be ound in publica ions [10], [11] and [12], due o hei scope
and gene al knowledge hey a e no gi en he e. S eady low was sol ed wi h a ee wa e su ace,
one incomp essible luid (wa e ) wi h densi y 1000 kg/m3 and kinema ic iscosi y 0,001 m2/s. I was
conside ed wi h a su ace ension 0,073 N/m. The h eshold de ia ion in he calcula ion was se
o 2% [2].
app oach sha
l
h
wei
Q
ou le sha
L
z
x
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The coo dina e sys em has been chosen so ha x coo dina e de ined he leng h o he sha
and c es o wei in he low di ec ion, he y coo dina e de ined he wid h o he sha and he wei
and he z di ec ion he heigh o he sha .
Bounda y condi ions o se e al ypes a e desc ibed in Figu e 2. The bounda y condi ion a
he in low (bo om o he app oach sha ) o 2D simula ion was en e ed as he in low eloci y (V)
co esponding o he speci ic discha ge o he gi en s a e. The bounda y condi ion a he in low o
he 3D simula ion was en e ed as a p essu e (P). The ad an age o en e ing he eloci y is he
di ec calcula ion, he disad an age is ela i ely uns able solu ion. The ad an age o en e ing he
p essu e is a s able solu ion, he disad an age is he indi ec (i e a i e) calcula ion. The wall
bounda y condi ion (W) wi h a hyd aulically smoo h su ace was speci ied on all solid walls. The
symme ic bounda y condi ion (S) was speci ied a he junc ion o he ne wo k blocks. The ee
ou low (O) was en e ed a he ou low sec ion o he ou le sha and a he ai bounda ies (Figu e
2). The ini ial condi ion was he hyd os a ic p essu e dis ibu ion along he heigh o he calcula ion
space om he measu ed wa e le el. The ini ial wa e le el was en e ed as a ho izon al le el o e
he wei c es .
Fig. 2 – Scheme o calcula ion a ea and en e ing bounda y condi ions o 2D model (le ) and o
3D model ( igh )
Du ing he calcula ions, h ee analyses we e pe o med o de e mine he minimum
equi emen s o he models so ha he esul s we e conclusi ely and epea able.
The i s analysis conce ned he in luence o he posi ion o he s uc u ed ec angula mesh
agains he solid walls, when i was shown ha he mu ual posi ion in luences he esul s. The
e ec on he wa e su ace le el was up o 0.004 m, he change was mainly e lec ed in he eloci y
ield nea he walls. Fo easons o epea abili y, a posi ion was chosen whe e he nodes o he
ne wo k co esponded o he su ace o he solid wall.
The second analysis conce ned he e ec o cell size on he esul s. A 2D model wi h a k-ω
u bulence model was used o he analysis. The leng h o he cell edges in he calcula ion a ea
was g adually educed un il he wa e le el did no change by mo e han 0.001 m. This s a e
occu ed a a cell size leng h (in he x and z di ec ion) o 0.005 m in he whole calcula ion a ea. In
he y di ec ion, he size o he cells in he 3D model was se o 0.010 m o e he whole wid h o he
a ea. Guidelines o good mesh quali y speci ied by FLOW-3D was sa is ied. The maximum aspec
z
x
y
z
x
y
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a io wi hin a single cell was less han 3, he maximum adjacen cell size a io was less han 1.25,
he maximum in e -block cell size a io was less han 2, and mesh planes coincide in he
bounda ies.
The hi d analysis conce ned he minimum equi ed sha heigh . A 2D model was used o
he analysis, whe e he de elopmen o he eloci y ield and he in luence o he sha leng h on
he wa e su ace le el we e moni o ed. I u ned ou ha he heigh o he app oach sha 2 m
below he wei c es was su icien due o achie e a cons an eloci y dis ibu ion along he sha
heigh . I was also de e mined ha he heigh o he ou le sha 0.100 m below he wei c es will
su ice, when he posi ion o he ne wo k bounda y does no a ec he wa e su ace le el abo e
he wei .
The o al numbe o cells in he 2D model was 183 911, o which 46 366 we e ac i e. In he
3D model, he o al numbe o cells was 4 114 141, all cells we e ac i e. The s abiliza ion ime was
de e mined based on he change in wa e su ace le el o e ime. The c i e ion was a change o
wa e su ace le el smalle han 0.001 m du ing 10 s. Flow was s abilized in 20 s a he la es .
A s anda d desk op compu e was used o he calcula ions (In el i7, 4 co es, 4,6 GHz, 8 GB RAM).
The calcula ion ime las ed in he case o a 2D model in he ange o 3 o 4 hou s, in he case o
a 3D model 1 o 2 days, depending on he o e low heigh .
EVALUATION AND COMPARISON
Wa e le els in he longi udinal plane o symme y o 2D and 3D models and wa e le els in
he en i e app oach sha we e e alua ed using RANS u bulence models k-ω, k-ε and he LES
model. The e alua ion was pe o med in MS Excel. The alues we e hen compa ed wi h he
measu ed alues [3]. In summa y, he expe imen al model has he leng h o he wei 0.650 m,
wid hs o he wei as well as he sha b = 0.500 m, and heigh o he sha 4.02 m. Discha ge was
0.090 m3/s. The b oad-c es ed wei and he igh downs eam wall o he wei we e made om
polyme hyl me hac yla e wi h a hickness o 0.010 m. The le downs eam wall was made o
wa e p oo plywood wi h a hickness o 0.021 m. Wa e su ace was measu ed by he poin gauge
and discha ge by he elec omagne ic lowme e . F ee o e low was achie ed. The space
unde nea h he nappe was ully ae a ed.
Figu e 3 shows measu ed and calcula ed (2D and 3D) wa e su ace longi udinal p o ile
( u bulence model k-ω). Figu e 4 shows measu ed wa e su ace longi udinal p o ile and calcula ed
(3D model) by RANS u bulence models k-ω, k-ε and he LES model.
F om he compa ison o calcula ed and measu ed wa e su ace p o iles shown in Figu e 3
and Figu e 4 is isible he quan i a i e di e ence. All nume ical models unde es ima e he wa e
su ace le el. The di e ence be ween he calcula ed and measu ed alues is up o −0,016 m. F om
Figu e 3, he wa e su ace p o ile in he 3D model is quali a i ely simila o he measu ed. F om
he abo e, he low simula ion using a 3D model be e cap u es measu ed wa e su ace han
using a 2D model, which is due o he inclusion o ic ion agains he side walls in 3D model. F om
Figu e 4 i is isible ha he k-ω model desc ibes wa e su ace p o ile he mos accu a ely.
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Fig. 3 – Longi udinal wa e su ace p o ile, u bulence model k-ω, in luence o model dimension
Fig. 4 – Longi udinal wa e su ace p o ile, 3D model, in luence o u bulence model
Figu e 5 shows he ela i e e o o he calcula ed wa e su ace le el p o iles o he
indi idual u bulence models, whe e hM is measu ed wa e su ace le el and hV is calcula ed o he
indi idual models.
F om Figu e 5 i is isible ha o x < 0.15 m he u bulence models show a ela i e e o up
o 10%, in he ange 0.15  x  0.65 up o 18 %.
0.05
0.10
0.15
0.20
0.25
0.30
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70
z[m]
x[m]
2D
3D
measu ed
0.05
0.10
0.15
0.20
0.25
0.30
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70
z[m]
x[m]
k-w
k-e
LES
measu ed
k-
e
LES
k-
w

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Fig. 5 - Rela i e e o o he calcula ed wa e su ace p o iles o indi idual models
The wa e su ace le els in he whole app oach sha calcula ed by 2D and 3D model we e
e alua ed in he p og am SMS 12.3. Figu e 6 shows he isolines o he wa e su ace le el ( ela i e
o he wei c es le el) calcula ed by 2D and 3D model using he k-
w
model o u bulence and
isolines om he measu ed alues (linea in e pola ion on a iangula mesh) Chyba! Nenalezen
zd oj odkazů..
Fig. 6 - Wa e su ace le el abo e wei c es [m] in he app oach sha , le – 2D model, middle –
3D model and igh – measu ed, he igh bounda y o he a ea de ines he ups eam edge o he
wei c es
-0.20
-0.15
-0.10
-0.05
0.00-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
(hV-hM)/hM[-]
x[m]
k-w
k-e
LES
k-
w
k-
e
LES
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Fig. 7 – De ia ion [%] o compu ed alues o wa e su ace le el abo e he wei c es in he
app oach sha om measu ed alues, le – 2D model, igh – 3D model, he igh bounda y o he
a ea de ines he ups eam edge o he wei c es
In Figu e 6 he e a e isible di e ences in he wa e su ace le els o he calcula ed 2D and
3D models compa ed o he measu ed. In he Figu e 7 he e is isible he de ia ion o he
compu ed alues o he wa e su ace le el abo e he wei c es in he app oach sha om
measu ed alues.
In he case o 2D model, he wa e su ace le el is cons an ac oss he en i e wid h o he
in low sha . The wa e le el does no co espond quan i a i ely and quali a i ely o he measu ed
wa e le el. The gi en shape is simila o he condi ions a lowe o e low heigh s o la ge leng hs
o he app oach sha [2]. The di e ence be ween he calcula ed and measu ed wa e su ace le el
is up o −0.033 m (de ia ion −12 %). The la ges de ia ions a e a he ups eam edge o he wei
c es . In he case o 3D model, he esul s a e quan i a i ely di e en , bu quali a i ely mo e
simila o hose measu ed. Nea he app oach sha walls, he wa e su ace le el is lowe , in he
middle o he sha i is highe . The di e ence is up o 0.006 m. The di e ence be ween he
calcula ed and measu ed wa e su ace le el is up o −0.030 m (de ia ion −11 %). The la ges
de ia ions a e a he ups eam edge o he wei c es .
CONCLUSION
The nume ical models in he case o low o e a b oad-c es ed wei wi h app oach sha
made i possible o de e mine he low cha ac e is ics ela i ely quickly. The use o solid blocks in
modelling allows o quick model c ea ion, bu a he cos o la ge numbe o inac i e cells and
equen non-ma ching o he su ace wi h he cell edges. Modelling wi hou hei use seems o be
mo e sui able in e ms o accu acy and complexi y o calcula ion. I is necessa y o obse e a
su icien heigh o he app oach and ou le sha s so ha he in luence o he inpu o bounda y
condi ions, he size and posi ion o he cells and su icien simula ion ime o s abilize he low do
no show.
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Quali a i e ag eemen o he calcula ed wa e su ace le el p o ile by 2D and 3D models
wi h RANS u bulence models k-ω, k-ε and LES model is ela i ely good, bu quan i a i e
ag eemen is insu icien . The wa e su ace le el calcula ed by nume ical models in he sec ion o
he app oach sha and in he sec ion o wei c es is signi ican ly unde es ima ed in he whole
longi udinal plane o symme y agains o he measu ed. The ela i e e o o he wa e su ace
le el abo e he app oach sha is up o 10% and abo e he wei c es up o 18%. The RANS
models calcula e he wa e su ace le el p o ile mo e accu a ely han he LES model. Simula ion
using a 3D model gi es a quali a i ely be e in o ma ion abou he wa e su ace le el in he
app oach sha han he 2D model. The calcula ed wa e su ace p o iles using he RANS
u bulence models k-ω and k-ε a e e y simila .
ACKGNOWLEDGEMENT
The a icle was c ea ed wi h he suppo o he p ojec FAST-J-17-4577. In luence o he
app oach sha geome y on he capaci y o b oad-c es ed wei and p ojec FAST-S-18-5084 Flow
wi h wakes in cons uc ions.
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