© Acta hydrotechnica 20/33 (2002), Ljubljana
ISSN 0352-3551
329
UDK / UDC: 519.61/.64:627.8(282.249 Soč a) Prejeto / Received: 17.7.2002
Strokovni prispevek – Professional paper Sprejeto / Accepted: 27.11.2002
MATEMATIČ NO MODELIRANJE TOKA SOČ E NA OBMOČ JU IZTOKA
HIDROELEKTRARNE  PLA VE II
MATHEMATICAL MODELLING OF THE SOČ A RIVER FLOW IN THE
AREA OF THE PLA VE II POWER PLANT OUTFLOW
Matjaž Č ETINA, Mario KRZYK
Pri turbinskem iztoku hidroelektrarne (HE) Plave II se struga Soč e nekoliko razširi, zato na tem
območ ju obstaja nevarnost zasipavanja s prodom. S primerno oblikovanim talnim pragom tik
gorvodno od iztoka je treba pri prodonosnih pretokih tok reke Soč e preusmeriti tako, da poveč ane
hitrosti ob desnem bregu izpirajo prod in prepreč ujejo recirkulacijo. Dvodimenzijski hidravlič ni
izrač un je obravnaval odsek Soč e v dolžini približno 800 m. Za umerjanje modela smo uporabili
enodimenzijske rač une in podatke o gladinah visokih voda za stanje med gradnjo, ko je bila
gradbena jama strojnice HE Plave II zašč itena z zač asnim visokovodnim nasipom. Razmere pred
izgradnjo nasipa pa so nam služile kot referenca za primerjavo s konč nim stanjem po odstranitvi
pomožne pregrade in izgradnji prodnega praga. Pri nižjih pretokih Soč e (Q = 1 00, 200 in 300 m
3
/s)
je bilo raziskanih več različ ic oblike praga. Za konč no predlagano rešitev pa je znač ilna lomljena
oblika v tlorisu, zniževanje krone praga proti desni brežini in hidravlič no ugodno oblikovana
priključ itev praga na desno brežino. Preverili smo bile tudi hidravlič ne razmere  pri visokih vodah.
Pri stoletnem pretoku Q
1 00
 = 271 8 m
3
/s potek gladin na območ ju hiš na levem bregu gorvodno od
pragu kaže, da talni prag tudi v primeru zaproditve do krone razmer, v primerjavi s stanjem pred
zač etkom izgradnje strojnice HE Plave II, ne poslabšuje zaznavno.
Ključ ne besede: dvodimenzijsko hidrodinamič no modeliranje, model PCFLOW2D, Soč a, iztok HE
Plave II, talni prag, prodonosnost, sedimentacija
Due to the widening of the Soč a riverbed near the Hydro Power Plant Plave II outflow, there is a
possibility of sedimentation. It is necessary to build a bottom sill just upstream of the outflow in
order to redirect the flow of the Soč a River in the way that increased velocities near the right bank
could flush bed-load and prevent recirculation. A two-dimensional hydraulic computation was
applied on the 800 m long Soč a River reach. The results of one-dimensional computations and
measured water elevations near the temporarily-built flood levee in the vicinity of the HPP Plave II
power house were used for the model calibration. At lower discharges of the Soč a River (Q = 1 00,
200 and 300 m
3
/s), several variants of the proposed bottom sill were investigated. As a final
solution,  a “broken-line” weir with a lowered crest near the right bank and a hydraulically
optimized connection to the bank were suggested. In addition, the high water stages were also
investigated. Even when the available sedimentation volume is fully filled with sediment to the weir
crest elevation, the water surface elevations near the houses at the left bank just upstream of the
weir were not significantly higher at the flood discharge of Q
1 00
 = 271 8 m
3
/s.
Key words: twodimensional hydrodynamic modelling, PCFLOW2D code, the Soč a River, HPP
Plave II outflow, bottom sill, bed load transport, sedimentation

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
330
1 . UVOD
Pri projektiranju iztoka iz hidroelektrarne
(HE) Plave II je bilo treba upoštevati veliko
prodonosnost reke Soč e. Območ je iztoka (slika
1) bi bilo lahko ogroženo z odlaganjem plavin,
ki jih s seboj nosi reka, kar bi negativno
vplivalo na delovanje hidroelektrarne. Zaradi
dviga dna struge bi se dvignila tudi gladina
spodnje vode, kar bi povzroč ilo zmanjšanje
moč i hidroelektatrne, v ekstremnih primerih pa
bi lahko prišlo tudi do nezaželenega
zasipavanja iztoka. Zato je bil  med zoženim
delom struge reke Soč e dolvodno od mostu v
Desklah in iztokom iz HE Plave II v letu 2001
zgrajen kamnito-betonski prag, ki ima dvojni
namen. Z ustrezno izbrano koto krone preliva
se je za pragom ustvaril prodni zadrževalnik,
ki pri visokih vodah vsaj delno ustavi
transportirani material. Pri določ anju kote
krone praga je bilo treba upoštevati gorvodni
dvig gladine vode, ki bi lahko poplavila
stanovanjske objekte na levem bregu Soč e v
Desklah. Druga vloga praga pa je, da pri
obratovalnih pretokih usmerja tok proti iztoku
iz HE, kjer poveč ane hitrosti vodnega toka
prepreč ujejo odlaganje rinjenih plavin.
1 . INTRODUCTION
At the design process of the outflow from
the Plave II hydro power plant (HPP) it was
necessary to take into account the large bed-
load transport capacity of the Soč a River. The
area of the Soč a River near the outflow (Figure
1) could be affected by sedimentation and the
power plant operation could become less
effective. The rise of the river bottom would
increase the tailwater surface elevations. As a
consequence, the HPP power capacity would
be reduced or the plant operation completely
stopped by the sedimentation of the outflow.
To prevent such a situation, a concrete bottom-
sill was built in 2001 in the Soč a riverbed just
upstream of the HPP Plave II outflow. The
first aim of the sill was to form a sediment
retention upstream, where bed-load material
could be partly trapped at flood discharges.
The bottom-sill crest elevations were
determined by taking into account the
backwater effect, which could cause floods at
the houses on the left bank of the Soč a River
near the village of Deskle. The second aim of
the bottom-sill was to redirect the flow of the
Soč a River towards the HPP outflow, where
bed-load deposition should be prevented by
increased flow velocities.
Slika 1. Položaj obravnavanega odseka Soč e z vrisanim območ jem matematič nega modela.
Figure 1 . Layout of the Soč a River reach with the area of the mathematical model.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
331
Za reševanje podobnih problemov se v
hidrotehnični praksi običajno uporabljajo
enodimenzijski matematič ni modeli. Zaradi
znač ilne oblikovanosti struge reke Soč e na
obravnavanem območ ju ter zaradi izbranega
položaja in oblike praga so bile prič akovane
znatne spremembe gladine tudi v preč ni smeri.
Nastal je poudarjeno neenakomeren tok z
izrazito več jimi hitrostmi ob desni brežini. Z
enodimenzijskimi modeli takšnih
neenakomernih razporeditev hitrosti po
preč nem preseku ne bi bilo mogoč e ustrezno
ponazoriti.
 Zato je bil za simulacijo toka reke Soč e č ez
prag in ob iztoku iz HE Plave II izbran
dvodimenzijski hidrodinamič ni matematič ni
model PCFLOW2D. S tem modelom lahko
napovemo potek gladine tudi v preč ni smeri in
izrač unamo globinsko povpreč ne hitrosti  toka
v vodoravni ravnini. S pomočjo analize
tokovnih razmer tako pri obratovalnih kot tudi
pri visokovodnih pretokih je bil ob
sodelovanju projektantov (Fazarinc, 2001;
IBE, 1997) določ en položaj, oblika in višinski
položaj praga.
Podrobna simulacija toka nad samim
pragom bi sicer zlasti pri nizkih pretokih
zahtevala uporabo zahtevnejšega
tridimenzionalnega modela. Vendar nas je v
okviru opisane študije zanimalo predvsem
globinsko povprečno hitrostno polje tik
dolvodno od praga in vpliv na gladine
gorvodno, zato je bil izbran dvodimenzijski
model PCFLOW2D, ki je bil že uspešno
uporabljen za simulacijo podobnega primera
toka preko talnih pragov v kajakaški progi v
Tacnu (Č etina & Rajar, 1993).
2. MATEMATIČ NI MODEL
2.1 OSNOVNE ENAČ BE
V dvodimenzijskem matematič nem modelu
PCFLOW2D so bile za obravnavani primer
reke Soč e upoštevane kontinuitetna (1) in
dinamič ni enač bi v x in y  smeri (enač bi (2) in
(3)) za stalni globinsko povprečni tok.
Koeficient efektivne viskoznosti υ
ef
 določ imo
To solve similar problems in hydraulic
engineering, one-dimensional mathematical
models are usually used. Due to the specific
Soč a riverbed configuration and the chosen
layout and shape of the bottom-sill, significant
changes of the water surface elevations were
expected in the lateral direction, too. A
pronounced non-uniform flow with higher
velocities near the right bank occurred.  One-
dimensional models would not be able to
simulate such non-uniform velocity
distributions in river cross sections in an
appropriate way.
 The two-dimensional PCFLOW2D
mathematical model was chosen to simulate
the Soč a River flow over the bottom sill near
the Plave II HPP outflow. The model could
predict changes of water surface elevations in
longitudinal and lateral directions, and
calculate the depth-averaged velocity field.
The detailed analysis of the flow at operational
and flood discharges helped the designers
(Fazarinc, 2001; IBE, 1997) to determine the
final layout, shape and crest elevations of the
bottom-sill.
At low discharges, a more sophisticated
three-dimensional model would be needed to
simulate flow details over the bottom sill. But
since the main purpose of the study was to
simulate the depth-averaged velocity field
downstream of the sill and a backwater effect
upstream, the two-dimensional model
PCFLOW2D was chosen. It has already been
successfully used for the study of similar flow
over the bottom weirs in the Tacen kayak
racing channel  (Č etina & Rajar, 1993).
2. MATHEMATICAL MODEL
2.1 BASIC EQUATIONS
For the case of the Soč a River flow in the
PCFLOW2D two-dimensional mathematical
model, the continuity (1) and dynamic
equations in the x and y directions (equations
(2) and (3)) for steady depth-averaged flow
were used. The coefficient of effective

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
332
s pomoč jo enač be (6) in dveh dodatnih
transportnih enač b (4) in (5) za k in ε , kar je
znano kot t. im. globinsko povpreč na verzija
k -ε modela turbulence (Rodi, 1980).
Podroben opis enač b in pomen posameznih
č lenov v njih je možno najti v literaturi (npr.
Č etina, 1989; Č etina & Rajar, 1993).
turbulent viscosity υ
ef
 was determined by
equation (6), and the two additional transport
equations (4) and (5) for k and ε which are
known as the depth-averaged versions of the
k -ε turbulence model (Rodi, 1980). A more
detailed description of the equations and
individual terms can be found in the literature
(e.g. Č etina, 1989; Č etina & Rajar, 1993).
0
) ( ) (
= +
y
hv
x
hu
∂
∂
∂
∂
(1)
    ) ( ) (
) ( ) (
3
4
2 2
2
2
y
u
h
y x
u
h
x
h
v u u
ghn
x
z
gh
x
h
gh
y
huv
x
hu
ef ef
b
∂
∂
υ
∂
∂
∂
∂
υ
∂
∂
∂
∂
∂
∂
∂
∂
∂
+ +
+
− − − = + (2)
      ) ( ) (
) ( ) (
3
4
2 2
2
2
y
v
h
y x
v
h
x
h
v u v
ghn
y
z
gh
y
h
gh
y
hv
x
huv
ef ef
b
∂
∂
υ
∂
∂
∂
∂
υ
∂
∂
∂
∂
∂
∂
∂
∂
∂
+ +
+
− − − = + (3)
             
kv D
k
ef
k
ef
hP h c hG
y
k
h
y x
k
h
x y
hvk
x
huk
+ − + + = + ε
∂
∂
σ
υ
∂
∂
∂
∂
σ
υ
∂
∂
∂
∂
∂
) ( ) (
) ( ) (
(4)
      
v
ef ef
hP h
k
c hG
k
c
y
h
y x
h
x y
hv
x
hu
ε
ε ε
ε ε
∂
∂ε
σ
υ
∂
∂
∂
∂ε
σ
υ
∂
∂
∂
ε ∂
∂
ε
+ − + + = +
2
2 1
) ( ) (
) ( ) (
  (5)
ε
υ υ υ υ
µ 2
k
c
t ef
+ = + =                                                 (6)
Oznake pomenijo: h - globina vode, u in v -
komponenti hitrosti v x in y smeri, z
b
 - kota
dna, n - Manningov koeficient hrapavosti, υ
ef
,
t
υ in υ - kinematič ni koeficienti efektivne,
turbulentne in molekularne viskoznosti, g -
gravitacijski pospešek, k - turbulentna
kinetič na energija na enoto mase in ε - stopnja
njene disipacije. Izrazi za G (produkcija k
zaradi vodoravnih gradientov hitrosti) ter P
kv
 in
P
v ε
 (izvorna č lena zaradi trenja ob dno) so
skupaj s konstantami v modelu turbulence (c
D
,
c
µ , c
1
, c
2
, σ
k
 in σ
ε
) podani v literaturi (Rodi,
1980).
Notations: h – water depth, u and v – velocity
components in the x and y directions, z
b
 -
bottom elevations, n – Manning’s friction
coefficient, 
ef
υ , 
t
υ and υ - kinematic
coefficients of  the effective, turbulent and
molecular viscosity, g - gravity acceleration, k
- turbulent kinetic energy per unit mass and ε
- the rate of its dissipation. The expressions of
G (the production of k due to horizontal
velocity gradients), P
kv
 and P
v ε
 (source terms
due to the bottom friction), as well as the
values of the standard turbulent constants (c
D
,
c
µ , c
1
, c
2
, σ
k
 and σ
ε
), can be found in the
literature (Rodi, 1980).

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
333
2.2 ROBNI POGOJI
Pri reševanju enač b (1) do (5) potrebujemo
robne pogoje na vseh štirih straneh rač unskega
področ ja, saj gre zaradi vključ enih difuzijskih
č lenov z efektivno viskoznostjo za eliptič en tip
problema. Za konkreten primer toka Soč e smo
upoštevali spodaj navedene robne pogoje, pri
č emer so hitrosti v usmerjene v y smeri vzdolž
toka, hitrosti u pa v smeri x preč no na tok.
Na gorvodnem robu modela so predpisane
hitrosti. Vtoč na hitrost v se v vsaki iteraciji
izrač una tako, da vtok v model ustreza
obravnavanemu stalnemu pretoku Q
0
, preč na
hitrost u pa se izrač una ob pogoju 0 / = y u ∂ ∂ .
Ker so popravki vtoč nih hitrosti enaki nič ,
morajo biti na vtoku enaki nič tudi gradienti
popravkov gladin h' v vzdolžni smeri,
∂ ∂ hy '/ = 0 (Patankar, 1980; Č etina, 1988). Na
dolvodnem robu smo podali v preč ni smeri
vodoravno koto gladine z ter upoštevali, da so
vzdolžni gradienti hitrosti enaki nič , torej
∂ ∂ vy / = 0 in ∂ ∂ uy / = 0. Ustrezne kote z za
spodnji rob dvodimenzijskega modela smo
povzeli iz poroč ila IBE (1997). Na vseh trdnih
robovih so preč ne hitrosti enake 0, za vzdolžne
pa smo upoštevali veljavnost logaritemskega
stenskega zakona (Č etina, 1988; 1989).
Tudi za k in ε potrebujemo robne pogoje
na vseh štirih robovih rač unskega področ ja.
Podrobneje so navedeni v ustrezni literaturi
(Rodi, 1980).
2.3 METODA REŠEVANJA IN
RAČ UNALNIŠKI PROGRAM
PCFLOW2D
Sistem nelinearnih parcialnih diferencialnih
enač b (1) - (6) rešujemo numerič no z metodo
konč nih prostornin (Patankar, 1980; Č etina,
1989). Njene temeljne značilnosti so
premaknjena numerična mreža, hibridna
shema (kombinacija centralnodiferenč ne in
sheme gorvodnih razlik) ter iteracijsko
reševanje na podlagi popravkov globin. V
primeru upoštevanja nestalnega toka je
uporabljena polna implicitna shema (Č etina,
1988).
2.2 BOUNDARY CONDITIONS
To solve the elliptic equations (1) to (5) that
include terms with effective viscosity,
boundary conditions were needed on all four
sides of the computational domain. If we
consider that velocities v are directed
streamwise along the y axis, and velocities u,
meanwise along the x axis, in the case of the
Soč a River the following boundary conditions
were taken into account.
 At the upstream end of the model, the
velocities were prescribed. In each iteration,
the inflow velocity v was computed according
to the steady discharge Q
0
, while the lateral
velocities u were determined by the condition
0 / = y u ∂ ∂ . Since inflow velocity corrections
are zero, the longitudinal gradients of depth
corrections h' at the upstream end are also
zero, ∂ ∂ h y '/ = 0 (Patankar, 1980; Č etina,
1988). At the downstream end, the horizontal
water surface elevation was prescribed, and
the longitudinal velocity gradients were zero,
∂ ∂ v y / = 0 and ∂ ∂ u y / = 0. The appropriate
surface elevations, z, at the downstream end of
the two-dimensional model, were obtained
from the IBE report (1997). At all solid
boundaries, the normal velocities were 0 and
the longitudinal velocities were computed
from the logarithmic law of the wall (Č etina,
1988; 1989).
The boundary conditions for k and ε that
are also needed on all four boundaries of the
computational domain can be found in Rodi
(1980).
2.3 METHOD OF SOLUTION AND *THE
PCFLOW2D COMPUTER CODE
The set of non-linear partial differential
equations (1) – (6) was solved numerically by
the finite volume method (Patankar, 1980;
Č etina, 1989). The main characteristics of the
method are: a staggered numerical grid; a
hybrid scheme (a combination of central-
difference and upwind scheme) and an
iterative procedure of depth corrections. In the
case of unsteady flow computations, a fully
implicit scheme is used (Č etina, 1988).

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
334
Program PCFLOW2D, ki je prirejen za
uporabo na osebnih rač unalnikih, je bil razvit
na UL FGG v Ljubljani (Č etina, 1988).
Program se stalno izpopolnjuje z možnostjo
upoštevanja novih vrst robnih pogojev, z
vgradnjo novejših numerič nih metod ter z
razvojem prijaznejših uporabniških
vmesnikov. Simulacije za Soč o smo izvedli na
rač unalniku s procesorjem Pentium III s
frekvenco delovanja 900 MHz, tako da so
kljub numerič ni zahtevnosti posamezni rač uni
trajali le dobre pol ure.
Za vsak računski pretok program
PCFLOW2D sam poišče spremenljive
geometrijske meje rač unskega področ ja. Za
dosego konvergentne rešitve je bilo povpreč no
potrebnih okrog 3000 iteracij pri relativni
napaki, ki ne presega enega odstotka
rač unskega pretoka skozi področ je.
3. UMERJANJE MODELA
3.1 VHODNI PODATKI
3.1.1 GEOMETRIJSKI PODATKI
Za pripravo matematič nega modela struge
reke, ki je bil podlaga za pridobivanje
informacij o kotah dna v posameznem
rač unskem polju, so bili uporabljeni
razpoložljivi geodetski podatki, katerih viri so
podrobno navedeni v strokovnem poroč ilu
(Č etina & Krzyk, 2001). Za odsek gorvodno
od praga je bilo na voljo 13 preč nih profilov
struge reke Soč e, ki so bili posneti na
medsebojni razdalji od 13 do 78 m v skupni
dolžini 325 m in podani s 173 geodetsko
izmerjenimi točkami, in situacijski nač rt
varovalnega nasipa gradbene jame strojnice
HE Plave II. Za odsek dolvodno je bil za stanje
pred izgradnjo praga upoštevan enakomeren
padec dna v dolžini približno 200 m do kote
75.0 m n.m. Za stanje po izgradnji praga pa je
bilo upoštevano poglobljeno dno struge Soč e
do kote 75.0 m n.m. Širina poglobljenega
korita je 45 m, brežine obeh bregov so
oblikovane v naklonu 1:2, dolžina poglobitve
je 860 m dolvodno od iztoka HE Plave II. V
primerih, kjer je bila simulirana zapolnitev
prodne lovilne jame za pragom z odloženimi
The PCFLOW2D computer code was
developed at the UL FGG in Ljubljana
(Č etina, 1988), and it was adopted to run on
personal computers. Recently additional
boundary conditions, new numerical methods
and user-friendly interfaces have been
implemented. The simulations of the Soč a
River flow were performed on the PC
computer, with a Pentium III processor
running at 900 MHz. In spite of the numerical
complexity of the calculations, they took only
about half an hour of the computer time.
For different discharges, the PCFLOW2D
program is capable of adopting the boundaries
of the computational domain automatically. To
achieve the convergent solution, an average
number of about 3000 iterations was needed to
lower the numerical error below 1% of the
total discharge.
3. MODEL CALIBRATION
3.1 INPUT DATA
3.1.1 GEOMETRIC DATA
To prepare the digital terrain model, which
has served as a basis for the information about
bottom elevations in numerical cells, geodetic
data from different sources were used (Č etina
& Krzyk, 2001). For the 325 m long Soč a
River reach upstream from the bottom-sill, 13
cross-section profiles were given at distances
from 13 to 78 m that were surveyed with 173
geodetic points. The layout of the protective
levee of the HPP Plave II powerhouse
building-ground was also available. For the
downstream reach before the bottom-sill
construction, the uniform bottom slope was
supposed to be a length of 200 m to the bottom
elevation of 75,0 m above sea level. After the
bottom-sill construction, an excavation of
sediments to the bottom elevation of 75,0 m
above sea level and a river training of the 860
m long reach downstream of the HPP Plave II
outflow was performed. The channel width of
45 m and the bank slope of 1:2 was taken into
account. In the cases where the fulfilment of
the sediment retention capacity upstream of

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
335
rinjenimi plavinami, je bilo predpostavljeno
vodoravno oblikovano dno od praga do
zoženega dela struge reke Soč e.
Geodetske podatke v grafič ni in številč ni
obliki je bilo treba ustrezno obdelati in
pripraviti v obliki, ki bi bila najbolj primerna
za nadaljnjo pripravo rač unalniškega modela
struge. S pomoč jo programa QuickSurf, ki je
dodatek grafič nemu programu AutoCAD, in s
pomoč jo lastnih pomožnih programov za delo
z geodetsko podanimi točkami, je bila
oblikovana podatkovna baza izmerjenih ali
projektiranih toč k dna struge in brežin reke
Soč e ter predvidenega praga. Na podlagi
izbranega nač ina interpolacije je nato program
QuickSurf oblikoval površino struge reke in
določ il kote dna vseh toč k numerič ne mreže.
Uporabljena je bila razmeroma gosta
neenakomerna numerična mreža z 291
toč kami v vzdolžni y smeri, kjer je bila
razdalja ∆ y = 2 m na gorvodnem delu
obravnavanega območ ja dolžine 445 m in
več ja ∆ y = 4 m na območ ju 120 m dolvodno
od praga do konca modela. Numerič na mreža
je bila v preč ni x smeri enakomerna s 86
toč kami (∆ x = 2 m). Dolžina rač unskega
področ ja (vzdolž toka) je 712 m, širina pa 169
m (slika 2). V strugi je prikazan tudi položaj
praga, katerega prerez podaja slika 3.
the bottom-sill was simulated, the horizontal
river bottom was approximated from the sill to
the upstream narrow of the Soč a River.
The raw geodetic data in graphical and
numerical form had to be processed for the
further preparation of the digital terrain model.
With the use of the QuickSurf and AutoCAD
graphic packages and our own intermediate
programs to manipulate with geodetic points, a
large data base of measured and designed
terrain and river bed (including different
shapes of the bottom-sill) points was formed.
On the basis of different interpolation
methods, the QuickSurf package was capable
of forming the riverbed surface, and, as a final
result, the bottom elevations at all points of the
numerical mesh were determined. A relatively
dense non-uniform numerical mesh with 291
points in the longitudinal y direction was used.
The chosen space step at the 445 m long
upstream part of the model was ∆ y = 2 m,
while the space step increased to ∆ y = 4 m at
the downstream end of the model. In the
lateral x direction, a uniform mesh had 86
points with the space step ∆ x = 2 m.  The total
length of the computational domain was 712 m
and the total width was 169 m (Figure 2). The
cross section of the sill which is situated at the
riverbed can be seen from Figure 3.
x
y
Q
N
Slika 2. Rač unsko področ je in uporabljena numerič na mreža.
Figure 2. Computational domain and the numerical mesh.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
336
Cross-section "B" Prerez "B"
10
74
0
Prerez "C"
76
75
77
78
74
0
76
75
78
77
20 30
Cross-section "C"
10 20 30
Prerez "A"
0
74
75
76
77
78
79
10
A
75.0
B
75.0
C
Cross-section "A"
20
75.0
76.0
79.0
30
80.0
HPP outflow
Iztok HE
Slika 3. Preč ni prerez praga.
Figure 3. Cross-section of the sill.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
337
3.1.2 HIDROLOŠKI IN HIDRAVLIČ NI
PODATKI
Hidrološki podatki o pretokih so povzeti po
poroč ilu IBE (1997). V tem poroč ilu so
obdelani pretoki Soč e na območ ju predvidene
elektrarne Plave II na podlagi hidrološke
študije povodja reke Soč e, ki jo je izdelal
Vodnogospodarski inštitut Ljubljana (VGI,
1982). Pri analizi toka č ez prag nad iztokom iz
hidroelektrarne so bili upoštevani naslednji
pretoki: 100, 200 in 300 m
3
/s ter pretok s
povratno dobo 100 let, ki znaša po omenjeni
študiji 2718 m
3
/s. Poleg glavnih pretokov v
strugi Soč e so bili pri nižjih pretokih dodatno
upoštevani še iztoki iz HE Plave II do
vrednosti instaliranega pretoka (105 m
3
/s).
3.1.2 HYDROLOGIC AND HYDRAULIC
DATA
The hydrologic data about discharges was
taken from the IBE report (1997). The
characteristic discharges of the Soč a River in
the HPP Plave II cross-section were elaborated
on the basis of the hydrologic study of the
Soč a River basin, which was performed by the
Water Management Institute in Ljubljana
(VGI, 1982). At the flow analysis over the sill,
the following discharges were taken into
account: 100, 200 and 300 m
3
/s, and the flood
discharge 2718 m
3
/s, which has a return period
of 100 years. At lower discharges, the
additional outflow from the HPP Plave II in
the amount of 105 m
3
/s was also considered.
Tudi podatki o kotah gladine vode so bili
povzeti po poroč ilu IBE (1997). Gladine pri
enodimenzijskem rač unu zajezitve med HE
Solkan, HE Plave I in HE Plave II so bile
določ ene s pomoč jo programa DRAGLA iz
programskega paketa HIDRO90 (Širca, 1990).
Izrač uni so bili opravljeni na podlagi 42
preč nih profilov, ki jih je leta 1982 posnel
Geodetski zavod Maribor, saj novejših izmer
geometrije struge ni bilo na voljo. Umerjanje
enodimenzijskega modela je bilo opravljeno
na podlagi dveh podatkov o gladinah: kote
gladine v akumulaciji Solkan in kote v profilu
HE Plave I, in sicer za pretoke med 200 in
2500 m
3
/s. Za različ ne pretoke so bili ustrezno
določ eni koeficienti hrapavosti. Rezultat teh
rač unov so bile tudi kote gladine vode v
posameznih profilih. Kote gladine v profilu
P40 na stacionaži km 13 + 017 (sliki 4 in 5) so
bile uporabljene pri določ anju dolvodnega
robnega pogoja za dvodimenzijski
matematični model. Zadnji profil
dvodimenzijskega modela leži približno 200 m
dolvodno, zato so kote v tem profilu nekaj
nižje kot v P40, določ ene pa so bile s
poskusnimi rač uni, tako da v P40 dobimo
ustrezno koto (Č etina & Krzyk, 2001).
Uporabljeni dvodimenzijski matematič ni
model nam dopušča možnost podajanja
različ nih vrednosti koeficienta hrapavosti po
posameznih podpodroč jih. Vendar bi za
njegovo natančno določ anje potrebovali
The data about water surface elevations
were also taken from the IBE report (1997).
The one-dimensional calculations of the
backwater effect between the HPP Solkan, the
HPP Plave I and the HPP Plave II were
performed using the DRAGLA computer code
from the HIDRO90 program package (Širca,
1990). Due to a lack of more recent data, the
calculations were made on the basis of 42
cross-sectional profiles that had been
measured by the Geodetic service, Maribor, in
1982. For discharges between 200 to 2500
m
3
/s, the calibration of the one-dimensional
model was performed by the comparison with
measured water surface elevations in the
Solkan reservoir and the HPP Plave I cross-
section. With calibrated friction coefficients,
the water surface elevations were computed at
different discharges. The water surface at the
P40 cross-section (km 13 + 017, Figures 4 and
5), served as the downstream boundary
condition for the two-dimensional
mathematical model.   Since the last cross-
section of the two-dimensional domain was
located about 200 m downstream from P40,
the boundary water surface elevations were
slightly lower, and obtained by trial and error
calculations (Č etina & Krzyk, 2001).
In the two-dimensional model which was
used, it was possible to prescribe different
friction coefficients for the sub-areas. To
determine the coefficients more accurately,

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
338
terenske meritve hitrosti in gladin pri visokih
vodah, ki bi bile bolj natanč ne in bolj
zanesljive od razpoložljivih. V študiji toka
reke Soč e z enodimenzijskim matematič nim
modelom so bile uporabljene okvirne
vrednosti koeficienta hrapavosti po Manningu
v mejah n = 0,041 - 0.059 sm
−13 /
 (IBE, 1997),
ki so bile določ ene za pretoke od 200 do
1600 m
3
/s. Glede na te vrednosti, znač ilnosti
struge reke Soč e na obravnavanem odseku s
predlaganimi vrednostmi iz literature in
predvsem glede na dosedanje izkušnje z
nekoliko nižjimi koeficienti hrapavosti pri
dvodimenzijskih modelih, so bile za zač etno
umerjanje uporabljene vrednosti za n med 0.03
in 0.05 sm
−13 /
.
3.2 POSTOPEK UMERJANJA
Umerjanje matematič nega modela pomeni
predvsem postopek, pri katerem določ imo
porazdelitev koeficientov hrapavosti v strugi.
Po določitvi okvirnih fizikalnih mej
(podpoglavje 3.1.2) smo toč nejšo vrednost
koeficienta hrapavosti za obravnavano
območ je določ ili z upoštevanjem naslednjih
dejavnikov: terenskega ogleda, naših
dosedanjih izkušenj, primerjave rezultatov
dvodimenzijskega izrač una z enodimenzijsko
izrač unanimi kotami z modelom DRAGLA in
primerjave izrač unanih kot gladine vode z
merjenimi vrednostmi.
Pri pretokih do 500 m
3
/s smo umerjanje
izvedli predvsem na podlagi primerjave gladin
z ustreznimi enodimenzijskimi rač uni IBE
(1997). Za določ itev koeficienta hrapavosti pri
višjih pretokih pa smo se najbolj opirali na
podatke Soških elektrarn o izmerjenih
gladinah ob nasipu gradbene jame pri visoki
vodi v oktobru 2000, ko je pretok dosegel Q =
1700 m
3
/s. Pri umerjanju so bili smiselno
uporabljeni tudi razpoložljivi podatki
Vodnogospodarskega inštituta o sledovih
visoke vode na levem bregu, predvsem za
približno ugotavljanje poteka gladin tudi v
preč ni smeri. Konč ni rezultat umerjanja pri Q
= 1700 m
3
/s in pri upoštevanju nasipa
gradbene jame je razviden iz slik 4 (hitrostno
polje) in 5 (kote gladin v preč nih in podolžnih
additional field measurements of velocities
and surface elevations at flood discharges were
needed. One-dimensional calculations of the
Soč a River flow were performed with the
Manning’s friction coefficient values from n =
0,041 to 0,059 sm
−13 /
 (IBE, 1997) for
discharges between 200 and 1600 m
3
/s.
According to these values, the characteristics
of the Soč a riverbed on the investigated area,
proposed values from the literature and our
previous experiences with the behaviour of
two-dimensional models, which usually
require lower friction coefficients, we started
the calibration process with values between
n = 0.03 and 0.05 sm
−13 /
.
3.2 CALIBRATION PROCESS
The main aim of the calibration process was
to find out the distribution of friction
coefficients in the riverbed.   After the rough
limits were set (chapter 3.1.2), more accurate
values of the roughness coefficients were
determined by taking into account the
following factors: field examinations; the
comparison between the results of the two-
dimensional model with water surface
elevations obtained by the one-dimensional
DRAGLA model and the comparison between
the computed and measured water levels.
At discharges below 500 m
3
/s, the
calibration was mainly performed by the
comparison of water levels with one-
dimensional computations (IBE, 1997). To
determine friction coefficients at higher
discharges, we relied on the data about water
levels near the protective levee that were
recorded by the “Soč a River Hydro Power
Plants” company at the flood discharge Q =
1700 m
3
/s. During the calibration process,
available data about flood signs at the left
bank, obtained by the Water Management
Institute, were also taken into account. They
were mainly used to assess the shape of the
water surface in the lateral direction.  The final
result of the calibration process at Q = 1700
m
3
/s and for the case with the protective levee
can be seen from Figure 4 (velocity field) and
Figure 5 (computed and measured water

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
339
profilih vzdolž ravnih linij z vrisanimi
merjenimi gladinami). Na levem bregu je bila
med profiloma P40.4 in P40.5 izmerjena kota
gladine z = 85.05 m in izrač unana z = 85.06 m,
med profiloma P40.6 in P41 pa izmerjena z =
85.35 m in izrač unana z = 85.34 m. Merjene in
izrač unane kote gladin na desnem bregu so
bile z = 85.25 m in z = 85.27 m (približno v
profilu P40.5) ter z = 85.41 m in z = 85.47 m
(med profiloma P40.6 in P41). Na podolžnem
profilu gladin sta med preč nima profiloma
P40.6 in P41 vnešeni izmerjeni koti z = 85.35
m (levo) in z = 85.41 m (desno), izrač unani
koti blizu levega  in desnega brega pa sta bili z
= 85.32 m in z = 85.46 m. Izbrana koeficienta
hrapavosti, ki smo jih nato uporabili pri
nadaljnjih rač unih,  sta bila n = 0.05 sm
−13 /
 za
pretoke do Q = 500 m
3
/s in n = 0.04 sm
−13 /
 za
višje pretoke.
Toč nost uporabljenega modela je, glede na
to, da lahko zajamemo tok v dveh dimenzijah,
razmeroma dobra. Pri točnih vhodnih
hidroloških, hidravličnih in geometrijskih
podatkih ocenjujemo, da bi bila toč nost
modela na obravnavanem odseku pri nizkih
pretokih do Q = 500 m
3
/s v mejah ± 5 cm, pri
stoletnem pretoku Q
1 00
 pa v mejah ± 10 cm.
To je hkrati tudi natanč nost, s katero lahko
ugotavljamo vplive relativnih sprememb
gladine zaradi različ nih ukrepov v strugi
(predvsem oblike in višine praga ter
poglobitve struge) glede na izbrano referenč no
stanje.
Vendar pa največ jo napako vnesemo v
rač un z netoč nimi vhodnimi podatki. Kot je
pokazalo umerjanje, imamo v primeru Soč e na
območju iztoka HE Plave II, kljub
geometrijskim in geodetskim podatkom o
topografiji struge in terena, na voljo premalo
natanč ne podatke o spodnjem robnem pogoju
ter relativno malo zanesljivih meritev gladin (v
samo štirih toč kah). Ker pa gre za relativno
kratek odsek, na podlagi umerjanja
ocenjujemo, da je točnost izrač unanih
absolutnih kot na odseku med ± 15 cm pri Q =
1700 m
3
/s in ± 20 cm pri Q
1 00
 = 2718 m
3
/s.
surface elevations in cross-section and
longitudinal profiles along straight lines). At
the left bank, between cross-sections P40.4
and P40.5, measured and computed water
surface elevations were z = 85.05 m and z =
85.06 m, respectively, and between cross-
sections P40.6 and P41, z = 85,35 m
(measured) and z = 85.34 m (computed). The
measured and computed water surface
elevations at the right bank were z = 85.25 m
and z = 85.27 m (approx. in the cross-section
P40.5) and z = 85.41 m and z = 85.47 m
(between the cross-sections P40.6 and P41). At
the longitudinal water surface profile, the
measured elevations z = 85.35 m (left) and z =
85.41 m (right) between cross-sections P40.6
and P41 are marked, while the computed
values near the left and right banks were z =
85.32 m and z = 85.46 m, respectively. The
calibrated friction coefficients for further
computations were n = 0.05 sm
−13 /
for
discharges below 500 m
3
/s, and n = 0.04 sm
−13 /
for higher discharges.
Since it was possible to simulate flow in
two dimensions, the accuracy of the model is
relatively high. If the input hydrologic,
hydraulic and geometric data were exact, the
accuracy of the model would be ± 5 cm at
lower discharges, and below 500 m
3
/s and
± 10 cm at Q
1 00
= 2718 m
3
/s. With this
accuracy it is possible to investigate changes
of water levels, relative to the reference state,
due to different training measures in the
riverbed (particularly the shape and height of
the bottom-sill and the deepening of the river
bed).
But the largest error could be introduced by
the inexact input data. The calibration process
in the case of the Soč a River near the HPP
Plave II outflow showed the lack of certain
data. In spite of geometric and geodetic
riverbed and terrain data, more accurate water
levels at the downstream end of the model and
more reliable measurements during floods
would be needed (water levels in 4 points were
available only). But since the simulated reach
is relatively short, the calibration process
suggests that the accuracy of the computed
absolute water surface elevations are between
± 15 cm at Q = 1700 m
3
/s and ± 20 cm at Q
1 00
= 2718 m
3
/s.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
340
P40.4
km 13+364
P41
P40.6
P40.5
Detajl
P40.3
P40.2
P40.1
km 13+017
P40
P40.4
P40.3
P40.2
P40.1
Dolžin, Length
Hitrosti, Velocity
Merilo, Scale
Merilo, Scale
Hitrosti, Velocity
Dolžin, Length
Slika 4. Umerjanje modela – izrač unano hitrostno polje pri pretoku Q = 1700 m
3
/s.
Figure 4. Model calibration – computed velocity field at the discharge Q = 1 700 m
3
/s.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
341
P40.5
P41
km 13+364
P40.6
P40.4
P40.3
P40.1
P40.2
km 13+017
P40
P40.5
P41
km 13+364
P40.6
P40.4
P40.3
P40.1
P40.2
km 13+017
P40
84.5
84.7
84.5
84.9
85.1
Measurement in October 2000 *
84.9
84.7
85.1
85.5
85.3
85.5
85.3
84.5
*
84.9
84.7
85.1
85.5
85.3
*
84.5
*
85.1
84.5
84.9
84.7
85.5
85.3
84.7
84.9
84.7
84.9
84.5
84.7
84.5
84.9
85.1
84.9
84.5
84.7
85.1
84.7
84.5
84.9
85.3
85.5
85.1
84.7
84.9
84.5
84.7
85.5
85.3
84.9
85.1
84.5
84.7
85.5
85.3
84.9
85.1
84.5
Meritev oktober 2000
*
*
*
Slika 5. Umerjanje modela – primerjava izrač unanih in merjenih gladin pri Q = 1700 m
3
/s.
Figure 5. Model calibration – comparison between computed and measured water surface
elevations at Q = 1 700 m
3
/s.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
342
4. REZULTATI IZRAČ UNOV
Po že opisanem umerjanju modela pri
pretoku Q = 1700 m
3
/s je bil ob upoštevanju
nasipa gradbene jame in brez poglobitve struge
dolvodno od iztoka izveden še primer rač una s
Q
1 00
 = 2718 m
3
/s. Ta rač un smo potem
uporabili kot referenč no stanje, s katerim so
bili primerjani konč ni rač uni. Kot konč no
stanje  je bila upoštevana odstranitev nasipa
gradbene jame, različ ne različ ice položaja in
oblike talnega pragu in horizontalna
poglobitev struge dolvodno do kote 75.00 m,
dokler se ne vklopi v obstoječ e dno struge.
Pregled vseh simuliranih primerov in
izrač unanih hitrostnih polj je podan v poroč ilu
Č etina & Krzyk (2001), kjer je podrobno
opisana tudi izbira konč ne oblike talnega
praga. V tem poroč ilu so na slikah 6 do 10
grafič no prikazani samo nekateri najznač ilnejši
rezultati za konč no obliko pragu in pretoke
100, 200, 300 in 2718 m
3
/s.
4.1 POLJE HITROSTI
Izrač unani vektorji globinsko povpreč nih
hitrosti za posamezne primere nam povedo, ali
je rešitev s talnim pragom ustrezna za
preprečitev odlaganja rinjenih plavin na
območ ju iztoka HE Plave II. Na slikah 6, 7 in
8 so vektorji izrisani v vsaki drugi toč ki
numerič ne mreže tako v x kot tudi y smeri. Pri
pretoku Soč e Q = 100 m
3
/s brez delovanja
turbine (slika 6) ter pretoku Q = 300 m
3
/s ob
hkratnem delovanju turbine (+ 105 m
3
/s, slika
7) so lepo vidne znač ilnosti toka preko praga.
Zlasti v znižanem delu ob desnem bregu so
hitrosti povečane, tako da prepreč ujejo
morebitno odlaganje rinjenih plavin.
Za oceno strižnih napetosti ob dnu pri
različ nih pretokih so pomembne velikosti
hitrosti. Pri Q = 100 m
3
/s največ je globinsko
povpreč ne hitrosti nastopajo na hrbtu praga ob
desnem bregu in znašajo približno 1.6 m/s
(slika 6), v preostalem delu struge pa je
razpored hitrosti bolj enakomeren z
vrednostmi med 1.1 in 1.3 m/s. Pri
Q
1 00
 = 2718 m
3
/s pa so hitrosti med 3 in 4 m/s
(slika 8). Najmanjše ali celo povratne
gorvodno usmerjene hitrosti pa nastopajo na
recirkulacijskih območ jih (največ je je
dolvodno od talnega praga ob levem bregu),
kjer je mogoč e prič akovati odlaganje rinjenih
plavin.
4. COMPUTATIONAL RESULTS
After the model had been calibrated at Q =
1700 m
3
/s, the case with protective levee and
without riverbed deepening could be computed
at a discharge of Q
1 00
= 2718 m
3
/s. This case
was chosen as a reference state to be compared
with final computations. In the final state, the
removal of the protective levee, different
layouts and shapes of the bottom-sill and the
horizontal riverbed profile downstream to the
bottom elevation of 75.00 m above sea level
were considered. The review of all simulated
cases and computed velocity fields can be
found in the report Č etina & Krzyk (2001),
where the choice of the final shape of the
bottom-sill is explained in detail. Here only,
some most significant graphical results for the
final shape of the bottom-sill are shown in
Figures 6 to 10, for the discharges 100, 200,
300 and 2718 m
3
/s.
4.1 VELOCITY FIELD
The computed velocity vectors for different
cases could indicate whether the solution with
the bottom-sill is appropriate to prevent
sedimentation in the area of the HPP Plave II
outflow. In Figures 6, 7 and 8, the velocity
vectors were plotted in every second point of
the numerical grid in the x and y directions,
respectively.  At the Soč a River discharge Q =
100 m
3
/s without turbine operation (Figure 6)
and at the discharge Q = 300 m
3
/s, with turbine
operation (+ 105 m
3
/s, Figure 7), the
characteristics of the flow over the sill could
be clearly seen. The higher velocities near the
right bank over the lower part of the sill
prevented sedimentation.
Magnitudes of the velocities were important
in assessing bottom shear stresses. At Q = 100
m
3
/s, the maximum values of depth-averaged
velocities near the right bank on the
downstream part of the sill were about 1,6 m/s
(Figure 6). In another part of the river, the
velocity distribution was more uniform, with
values between 1,1 and 1,3 m/s. At Q
1 00
=
2718 m
3
/s, velocities were between 3 and 4
m/s (Figure 8).  Minimum or even upstream-
directed velocities could be indicated in the
recirculation zones. In the largest recirculation
zone, just downstream of the sill near the left
bank, sedimentation could be expected.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
343
P40.2
P40.1
Detajl
P40.3
P40.4
P40.2
P40.6
km 13+364
P41
P40.5
P40.4
P40.3
km 13+017
P40
Hitrosti, Velocity
Dolžin, Length
Merilo, Scale
P40.1
Merilo, Scale
Dolžin, Length
Hitrosti, Velocity
Slika 6. Izrač unano hitrostno polje pri pretoku Soč e Q = 100 m
3
/s brez delovanja turbine.
Figure 6. Computed velocity field at the Soč a River discharge Q = 1 00 m
3
/s without turbine
operation.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
344
P40.4
P40.1
P40.3
Detajl
P40.4
P40.2
P41
P40.6
km 13+364
P40.5
P40.3
km 13+017
Merilo, Scale
P40
Hitrosti, Velocity
Dolžin, Length
P40.1
P40.2
Merilo, Scale
Dolžin, Length
Hitrosti, Velocity
Slika 7. Izrač unano hitrostno polje pri pretoku Soč e Q = 300 m
3
/s in delovanju turbine (+ 105 m
3
/s).
Figure 7. Computed velocity field at the Soč a River discharge Q = 300 m
3
/s with turbine operation
(+ 1 05 m
3
/s).

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
345
km 13+364
P41
P40.6
P40.5
P40
km 13+017
Hitrosti, Velocity
Dolžin, Length
Merilo, Scale
Dolžin, Length
Hitrosti, Velocity
P40.4
P40.3
P40.2
P40.1
Merilo, Scale
P40.4
P40.2
P40.1
P40.3
Detajl, Detail
Slika 8. Izrač unano hitrostno polje pri pretoku Soč e Q
1 00
 = 2718 m
3
/s.
Figure 8. Computed velocity field at the Soč a River discharge Q
1 00
 = 271 8 m
3
/s.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
346
4.2 KOTE GLADIN
Poleg hitrostnih polj nas kot konč ni rezultat
izrač unov zanimajo tudi kote gladine pri
različ nih visokovodnih pretokih in predvidenih
spremembah v strugi Soč e na območ ju iztoka
HE Plave II. Ker dvodimenzijski model
omogoč a izrač un gladin v vsaki “aktivni”
celici razmeroma goste numerič ne mreže
(aktivnih je približno polovica od skupno 86 x
291  toč k, kar je še vedno okrog 12 500 toč k),
lahko postane spremljanje številč nih podatkov
o kotah gladine zelo zamudno in nepregledno.
Zato so rezultati prikazani grafično v
karakteristič nih preč nih in podolžnih profilih,
izometrič na slika  pa omogoč a lažjo prostorsko
predstavo o poteku gladine.
Primerjava kot gladin pri Q
1 00
 = 2718 m
3
/s
med konč nim in referenč nim stanjem (slika 9)
kaže, da konč no stanje po izgradnji talnega
praga zaradi hkratne odstranitve nasipa
gradbene jame ne poslabšuje hidravlič nih
razmer gorvodno. Rač uni so pokazali, da se bo
kota pri hišah ob levem bregu približno 100 m
gorvodno od  profila P41 znižala z 88.67 m
n.m. med gradnjo strojnice HE Plave II na
88.23 m n.m. po odstranitvi nasipa. Torej bo
kljub že upoštevanem zasipavanju dna do kote
krone praga celo za 0.44 m nižja kot med
gradnjo strojnice.
 Izometrič na slika gladine pri Q
1 00
 = 2718
m
3
/s je podana na sliki 10. Pri tem je treba
poudariti, da teren v suhih nesodelujoč ih
celicah zunaj struge ne predstavlja dejanske
topografije, temveč je umetno ravno odrezan,
da je bolje viden potek izrač unane gladine v
strugi.
4.2 WATER SURFACE ELEVATIONS
Besides velocity fields in the HPP Plave II
outflow area an interesting final result was
also the water surface elevations at various
flood discharges, and the different changes in
the Soča riverbed. Since, with a two-
dimensional model, it is possible to compute
the water surface in every “active” cell of a
relatively dense numerical grid (about half of
total 86 x 291  points were active,
approximately 12 500 points), the
interpretation of the numerical values about
water levels could become very unpleasant and
difficult to scan. For that reason the results are
shown graphically in characteristic cross-
sections and longitudinal profiles.   The
isometric view of the water surface is also
added.
The comparison of water surface elevations
at Q
1 00
 = 2718 m
3
/s for the final and reference
state (Figure 9) show that, in spite of bottom-
weir construction due to the simultaneous
protective levee removal, the hydraulic
conditions upstream would not be worsened.
The computations showed that the water
surface elevation near the houses at the left
bank approximately 100 m upstream from the
P41 profile would decrease from 88.67 m to
88.23 m above sea level after the protective
levee was removed. So it would be, in spite of
the sediment deposition up to the sill crest,
even 0.44 m lower than during the HPP Plave
II powerhouse construction.
The isometric view of the water surface at
Q
1 00
 = 2718 m
3
/s can be seen in Figure 10. It
was added to stress that the terrain elevations
in dry cells are not realistic, but cut off at
certain levels to improve the visibility of the
computed water surface in the river.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
347
P40
km 13+017
P40.1
P40.2
P40.3
P40.5
P40.6
P41
km 13+364 P40.5
P41
km 13+364
P40.6
P40.4
P40.3
P40.1
P40.2
km 13+017
P40
87.9
87.5
87.7
88.5
88.3
88.1
87.9
87.7
87.5
88.5
88.3
88.1
88.5
87.9
87.7
88.1
88.3
87.5
88.5
87.9
87.7
88.1
88.3
87.5
88.5
87.9
87.7
88.1
88.3
87.5
87.9
87.7
88.1
88.3
87.5
87.9
87.7
88.1
87.5
87.9
87.7
87.5
87.9
87.7
87.5
Merilo Merilo
88.3
88.1
87.7
87.5
87.9
88.5
88.1
87.7
87.9
87.5
P40.4
88.5
88.3
88.1
87.7
87.9
87.5
88.1
87.7
87.9
87.5
Slika 9. Primerjava gladin  pri pretoku Soč e Q = 2718 m
3
/s med referenč nim in konč nim stanjem.
Figure 9. The comparison of water surface elevations between the reference and final state at the
Soč a River discharge Q = 271 8 m
3
/s.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
348
Slika 10. Izometrič na slika izrač unane gladine pri pretoku Soč e Q = 2718 m
3
/s.
Figure 1 0. Isometric view of the computed free surface at Q = 271 8 m
3
/s.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
349
5. ZAKLJUČ KI
Z dvodimenzijskim globinsko povpreč nim
modelom PCFLOW2D je bil prerač unan odsek
Soč e na območ ju turbinskega iztoka HE Plave
II. Potem, ko je bil model zadovoljivo
umerjen, je bilo s simulacijami pri nižjih
pretokih Soče preverjenih več različ ic
poglobitev struge ter tlorisnega in višinskega
položaja talnega praga gorvodno od iztoka.
Ocenjen je bil tudi vpliv praga na gladine pri
visokovodnem pretoku Q
1 00  
= 1718 m
3
/s.
    Rezultati rač unov pri nizkih pretokih od
100 do 300 m
3
/s z upoštevanjem delovanja
turbine (+ 105 m
3
/s) ali brez so pokazali, da
predlagana rešitev s talnim pragom razmeroma
uč inkovito rešuje problem odlaganja rinjenih
plavin na območ ju turbinskega iztoka. Zaradi
padanja kote krone praga proti desnemu bregu
so namreč hitrosti dolvodno ob desnem bregu
poveč ane, kar prepreč uje recirkulacijo vode in
s tem odlaganje plavin. Variantni izrač uni
hitrostnega polja so tudi omogoč ili
optimizacijo konč ne oblike praga (sliki 2 in 3).
Druga pomembna ugotovitev je, da
predvideni prag pri visokih pretokih Soč e
(Q
1 00
 = 2718 m
3
/s) ne poslabšuje hidravlič nih
razmer gorvodno, glede na obstoječ e
referenčno stanje pri zgrajenem nasipu
gradbene jame. Po izgradnji praga in
dokonč anju strojnice je bil namreč nasip
odstranjen, ustrezno pa je bila urejena tudi
desna brežina.
Računi razpoložljivih prostornin za
odlaganje rinjenih plavin gorvodno od praga
so pokazali vrednosti okrog 10 000 m
3
/s. To
seveda ne zadostuje za zadrževanje letnih
količ in rinjenih plavin Soč e, ki so bistveno
višje, še zlasti po zadnjem potresu v Posoč ju in
katastrofalnem drobirskem toku v Logu pod
Mangartom. Del proda bo zato potoval preko
praga in se odlagal ob levem bregu v zatišnem
delu korita. Zato bo potrebno obč asno č išč enje
tega odseka, ki ga bo mogoč e izvesti deloma s
hidravlič nim izpiranjem ob denivelaciji HE
Solkan in deloma z neposrednim bagranjem
korita. Na primernih lokacijah gorvodno od
obravnavanega odseka je treba predvideti
lovilne jame za prod.
5. CONCLUSIONS
The reach of the Soč a River in the area of
the HPP Plave II powerhouse outflow was
computed using the two-dimensional depth-
averaged model PCFLOW2D. After the model
had been satisfactorily calibrated, several
variants of the riverbed deepening with
different layouts and crest elevations of
bottom-sill were simulated at low discharges.
The influence of the sill on the water levels
upstream was also assessed at the flood
discharge Q
1 00 
= 2718 m
3
/s.
The computational results at low discharges
from 100 to 300 m
3
/s, with (+ 105 m
3
/s) or
without the turbine operation, proved that the
bottom sill could efficiently prevent the
deposition of bed-load in front of the outflow.
Due to the lower crest elevations on the right
part of the sill, the higher velocities near the
right bank prevent water recirculation, and
thus the deposition of sediments. Several
alternative computations of the velocity field
enabled the optimisation of the final shape of
the sill (Figures 2 and 3).
Secondly, at high discharges of the Soč a
River (Q
1 00
 = 2718 m
3
/s), the designed sill did
not worsen the hydraulic conditions upstream
according to the  reference state with the
protective levee of the powerhouse. This is
because after the bottom-sill construction and
the finishing of the powerhouse building, the
levee was removed. The trained right bank
also improved the situation.
The retention capacity for sediment
deposition upstream of the sill was computed
to be about 10 000 m
3
. Certainly this is not
enough to stop the evidently higher sediment
inflow of the Soč a River, especially after the
last earthquake disaster in the Posoč je region
and the catastrophic debris flow at the
upstream-located village of Log pod
Mangartom. Part of the bed-load is thus
expected to flow over the sill and to deposit
just downstream near the left bank of the Soč a
River.   Occasionally it will be necessary to
remove the deposited material, either with
hydraulic flushing b y lowering the
downstream water level, or by excavation.
Some additional sediment retention basins at
the upstream locations are also needed.

Č etina, M., Krzyk, M.: Matematič no modeliranje toka Soč e na območ ju iztoka hidroelektrarne Plave II –
Mathematical Modelling of the Soč a River Flow in the Area of the Plave II Power Plant Outflow
© Acta hydrotechnica 20/33 (2002), 329-350, Ljubljana
350
VIRI – REFERENCES
Č etina, M. (1988). Mathematical Modelling of Two-dimensional Turbulent Flows. Acta
hydrotechnica 5/6, 56 p. (in Slovenian with extended abstract in English).
Č etina, M. (1989). Uporaba k -ε modela turbulence pri rač unu toka vode s prosto gladino (Free
Surface Flow Computations Using a Depth-Averaged k -ε Turbulence Model). Proceedings of
the Slovenian Conference on Mechanics “Kuhljevi dnevi '89”, Rogla, 253-262 (in Slovenian).
Č etina, M., Rajar, R. (1993). Mathematical Simulation of Flow in a Kayak Racing Channel.
Proceedings of the 5
th
 International Symposium on Refined Flow Modelling and Turbulence
Measurements, Paris, 673-644.
Č etina, M., Krzyk, M. (2001). Dvodimenzionalni matematič ni model toka Soč e v območ ju iztoka
HE Plave II (Two-dimensional Mathematical Model of the Soč a River Flow in the Area of the
HEPP Plave II Outflow), University of Ljubljana, FGG, Report 76-KMTe/d-54, 16 p. (in
Slovenian).
Fazarinc, R. (2001). Idejna zasnova talnega praga nad iztokom HE Plave II (The Scheme of the
Bottom Sill Upstream of the Plave II HEPP Outflow). Osebna komunikacija (Personal
Communication, in Slovenian).
IBE (1997). HE Plave II – PGD (HEPP Plave II – Project Documentation). IBE, Ljubljana (in
Slovenian).
Patankar, S.V. (1980). Numerical Heat Transfer and Fluid Flow. McGraw-Hill Book Company,
197 p.
Rodi, W. (1980). Turbulence Models and Their Application in Hydraulics – A State of the Art
Review. IAHR Book Publication, Delft, 104 p.
Širca, A. (1990). Programski paket HIDRO90 – navodila za uporabo (Computer Code HYDRO 90
– User’s Manual). University of Ljubljana, FAGG, 86 p. (in Slovenian).
VGI (1982): Hidrološka študija Soč e (The Soč a River Hydrology), Water Management Institute
(VGI), Ljubljana (in Slovenian).
Naslov avtorjev - Authors' Addresses
izr. prof. dr. Matjaž Č etina, mag. Mario Krzyk
Univerza v Ljubljani - University of Ljubljana
Fakulteta za gradbeništvo in geodezijo - Faculty of Civil and Geodetic Engineering
Jamova 2, SI-1000 Ljubljana
E-mail: mcetina@fgg.uni-lj.si