J. Hyd ol. Hyd omech., 71, 2023, 3, 316–330
h ps://doi.o g/10.2478/johh-2023-0022
©2023 Ja omí Říha e al., published by Sciendo. This wo k is licensed unde he C ea i e
Commons A ibu ion-NonComme cial-NoDe i a i es 4.0 In e na ional Licence.
316
Simpli ied dispe sion analysis based on dye es s a a small s eam
Ja omí Říha*, Tomáš Julínek, S anisla Ko aška
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, Ve e i 95, 602 00 B no, Czech Republic.
E-mails: juli[email p o ec ed]b .cz; ko aska.s@ ce. u b .cz
* Co esponding au ho . E-mail: iha.j@ ce. u b .cz
Abs ac : The modelling o solid anspo in open channels equi es good knowledge abou pa ame e s ela ed o basic
p ocesses such as hyd odynamic dispe sion, ad ec ion and decay a es. Such pa ame e s a e usually de e mined by dye
es s. Nume ous ace s udies ha e been pe o med on labo a o y lumes and na u al i e s. Howe e , on-si e sampling is
o en di icul , expensi e and needs special appa a us. The main aim o he s udy was o jus i y simpli ied me hod based
on he moni o ing o he dye cloud shape in o de o de e mine bo h longi udinal and ans e sal dispe sion coe icien s.
In his s udy, ou dye es s we e ca ied ou on a small local s eam ( he Lipko sky) using Rhodamine WT luo escein dye
as a ace . The es s we e ca ied ou in such a manne ha bo h longi udinal and ho izon al ans e sal dispe sion da a
we e ob ained. Fo his pu pose, he isually de e mined ex en o he dye cloud was in e p e ed ia he analy ical solu ion
o he ad ec ion-dispe sion equa ion. The esul s ob ained by his simpli ied app oach indica ed ha he longi udinal
dispe sion coe icien Dx = 0.051–0.057 m2/s and he coe icien o ho izon al ans e sal dispe sion Dy = 0.00024–0.00027
m2/s. The me hod was jus i ied by co esponding oo mean squa e e o (RMSE) coun ing RMSE = 0.65–1.02 m o he
dye cloud cen e, RMSE = 1.87–2.46 m o he head and ail o he cloud and RMSE = 0.025–0.11 m o he cloud wid h,
he Nash-Su cli e e iciency coe icien s anged om 0.9 o 0.998. The compa ison o hese alues wi h empi ical
o mulae and o he ace s udies indica ed signi ican o e es ima ion o he men ioned alues o Dx, which can be
a ibu ed o he uni o m eloci y dis ibu ion along he wid h o Lipko sky S eam. Much be e ag eemen was achie ed
o Dy.
Keywo ds: Wa e quali y modelling; 2D i e mixing; Dye es ; Longi udinal and ans e sal dispe sion coe icien .
1. INTRODUCTION
S eam wa e quali y modelling has been widely used since
he 1970´s o assess cu en condi ions and he impac s o
p oposed measu es o wa e quali y imp o emen in open
channels (B own and Ba nwell, 1987; Chap a, 1997; C owde e
al., 2004; DHI, 2010; HEC-RAS, 2022). The applica ion o
models equi es eliable da a on pa ame e s desc ibing anspo
p ocesses such as hyd odynamic dispe sion, ad ec ion and decay
a es. Ad ec ion phenomena can be desc ibed by s anda d
hyd odynamics. The eliabili y o nume ical models p ima ily
depends on hei inpu pa ame e s, which include cha ac e is ics
such as he geome y o channel and hyd aulic s uc u es,
channel oughness, and anspo cha ac e is ics, o which he
mos in luen ial is hyd odynamic dispe sion. Also, he
implemen a ion o ini ial and bounda y condi ions is e y
impo an . E en i he alues o he men ioned pa ame e s may
be de e mined using a ious p edic i e echniques like gene ic
p og amming (Azama hulla and Wu, 2011; Riahi-Mad a e al.,
2009) o empi ical o mulae, he mos eliable echnique is
conside ed o be backwa d analysis ia he calib a ion o
nume ical models (Ani e al., 2009) using da a om ace
s udies. E en i a ailable li e a u e p o ides nume ous esul s o
ace s udies, new da a a e s ill needed o pa icula cases such
as small o winding s eams wi h a ious hyd aulic
cha ac e is ics. The ealisa ion and e alua ion o dye es s is
o en limi ed by he una ailabili y o sampling and app op ia e
measu ing de ices by which only poin samples o he dye may
be aken and p ocessed. Halmo á e al. (2014) summa ized he
esul s o ield expe imen s in small s eams and assessed he
coe icien o longi udinal dispe sion. Resul s om sal
expe imen s a di e en lows o a ious hyd ological and
ege a ion condi ions esul ed in he coe icien s o longi udinal
dispe sion in he ange om 0.2 o 2.5 m2/s. Shin e al. (2020)
used he eloci y and concen a ion in o ma ion collec ed om
ace es s in la ge-scale meande ing channels and calcula ed he
longi udinal and ans e se dispe sion coe icien s using he
wo-dimensional ad ec ion-dispe sion model. Pa k e al. (2020)
e alua ed wo di e en o ms o spa ially changing dispe sion
enso s employing bo h e ical eloci y p o iles and dep h-
a e aged low ields which enabled o es ima e he a i al
ime and peak concen a ion and o compa e bo h app oaches
used.
Dye es s (some imes e e ed as ace expe imen s) ha e
been ca ied ou in open channels o a ious sizes o de i e
dispe sion coe icien s and e i y empi ical o mulae (Kim,
2012; Leibundgu e al., 1993; Uyigue and Abah, 2020; Van
Mazijk, 1996; Velisko a and Kohu ia , 1992). Nadal e al.
(2021) in es iga ed and compa ed he alues o he longi udinal
dispe sion coe icien ob ained by using wo di e en
me hodologies. The i s me hod in ol ed applying a o mula
ha has been c ea ed ha con ains a ho ough desc ip ion o he
hyd odynamic pa ame e s measu ed wi h a hyd oacous ic
de ice, while he second me hod in ol ed injec ing a
conse a i e ace using he same me hodology as he non-ideal
chemical eac o heo y o low wi h dispe sion. E alua ion o
longi udinal dispe sion coe icien s ia in e se analysis usually
in ol es he modelling o he ad ec ion-dispe sion p ocess
(Boxall and Guyme , 2007; Julínek and Říha, 2017; Ma in e al.,
1999; Van Mazijk, 1996; Van Mazijk and Veling, 2005).
Comp ehensi e summa ies o expe imen ally measu ed da a o
he longi udinal dispe sion coe icien in na u al s eams a e
p o ided by (Zeng and Huai, 2014) and (Wang and Huai, 2016).
A emp s o use in e se modelling and gene ic algo i hms, and
Simpli ied dispe sion analysis based on dye es s a a small s eam
317
o apply neu al ne wo ks, we e made by (Sahay, 2013) and (Ani
e al., 2009). Han e al. (2019) ca ied ou an expe imen al s udy
ega ding he ela ionship be ween ans e se dispe sion and
di usion based on he esul s o dye es s pe o med in a
meande ing channel loca ed in he Andong Ri e Expe imen
Cen e (Sou h Ko ea). The coe icien o ans e se dispe sion
was de i ed o he meande ing channel. The in luence o he
ans e se mixing p ocesses consis s o mainly u bulen
di usion and dispe sion due o ans e se ci cula ions. The
ans e se dispe sion coe icien was s udied bo h unde
labo a o y condi ions (Gond e al., 2021; Han e al., 2019;
Okoye, 1971; Say e and Chambe lain, 1964; Seo e al., 2006)
and in na u al s eams (Jeon e al., 2007; Say e and Yeh, 1973).
The es s we e pe o med in o de o e alua e he ans e se
mixing coe icien unde di e en low condi ions. The
subs ance injec ed in o a s eam was moni o ed by measu ing
ace concen a ion in c oss-p o iles u he downs eam and
calib a ing a ans e se di usion model based on he
measu emen s. The es s allowed he inclusion o he in luence
o hyd aulic pa ame e s and channel geome y mainly in he
o m o empi ical o mulae o bo h longi udinal and ans e se
dispe sion coe icien s (Aghababaei e al., 2017; Deng e al.,
2001; Gond e al., 2021; Huai e al., 2018). Jung e al. (2019)
s udied he easibili y o using he eloci y-based me hod o
calcula ing he ans e se mixing coe icien o he wo-
dimensional con aminan anspo model o subs i u e he
concen a ion-based me hod in which he mixing coe icien is
calcula ed om he concen a ion cu es ob ained ia he ace
expe imen . I was ound ha he meande ing o he channel and
i s ibu a y con olled he ans e se mixing.
A a ie y o ace s may be applied du ing expe imen s (Field,
2003; Pujol and Sanchez-Cabeza, 1999). T ace s should ul il
ce ain equi emen s, such as high solubili y in wa e , easy
de ec ion, low backg ound concen a ion in na u al s eams,
conse a i e beha iou , negligible impac on he aqua ic
en i onmen , and low cos . F om his pe spec i e, luo escen
dyes a e he mos sui able and mos equen ly used ace s
(Feue s ein and Selleck, 1963). They include Fluo escein,
Lissamine FF, Rhodamine B and Rhodamine wa e ace (WT)
(Mc Cu cheon, 1989). Rhodamine WT is commonly used as a
dye ace , as i s p ope ies mee he a o emen ioned
equi emen s well (Ma in e al., 1999; USEPA, 1989). To
de e mine he iabili y o mapping spa ial pa e ns o dispe sion
in s eams wi h inc eased u bidi y, Leglei e e al. (2021)
pe o med an expe imen wi h a ying dye concen a ion and
u bidi y wi hin wo anks while ob aining ield spec a,
hype spec al, and RGB ( ed, g een, blue) pho os om a small
Unoccupied Ai c a Sys em. These da a se s we e subjec ed o
an op imal band a io analysis, which e ealed signi ican
connec ions be ween egionally a e aged e lec ance and
Rhodamine WT dye concen a ion o e ou di e en u bidi y
le els. The e o e, Rhodamine WT luo escein ace was selec ed
o his s udy, he e m “dye es ” is used h oughou he ollowing
ex . Bu dziakowski e al. (2021) s udied dispe sion o pollu an s
wi hin an en i onmen using, ace s Rhodamine WT and u anine.
De ec ion and calcula ions o ace concen a ion was done using
unmanned ae ial ehicle and a simple digi al came a. The me hod
enables ob aining in o ma ion on he ime o a i al, peak
concen a ion, and he dimensions o he dye cloud and i s
mo emen in he wa e en i onmen . Leglei e e al. (2019) based
de ec ion o dye cloud p og ession on a hype spec al imaging
sys em moun ed on UAV. Va ious de ec ion and g aphical
echniques a e used o he ep esen a ion o pollu ion anspo
bo h in su ace wa e s (Bu dziakowski e al., 2021) and
g oundwa e (Sadegh am e al., 2022).
Less eliable esul s may be ob ained om empi ical o mulae
applied o he de e mina ion o dispe sion pa ame e s (namely
he coe icien o dispe sion) aking in o accoun channel and
low cha ac e is ics (Top ak e al., 2004). Empi ical o mulae a e
de i ed based on he physics o dispe sion and a e alida ed
using da a ob ained om dye es s. The e o e, he esul s
ob ained using such empi ical equa ions should be used o
s eams wi h simila condi ions o hose o which hey we e
de i ed. Mos o he empi ical o mulae o he de e mina ion o
dispe sion coe icien s a e based on an equa ion in oduced by
Fishe (1967). Wi h espec o he pa icula hyd odynamic and
geome ical cha ac e is ics o a s eam, modi ied equa ions we e
p oposed by Liu (1977) and Seo and Cheong (1998). Deng e al.
(2001) implemen ed he local mixing coe icien in o Fische ´s
o iginal o mula. Longi udinal dispe sion iden i ied o na u al
s eams by dye es s was ecen ly published by (Uyigue and
Abah, 2020), while empi ical o mulae o he de e mina ion o
he coe icien o longi udinal dispe sion in small s eams we e
p oposed by Oli ei a e al. (2017), Mu phy e al. (2007),
Pe ucca e al. (2009), Tealdi e al. (2010) and Sokáč e al. (2020)
show he impo ance o ege a ion, ha ing ound ha mean
eloci y in channels wi h ege a ion can di e signi ican ly om
ha ound in non- ege a ed channels. Analy ical solu ions o
he ad ec ion-dispe sion equa ion we e p o ided by, e.g.
(Daněček e al., 2002; Van Genuch en and Al es, 1982) unde
simpli ied hyd odynamic condi ions (e.g. du ing uni o m
channel low). The applica ion o asymme ical s a is ical
dis ibu ions o he simula ion o solu e anspo in s eams was
s udied by (Sokáč e al., 2019).
Ad an ages and sho comings o me hods o he dispe sion
coe icien de e mina ion is in Table 1.
This s udy deals wi h longi udinal and ans e sal dispe sion
coe icien s ob ained ia he simpli ied e alua ion o dye es s.
The no el y o he e alua ion lies in i s simplici y, as he e is no
need o special equipmen and ad anced analysis p ocedu es
du ing he de e mina ion o dye concen a ion. The esul s we e
ob ained ia he analy ical solu ion o he ad ec ion-dispe sion
equa ion and compa ed wi h bo h empi ical equa ions and dye
es s ca ied ou by a ious au ho s.
2. METHODS
The me hodology consis s o acqui ing channel geome y and
cha ac e is ics (including g ain size o he bed) and low
condi ions using hyd ome ic measu emen . Fu he on
p epa a o y wo ks ollow including selec ion o he dye,
o ganisa ional issues and p elimina y analysis o he p oblem.
The dye es s and pa allel measu emen s p o ide geome ic
cha ac e is ics o he dye cloud a indi idual ime ins an s. A
inal e alua ion he dispe sion coe icien s a e de i ed and he
me hod p oposed is jus i ied ia e iciency coe icien s. The
me hodology is desc ibed g aphically using he lowcha in
Fig. 1.
2.1. Theo e ical conside a ions
The solu ion o he anspo o dissol ed ma e conce ns wo
sepa a e p oblems, namely open channel low and solu e
anspo .
In ou case he low was conside ed o be s eady and uni o m
du ing all dye es s. The wa e dep h and a e age eloci y in he
conside ed s eam each we e assumed o be app oxima ely con-
s an . This assump ion was jus i ied by he measu emen o he
wa e le el in p o iles 0 and D du ing he es s.
Ja omí Říha, Tomáš Julínek, S anisla Ko aška
318
Table 1. Compa ison o me hods o he dispe sion coe icien de e mina ion.
Me hod Ad an ages Disad an ages
Concen a ion based me hods
Taking single samples no special sampling equipmen
manual sampling
main enance and anspo o single
samples
labo a o y analyses
Con inuous measu emen
(one o mo e poin s in a p o ile)
con inuous empo al da a,
no manipula ion wi h samples
sampling appa a us necessa y,
need o skilled s a o da a
ans o ma ion and analysis
A ial pho o analyses iden i ica ion o cloud mo emen in
space and ime,
analysis o images using GIS
echniques
indi ec concen a ion de e mina ion wi h
less accu acy,
special equipmen and licenced pilo
necessa y
Dye cloud geome y obse a ion me hods
Manual measu emen iden i ica ion o cloud mo emen in
space and ime,
simple echnique,
no labo a o y analyses,
nume ous s a o cloud iden i ica ion,
applicable o smalle s eams,
limi ed accu acy
A ial pho o analyses desc ip ion o cloud mo emen in
space and ime
image analysis wi h GIS echniques
ae ial pho og aphy equipmen , licenced
pilo ,
S eam/ i e geome y and low condi ions
Empi ical equa ions easy o use,
no ield and labo a o y wo ks
app op ia e o mula o gi en s eam,
e y limi ed accu acy
Fig. 1. The lowcha ep esen ing p oposed me hodology.
T adi ionally, he p oblem is ea ed as one-dimensional (1D)
(Amb ose e al., 1996; Van Mazijk, 1996). Fo a 1D s eady uni-
o m open channel low in he x di ec ion he ollowing holds:
𝑄=𝐴𝑢=𝐴𝐶𝑅𝐽 (1)
whe e Q is discha ge, A is he low a ea, u is he mean eloci y
in he channel, R is he hyd aulic adius, J is he ene gy slope and
C is Chézy eloci y coe icien .
The calib a ion o he simple hyd aulic model based on ield
measu emen s indica ed ha low cha ac e is ics du ing all es s
p o ided only mino changes (see chap e s 2.3 and 3.2).
T anspo p ocesses in open channels a e desc ibed by he
ad ec ion-di usion equa ion (Fishe , 1967; Fishe e . al., 1979;
Fou ie , 1822; Knopman and Voss, 1987), which o he 1D low
(along x di ec ion) and 2D ho izon al dispe sion (in x, y
di ec ions) o conse a i e ma e eads (Singh e al., 2010; Van
Mazijk, 1996):
𝜕𝑐
𝜕𝑡+𝑢𝜕𝑐
𝜕𝑥−𝐷

𝜕

𝑐
𝜕𝑥

−𝐷

𝜕

𝑐
𝜕𝑦

=0 (2)
whe e c is solu e concen a ion, u is c oss-sec ional mean
eloci y, D
x
is longi udinal hyd odynamic dispe sion (dispe sion
coe icien ) including molecula di usion and in ol ing he
a ia ion o local eloci y ac oss he low p o ile, D
y
is he
ans e sal dispe sion coe icien , is ime and x, y a e spa ial
coo dina es.
Simpli ied dispe sion analysis based on dye es s a a small s eam
319
The ini ial condi ion exp esses he concen a ion a ime 0 = 0,
𝑐(𝑥,𝑦,𝑡)=𝑐(𝑥,𝑦). (3)
The Di ichle bounda y condi ion p esc ibes concen a ion in
he ups eam p o ile o s udied domain x = 0:
𝑐(0,𝑡)=𝑐(𝑡), (4)
whe e 𝑐(𝑡) is he known concen a ion. Al e na i ely, a Neu-
mann bounda y condi ion may be applied in he downs eam
bounda y p o ile x = L:
𝜕𝑐(𝐿,𝑡)
𝜕𝑥 =0. (5)
In Eq. (2), he key pa ame e s a e low eloci y u and coe i-
cien s Dx and Dy, which gene ally change o e ime and along
he s eam. In ou case, in o de o apply an analy ical solu ion,
i is assumed ha hese pa ame e s a e cons an in ime and along
he s eam axis.
In his s udy he longi udinal dispe sion coe icien Dx and
ho izon al ans e sal dispe sion coe icien Dy in he s eam
we e quan i ied by he analysis o he dye ex en iden i ied du -
ing he es s and ha ob ained om he calcula ion. Due o he
small dep h o he s eam, ideal mixing along he e ical was
aken in o accoun . Fo he case men ioned abo e, he analy ical
solu ion o Eq. (2) holds (Holly and Usseglio-Pola e a, 1984;
Van Mazijk, 1996):
𝑐(𝑥,𝑦,𝑡)=𝑀
4.𝜋.𝑡.ℎ𝐷.𝐷∙exp−󰇧󰇟𝑥−𝑢.𝑡󰇠
4.𝐷.𝑡 +𝑦
4.𝐷.𝑡󰇨 (6)
whe e MV is he mass o injec ed dye and h is he cons an wa e
dep h, x, y a e coo dina es (0,0 co esponds o he injec ion poin )
and is ime. F om Eq. (6), a e some manipula ion, i esul s
ha he cons an concen a ion (e.g. c = 0) a a gi en ime is
ep esen ed by an ellipse.
2.2. Empi ical equa ions
Nume ous s udies aimed a exp essing coe icien s Dx and Dy
using geome ic and hyd aulic cha ac e is ics o he s eam and
luid p ope ies (Seo and Cheong, 1998):
𝐷=𝑓(𝜌,𝜈,𝑢,𝑢∗,𝑤,ℎ), (7)
whe e ρ and ν a e luid densi y and iscosi y. The hyd aulic
cha ac e is ics a e mean eloci y u and shea eloci y u*, and he
geome ic cha ac e is ics a e he wid h o he channel w and
wa e dep h h. Dimensional analysis de ines he ela ionship
be ween he longi udinal dispe sion coe icien and he abo e-
men ioned cha ac e is ics. The in luence o he luid p ope ies
in a na u al s eam is p ac ically negligible (Seo and Cheong,
1998). In such s eams he ic ion losses and all channel
i egula i ies like con ac ions, expansions, e c. may be included
in he shea eloci y e m:
𝑢∗=(𝑔.ℎ.𝐽). (8)
whe e J is he ene gy line slope.
When app oxima ing R ≈ h o shallow low in a channel he
Chézy coe icien C in Eq. (1) may be exp essed using he Man-
ning o mula
𝐶=1
𝑛 ℎ
 (9)
and he Manning oughness coe icien n may be exp essed om
Eqs. (1) and (9) as ollows:
𝑛=1
𝑢 ℎ
 𝐽. (10)
Based on hese assump ions, Eq. (7) may be ew i en in di-
mensionless o m:
𝐷=𝑓󰇡𝑢
𝑢∗,𝑤
ℎ󰇢 (11)
Rela ion (11) has been aken in o accoun by a ious au ho s,
who ha e exp essed he longi udinal dispe sion coe icien ia
empi ical equa ions based on he esul s o labo a o y and ield
measu emen s. Fo he compa ison o ou esul s, h ee empi ical
o mulae ha e been selec ed which we e de i ed o condi ions
simila o hose in Lipko sky S eam (Table 2).
Table 2. Empi ical equa ions o Dx de e mina ion.
Au ho Empi ical equa ion
Fishe (1967) 𝐷=0.011𝑢𝑤
ℎ𝑢∗ (12)
Wang and Huai (2016) 𝐷=0.0798ℎ𝑢∗󰇡𝑤
ℎ
󰇢.󰇡𝑢
𝑢
∗󰇢 (13)
Oli ei a e al. (2017) 𝐷=0.744ℎ.𝑢.
𝑢∗.𝑤. (14)
The pa ame e s in he equa ions in Table 2 a e speci ied in he p e ious ex .
The ans e sal dispe sion coe icien Dy is also ela ed o he
hyd aulic and geome ic cha ac e is ics o an open channel
(Gond e . al., 2021):
𝐷=𝑓(𝑢,𝑢∗,𝑤,ℎ,𝜅,𝑆), (15)
whe e Sn is s eam sinuosi y and κ is he coe icien o he low
nonuni o mi y. In ou case, he low is conside ed uni o m, hus
κ
is equal o 0. The sinuosi y o Lipko sky S eam is negligible,
hus Sn is conside ed o be 1 o he selec ed each. Based on
hese assump ions, Eq. (15) may be ew i en in dimensionless
o m:
𝐷=𝑓󰇡𝑢
𝑢∗,𝑤
ℎ󰇢 (16)
Rela ion (16) has been aken in o accoun by he au ho s, who
exp essed he ans e sal dispe sion coe icien ia empi ical
equa ions based on he esul s o labo a o y and ield measu e-
men s (Table 3).
Ja omí Říha, Tomáš Julínek, S anisla Ko aška
320
Table 3. Empi ical equa ions o D
y
de e mina ion.
Au ho Empi ical equa ion
Fishe e al. (1979) 𝐷

=𝛼

ℎ𝑢
∗
(17)
Ru he o d (1994) 𝐷

=(0.15~0.30)ℎ𝑢
∗
(18)
Chau (2000) 𝐷

=0.18ℎ𝑢
∗
(19)
Huai e al. (2018) 𝐷

=ℎ𝑢
∗
⎣
⎢
⎢
⎢
⎡
0.69󰇡𝑢
𝑢
∗
󰇢
.

󰇧262+󰇡𝑢
𝑢
∗
󰇢

−31.8󰇡𝑢
𝑢
∗
󰇢󰇨+󰇧0.12󰇡𝑤
ℎ󰇢
.
󰇡𝑢
𝑢
∗
󰇢
.
𝑆

.
󰇨
󰇡𝑤
ℎ󰇢+0.222󰇡𝑢
𝑢
∗
󰇢−1.99⎦
⎥
⎥
⎥
⎤
(20)
The pa ame e s in he equa ions in Table 3 a e speci ied in he p e ious ex .
In compu e codes, he alue o he longi udinal dispe sion
coe icien is usually de e mined using he ollowing empi ical
equa ion (DHI, 2010):
𝐷= 𝛼ℎ𝑢
∗
(21)
whe e
α
is a dispe sion ac o , u* is shea eloci y and h is wa e
dep h.
2.3. The dye es
To de e mine he anspo pa ame e s, ou dye es s we e
ca ied ou a Lipko sky S eam loca ed in he no h o he Czech
Republic (Fig. 2). Lipko sky s eam is he igh bank ibu a y
o he Ticha O lice i e , he selec ed each o he es s is loca ed
jus ups eam o he con luence o bo h wa e bodies. The uppe
pa o he ca chmen is loca ed a he eas e n pa o he Jeseniky
moun ains a he K alicky Sneznik egion wi h na u ally p e-
se ed a ea. Close o he O lice i e he land is used o ag icul-
u al pu poses, Lipko sky s eam was egula ed a i s lowe pa .
The s eam is ela i ely small and s aigh (Fig. 3) wi h s eady
uni o m low du ing he es s i ing he abo e adop ed assump-
ions. Rela i ely uni o m dis ibu ion o he eloci y along he
c oss sec ion p o iding smalle longi udinal dispe sion is app o-
p ia e o obse a ions and "manual" moni o ing o he dye cloud
geome y. This also enables o comple e gene ally missing in o -
ma ion abou dispe sion cha ac e is ics in small na u al s eams.
The i s es (No. 1) was a ial one o p o ide p elimina y
in o ma ion abou he shape and eloci y o he dye cloud in he
s eam. Du ing he h ee ollowing es s (No. 2 o No. 4), he
measu emen s o he dye cloud dimensions we e ca ied ou . The
selec ed each o Lipko sky S eam was abou 60 m long wi h
cons an discha ge. The eloci y dis ibu ion and he discha ge
in he s eam we e de e mined by hyd ome ic measu emen s in
wo p o iles, 0 and D (Fig. 4). Ob ained alue o he discha ge
was Q = 0.45 m
3
/s wi h he mean eloci y being app oxima ely
u = 0.63 m/s. The dep h a e age eloci y a ied only sligh ly
ac oss he channel wid h and leng h (± 0.05 m/s).
The a e age wa e dep h was h = 0.25 ± 0.05 m, while he
wid h o he s eam was abou 3.0 o 3.2 m, he mean g ain size
o he s eam bed D
50
= 37 mm. The mean F oude numbe
F = 0.32 indica es subc i ical low in he s eam.
The Rhodamine WT dye was applied as an ins an aneous in-
jec ion in he cen eline o he s eam in p o ile 0 (Figs. 4, 5). The
mass o injec ed dye was M
V
=
2 g in all es s. The cen e o he
dye cloud was ma ked by a loa placed in o he s eam oge he
wi h he dye (Fig. 6).
As he ime-dependen spa ial sampling o he dye concen a-
ion is echnically a he di icul , an a emp was made o assess
dispe sion pa ame e s ia he simpli ied app oach o isually de-
e mining he ex en o he dye cloud p og essing along he
s eam. The a i al ime o he cloud on , he posi ion o he
cloud cen e and he end o he cloud, as well as he wid h o he
cen al pa o he cloud, we e simul aneously eco ded using
measu ing s a s du ing he expe imen a selec ed ime in e als.
In such a way, he p incipal axes o ellipses we e ob ained om
which he ellipses we e in e p e ed a indi idual ins an s (Fig. 7)
du ing he h ee es s (No. 2 o No. 4).
Fig. 2. The map o s udied a ea wi h he de ail o Lipko sky s eam.

Simpli ied dispe sion analysis based on dye es s a a small s eam
321
Fig. 3. Lipko sky S eam wi h he dye ( es No. 1).
Fig. 4. Schema ized s eam wi h injec ion poin and sampling p o iles.
Fig. 5. Dye injec ion.
Ja omí Říha, Tomáš Julínek, S anisla Ko aška
322
Fig. 6. App oxima ely ellip ical shape o he dye cloud wi h he loa (a ow) a he cen e o he cloud.
Fig. 7. Example o he esul ing calcula ion ob ained om he analy ical solu ion – es No. 3, ime 34 s. The ed ellipse schema ises he
obse ed dye ex en .
3. RESULTS
3.1. Measu ed geome ic cha ac e is ics o dye clouds
The measu emen s we e ela ed o he p o iles acco ding o
Fig. 4. The eadings o all geome ic cha ac e is ics we e ca ied
ou a he ins an s when he head o he dye cloud eached single
p o iles. The geome ic cha ac e is ics we e cloud head, cen e
and ail, and i s cen al wid h (Table 4). This enabled immedia e
compa ison o esul ing dis ances be ween indi idual es s and
es ima ing measu emen e o coun ing single decime es.
F om he geome ic da a in Table 4 he schema ic ex en o he
dye a indi idual ime ins an s was plo ed o each o he h ee
es s (Figs. 8 o 10); he igu es also con ain calcula ed esul s.
3.2. Backwa d analysis
F om Eq. (9) he Manning oughness coe icien n = 0.045 has
been de e mined which i s well he alues published e.g. by
(Noss and Lo ke, 2016) o simila ela i e oughness.
The dispe sion coe icien s D
x
and D
y
we e de e mined by
backwa d analysis ca ied ou by i ing he obse ed ellipses
wi h he analy ical solu ion ob ained om Eq. (6) (Fig. 7). The
mass o injec ed dye M
V
= 2 g. The ial-and-e o me hod was
used. Fo he backwa d analysis, he “no dye” zone was ep e-
sen ed by a concen a ion smalle han 10–5 mg/L, as his is he
minimum de ec able Rhodamine concen a ion acco ding o
Sma and Laidlaw (1977). This alue was also conside ed o be
he isibili y limi o a simila dye du ing a es ca ied ou wi hin
he s udy (Julínek and Říha, 2017). The calcula ion was ca ied
ou using Excel so wa e, whe e dye “ellipses” o indi idual
ime ins an s we e clea ly isible. They we e i ed wi h ellipses
aken om he ield obse a ions (Figs. 6, 7). In all es s, i = 7 in-
s an measu emen s o geome ic cha ac e is ics we e ca ied ou .
Using he ial-and-e o me hod, he measu ed low eloci y
u [m/s] was e i ied based on he loca ion o he cen e o he dye
clouds whe e he minimum o he oo mean squa e e o
(RMSE) was used as he c i e ion:
𝑅𝑀𝑆𝐸

= ∑
(







)




= min. (22)
whe e x
ci
and 𝑥





a e he p edic ed and obse ed x coo dina es o
he dye cloud cen e (Table 4).
The calib a ion o he a e age longi udinal dispe sion coe i-
cien D
x
[m
2
/s] was pe o med based on he measu ed loca ions
o he heads and ails o he dye clouds:
𝑅𝑀𝑆𝐸


= ∑
󰇟(








)

(







)

󰇠



= min. (23)
whe e 𝑥

and 𝑥





a e he p edic ed and obse ed x coo dina es
o he dye cloud heads, and 𝑥

and 𝑥





a e he p edic ed and
obse ed coo dina es o he ails (Table 4).
Simpli ied dispe sion analysis based on dye es s a a small s eam
323
Table 4. O e iew o dye cloud measu emen – obse ed cha ac e is ics o he clouds om he dye es s.
No. o he es P o ile (Fig. 4)
Cloud head Cloud cen e Cloud ail
Time Dis ance Dis ance Wid h Dis ance
[ s ] [ m ] [ m ] [ m ] [ m ]
2
0 0 0 0 0 0
7 5 3.5 0.7 1.6
A 12.5 10 7.1 0.8 4
B 24 20 13.5 1.05 7
BC 36 30 21.5 1.4 12
C 47 40 30 1.55 17
CD 58 50 37.5 1.65 26
D 68 60 44 2 28
3
0 0 0 0 0 0
6.5 5 2.8 0.5 1.5
A 12 10 7 0.7 4
B 22 20 14 0.9 6
BC 34 30 21.5 1.1 11
C 45 40 29 1.4 16
CD 56 50 34.5 1.6 24
D 67 60 42 1.9 29
4
0 0 0 0 0 0
5.5 5 3 0.7 2.5
A 10 10 7.5 0.9 4
B 21 20 14.5 1.1 7
BC 33.5 30 21 1.6 12
C 45.5 40 29.5 1.7 16
CD 57.5 50 36 1.85 25
D 68.5 60 43 2.1 31
The de e mina ion o he a e age ans e sal dispe sion coe i-
cien D
y
[m
2
/s] was based on he cen al wid hs o he dye clouds:
𝑅𝑀𝑆𝐸


= ∑
(







)




= min. (24)
whe e 𝑦

and 𝑦





a e he p edic ed and obse ed cen e wid hs
o a dye cloud (Table 4).
The esul ing mean eloci y and dispe sion coe icien s D
x
and D
y
a e lis ed in Table 5 oge he wi h he co esponding
RMSE.
G aphical compa isons o he measu ed and calcula ed esul s
o he h ee es s a e in Figs. 8 o 10, and an example o he
esul ing concen a ion dis ibu ion o he imes 34 s and 67 s is
shown in he 3D diag am in Fig. 11.
Table 5. The esul ing mean eloci y and dispe sion coe icien s D
x
and D
y
and co esponding RMSE.
Tes
No. u
[m/s] RMSE
u
[m] D
x
[m
2
/s] 𝛼
x
[–] 𝑅𝑀𝑆𝐸


[m] D
y
[m
2
/s] 𝛼
y
[–] 𝑅𝑀𝑆𝐸


[m]
2 0.634 1.02 0.051 1.75 2.43 0.00027 0.0093 0.025
3 0.629 0.65 0.057 1.96 2.19 0.00025 0.0086 0.065
4 0.630 0.76 0.052 1.79 1.87 0.00024 0.0083 0.11
Fig. 8. The esul s o backwa d analysis es No. 2 (blue ellipses – dye es , ed ellipses – calcula ion).
Ja omí Říha, Tomáš Julínek, S anisla Ko aška
324
Fig. 9. The esul s o backwa d analysis es No. 3 (blue ellipses – dye es , ed ellipses – calcula ion).
Fig. 10. The esul s o backwa d analysis es No. 4 (blue ellipses – dye es , ed ellipses – calcula ion).
Fig. 11. Spa ial in e p e a ion o he concen a ion dis ibu ion ob ained om he analy ical solu ion a imes 34 s and 67 s o es No. 3 (hal
o he clouds).