A Particle Dispersion Model For Analysis Of Two- Dimensional Mixing In Open Channels

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1 Cty Unversty of New Yor (CUNY) CUNY Academc Wors Internatonal Conference on Hydronformatcs A Partcle Dsperson Model For Analyss Of Two- Dmensonal Mxng In Open Channels Il Won Seo In Hwan Par Follow ths and addtonal wors at: Part of the Water Resource Management Commons Recommended Ctaton Seo, Il Won and Par, In Hwan, "A Partcle Dsperson Model For Analyss Of Two-Dmensonal Mxng In Open Channels" (2014). CUNY Academc Wors. Ths Presentaton s brought to you for free and open access by CUNY Academc Wors. It has been accepted for ncluson n Internatonal Conference on Hydronformatcs by an authorzed admnstrator of CUNY Academc Wors. For more nformaton, please contact AcademcWors@cuny.edu.

2 11 th Internatonal Conference on Hydronformatcs HIC 2014, New Yor Cty, USA A PARTICLE DISPERSION MODEL FOR ANALYSIS OF TWO- DIMENSIONAL MIXING IN OPEN CHANNELS IL WON SEO (1), INHWAN PARK (1) (1): Department of Cvl and Envronmental Engneerng, Seoul Natonal Unversty, Korea Non-Fcan dsperson model, 2D PDM (2D Partcle Dsperson Model) was developed to smulate the contamnant transport n both the ntal and the Taylor perod. The 2D PDM was based on the shear flow dsperson theory and conssted of two stages: the shear advecton and the vertcal mxng. The 2D PDM used the partcle to vsualze the soluble pollutant transport and analyzed the partcle mxng wthout determnaton of the dsperson coeffcent. The 2D PDM was appled to the straght channel and the meanderng channel for analyss of the conservatve pollutant mxng. In the straght channel, concentraton curves from the 2D PDM showed sewed dstrbuton n the ntal perod and then turned nto the Gaussan dstrbuton n the Taylor perod. The concentraton dstrbutons n the meanderng channel showed good agreement wth the tracer test results. 1. INTRODUCTION Contamnant transport n open channel flow was analyzed usng the Fcan dsperson model whch follows the Taylor s assumpton. The Taylor s assumpton ncludes that the Fcan dsperson model s avalable n the Taylor perod whch a balance between the advectve transport and the turbulent dffuson s reached (Taylor, [1]; Fscher et al., [2]). Durng the Taylor perod, concentraton dstrbuton shows the Gaussan dstrbuton whch has symmetrc shape n the unform flow. However, from the several feld measurements, concentraton dstrbuton showed the sewed dstrbuton. From the tracer test results n natural streams, Day [3] observed the non-fcan behavor of dsperson process and Nordn and Troutman [4] calculated the sewness coeffcent whch showed the dstorted shape of the concentraton curves due to channel rregulartes. Atnson and Davs [5] also measured sewed concentraton curves even though the tracer tests were conducted n the statstcally unform channel bed to test the mathematcal theory of dsperson. Alternatve models for the non-fcan dsperson process due to channel rregulartes were suggested to compensate the Fcan dsperson model. Bencala and Walters [6] developed the transent storage model to smulate solute transport n the pool-and-rffle stream and Czernuszeno et al. [7] compared the Fcan dsperson model and the dead-zone model whch showed the dstorted concentraton curves assocated wth the tals. Seo and Cheong [8] used the moment matchng method to calculate the parameters of the storage zone model and demonstrated dstorted concentraton curves whch were qute smlar wth the expermental

3 results. Deng et al. [9] developed the FRADE (FRactonal Advecton-Dsperson Equaton) wth revsng the Fc s law and smulated the long-taled dsperson processes n natural rvers. However, these models are based on the Fcan dsperson model and napproprate to apply n the ntal perod whch s stll not n equlbrum state between shear advecton and vertcal mxng. For demonstratng the asymmetrc concentraton dstrbuton n the ntal perod, the one-dmensonal analytc solutons and the non-fcan dsperson models were developed. Chatwn [10] derved the one-dmensonal analytc soluton of the pollutant mxng usng the edgeworth seres for the sewed dstrbuton n ntal perod and Schmd [11] used the Pearson type III dstrbutons. Based on physcal nterpretaton of pollutant mxng, Seo and Son [12] developed the SMM (Sequental Mxng Model) whch s the conceptual model for analyss of the contamnant transport. And, Jung and Seo [13] expanded the SMM to the 2D numercal model, TMM (Tme-splt Mxng Model) whch can be appled n both the ntal and the Taylor perod. However, these numercal models were developed to apply only n the straght channel. Therefore, the two-dmensonal numercal model whch can be appled n curved channels s necessary for the versatle applcatons of the non-fcan dsperson model. In ths research, the two-dmensonal partcle dsperson model (2D PDM) was developed to analyze the pollutant mxng n both the ntal and the Taylor perod wthout determnaton of the dsperson coeffcent. In straght channel, the 2D PDM was appled to nvestgate the asymmetrc concentraton curves durng the ntal perod. For the general applcaton, the 2D PDM was used n the meanderng channel. 2. DEVELOPMENT OF THE 2D PDM 2.1 Fcan dsperson model Soluble pollutant mxng n natural rvers s analyzed by usng the advecton-dffuson equaton as Eq. (1) ( uc) c c + = ε t x x x ( = 1, 2, 3 ) (1) where c s the tme-averaged concentraton; u s the velocty component; ε s the turbulent dffuson coeffcent. In natural rvers, the two-dmensonal depth-averaged advectondsperson equaton (2D ADE) n Eq. (2) s used for the analyss of the contamnant transport due to the rapd completon of the vertcal mxng. ( uc) C C 1 h + = ε u 0 jc dx3 t x x x xj h (, j = 1, 2 ) (2) where C s the depth-averaged concentraton; u s the depth-averaged velocty; h s the water depth; x 3 s the vertcal drecton. The addtonal transport term on the rght hand sde of Eq. (2) s assumed to be proportonal to the concentraton gradent by Taylor ([1]) as wrtten n Eq. (3)

4 1 h C u 0 c dx3 Dj h = (, 1, 2 xj j = ) (3) where D j s the dsperson tensor. However, D j ncorporatng the hydraulc propertes has dffcultes to determne due to the complex channel geometres and flow condtons. The 2D ADE model usng Eq. (3) s avalable only n the Taylor perod whch defned n Eq. (4) (Jung and Seo, [13]). 2 2 h W 0.4 < tt < 0.4 (4) ε ε z y where t T s the Taylor perod; W s the channel wdth. However, most open channel flow has long ntal perod whch maes the sewed concentraton dstrbuton due to the unbalance between the shear flow advecton and the vertcal mxng (Chatwn, [10]). Therefore, the non- Fcan dsperson model s necessary to compensate the lmtatons of the 2D ADE model. 2.2 Descrpton of the 2D PDM In ths study, the non-fcan dsperson model, 2D PDM was developed usng the physcal nterpretaton of the pollutant mxng by shear flow. In the 2D PDM, pollutant partcles were ntroduced to vsualze physcal mxng process accordng to the complcate flow varaton n open channels. The partcle dstrbuton n each tme step was converted to the concentraton feld for varous analyss. The 2D PDM s based on the shear flow dsperson theory and adopted the operator splt method whch dvdes the shear advecton stage and the turbulent dffuson stage as depcted n Fgure 1. In the shear advecton stage, partcles were separated by the vertcal velocty devatons n the longtudnal and transverse drectons. A partcle whch was ntroduced at ( ) ( ) ( ) x t was transported by the velocty as wrtten n Eq. (5). x t+ t = x t + u t (5) where x ( ) t s the partcle poston on the -th layer; u s the velocty on the -th layer; t s the tme step. The separated partcles accordng to the shear flow were mxed across the vertcal n the turbulent dffuson stage le Fgure 1 (b). For t, partcles were evenly dstrbuted n vertcal layers and the vertcal mxng was completed. After completon of the vertcal mxng, the number of partcle on each grd was converted to the concentraton feld usng Eq. (6) (, ) C x t (, ) mn x t = h x y L n x, t n x, t ( ) ( ) = 1 (6a) = (6b) where m s the mass for a sngle partcle; h s the water depth; grd sze; n s the number of partcle n the computatonal grd; the -th layer; L s the number of layer. x, y s the computatonal n s the number of partcle on

5 ( + ) n t t x x a) Shear advecton stage b) Turbulent dffuson stage Fgure 1. Conceptual descrpton of the 2D PDM For the shear advecton, the longtudnal partcle transport was determned usng Eq. (7a) (Rozovs, [14]) whch has the logarthmc dstrbuton, and Eq. (7b) whch has the lnear dstrbuton (Odgaard, [15]) was employed for the partcle dsplacement n transverse drecton. g z us( z) = u s ln κch h z y z 1 un( z) = un + 2vs h 2 ( 2m+ 1 h κus κch vs = us, m = = ) (7b) 2 * 2κ m r u g c where u s, u n s the stream-wse, span-wse velocty, respectvely; κ s the von Karman constant; C h s the Chezy coeffcent; r c s the radus of curvature; u * s the shear velocty; g s the gravty acceleraton. Transported partcles whch were out of the boundary were absorbed at the wall and excluded for the next computatons. z (7a) y 3. MODEL APPLICATIONS Wth the 2D PDM, the nstantaneously njected pollutant mxng n the artfcal channels was smulated to nvestgate the non-fcan dsperson durng the ntal perod. Smulaton condtons were lsted n Table 1 and a number of partcles were ntroduced at a pont. Table 1. Smulaton condtons for the 2D PDM Channels Q h W No. of No. of layer (m 3 /s) (m) (m) partcles ST ,000 M , Sewed concentraton dstrbuton n ntal perod 2D PDM was appled to the ST channel wth the smulaton condton n Table 1. Smulaton results of the 2D PDM was compared wth the analytc soluton whch has the Gaussan dstrbuton. In the unform flow, the analytc soluton of the 2D ADE s wrtten n Eq. (8).

6 ( ) 2 2 x Ut y M C( xyt,, ) = exp 4πt DD 4Dt L 4Dt L T T (8) where C( xyt,, ) s the concentraton; D L, D T s the dsperson coeffcent n each drecton; U s the unform velocty. C-x curves usng Eq. (8) were plotted wth the smulaton results n Fgure 2. The ntal perod ( t I ) n the ST channel s determned wth Eq. (9) (Jung and Seo, [13]) and the ntal perod sustaned for 18 sec n ths case. t I h > 0.4 (9) * u In the ntal perod, concentraton curves from the smulaton results show that asymmetrc dstrbutons and have large dscrepances wth the analytc soluton. After 20 sec, concentraton curves gradually change to the symmetrc dstrbuton. Shape of the concentraton curves was estmated wth the sewness coeffcent n Fgure 3. Immedately after the partcle njecton, the sewness coeffcent soared to 1.2 and rapdly decreased to about 0.3 durng the ntal perod. In the Taylor perod, the shear advecton and the vertcal mxng were achevng balance and the concentraton curves approached to the symmetrc dstrbuton whch follows the Fc s law. From the results, the 2D PDM was avalable both the ntal perod and the Taylor perod n natural rvers sec 2D PDM Anaytc soluton C (ppm) sec 15 sec 20 sec 30 sec 40 sec x (m) Fgure 2. Comparson of C-x curves between the smulaton results and the analytc soluton Sewness Tme (sec) Fgure 3. Tme evoluton of the sewness coeffcent from the smulaton results

7 3.2 Meanderng channels The 2D PDM was appled to the pollutant mxng n the M2 channel whch has rectangular cross secton and varous flow drecton. From the smulaton condton n Table 1, flow was smulated usng the hydrodynamc model developed by Song et al. [16] and Fgure 4(a) shows the flow analyss results. Wth changng flow drecton, njected partcles have possbltes to run nto the channel boundares. From the smulaton results of the 2D PDM n the M2 channel, mass conservaton was checed n Fgure 4 (b). Total mass was mantaned when the partcle cloud passed the cross-over part, but some partcles were absorbed to the boundary and total mass decreased to 98.4 % whch was nsgnfcant loss. U (m/s) nd apex Cross-over M/M st apex Tme step a) Velocty dstrbuton b) Change of total mass Fgure 4. Mass conservaton wth the absorbng boundary condton n the meanderng channel Fgure 5 shows the partcle dsperson and concentraton converson results n the M2 channel. Injected partcles were stretched n longtudnal drecton and concentraton cloud made a tal after passng the 1 st apex. The concentraton converson results were compared wth the two-dmensonal tracer test results whch were conducted by Seo and Par [17] n Fgure 6. From the comparson results, transverse averaged concentraton curves show good agreement wth the smulaton results on the 1 st apex, but the smulaton results on the 2 nd apex underestmated the expermental results. a) 15 sec C (ppm) b) 30 sec C (ppm) c) 45 sec C (ppm) Fgure 5. Smulaton results of 2D PDM n the M2 channel

8 a) 1 st apex 2D PDM Seo and Par [17] b) 2 nd apex C/C C/C Tme (sec) Tme (sec) Fgure 6. Comparson results of C-t curves 4. CONCLUSIONS The 2D PDM whch was avalable n both the Taylor and the ntal perod was developed to analyze contamnant transport n the open channel flow. Wth the shear flow dsperson theory, the 2D PDM was conssted of the shear advecton stage and the turbulent dffuson stage. Introduced partcles were transported usng the vertcal velocty profles n the shear advecton stage and the partcles were evenly dstrbuted to the vertcal layers to complete the vertcal mxng. At each tme step, partcle dstrbutons were converted to the concentraton feld for the quanttatve analyss. In the ST channel, smulaton results of the 2D PDM show the sewed dstrbuton n the ntal perod and has the large dfferences wth the analytc soluton. The sewness coeffcent of the smulaton results decreased to about 0.3 n the Taylor perod and the concentraton curves were approachng to the symmetrc shape. For the changng flow drecton, the absorbng boundary was adopted at the channel wall and the smulaton results of the 2D PDM were compared wth the tracer test results n the M2 channel. From the comparson results, the concentraton curves show good agreement wth the expermental results on the 1 st apex, but, on the 2 nd apex, smulaton results underestmated the expermental results. Acnowledgments Ths research was supported by a grant (11-TI-C06) from Advanced Water Management Research Program funded by Mnstry of Land, Infrastructure and Transport of Korean government. Ths wor was conducted at the Engneerng Research Insttute and the Integrated Research Insttute of Constructon and Envronment n Seoul Natonal Unversty, Seoul, Korea. REFERENCES [1] Taylor, G., The dsperson matter n turbulent flow through a ppe, Proceedngs of the Royal Socety of London, Seres A, Mathematcal and Physcal Scences, Vol. 223 (1954), pp [2] Fscher, H.B., Lst, J.E., Koh, R.C.Y., Imberger, J., and Broos, N.H. Mxng n nland and coastal waters, Academc Press, (1979). [3] Day, T. J., Longtudnal dsperson n natural channels, Water Resources Research, Vol. 11, No. 6 (1975), pp

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