Simplified dispersion analysis based on dye tests at a small stream

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).