Determination of Skm Mathematical Model for Estimation of Transverse Velocity Distribution in Compound Channels

Transcription

1 J. Basic. Appl. Sci. Res., 3(2s) , , TextRoa Publication ISSN Journal of Basic an Applie Scientific Research Determination of Skm Mathematical Moel for Estimation of Transverse Velocity Distribution in Compoun Channels E. Teymourei 1 *, G.A.Barani 2, H.Janfeshan 3, A. A. Dehghanie 4 1 Science an Research branch, Islamic Aza University, Civil Eng Kerman, Sirjan 2 Co-Professor in Civil Eg, Shahi Bahonar University, Kerman, Kerman 3 Civil Eng. Faculties of Islamic Aza University, Iran, Golestan, Gorgan 4 Department of Water Engineering, Gorgan University of Agricultural Sciences an Natural Resources, Golestan, Gorgan Receive: June Accepte: July ABSTRACT In compoun channels with be roughness, the existing methos for estimation of velocity istribution suffer from excess of errors which may be accounte for by consiering strong seconary flows cause by varie be roughness istribution. The present paper was formulate to etermine a general moel of velocity istribution where two novel an common analytic solutions were aopte for simple an trapezoial compoun channels in orer to epict their ifferences as well as importance of seconary flow. The 2-D numerical metho, SKM, was propose for obtaining velocity istribution in compoun channels; the metho is base upon Navier-Stokes equations. Comparing analytical results with experimental ata showe that SKM metho is capable of preicting transverse velocity istributions in compoun channels along with mutual effect of seconary flow. KEYWORDS: velocity istribution, SKM 2-D metho, seconary flows, ey flows. 1. INTRODUCTION Transverse velocity istribution an bounary-level shear stress are consiere as important aspects of river hyraulics. Rivers usually have compoun cross-sections in their ens so that the main channel of rivers gets over-flooe an water enters flooplains. Due to fast variations of flow epth an be roughness coefficient, transverse velocity istribution an bounary-level shear stress will be consierably uneven. In this situation, the main channel has a high velocity owing to high flow epth an low roughness coefficient while flow velocity in flooplains is rather lower owing to lower epth an higher roughness. The channels with compoun transverse cross-section are compose of one main channel an one or two floo be; therefore, such channels are terme as compoun or two-stage channels [1]. If compoun cross-section is compose of eep cross-section (main channel) an shallow cross-section (flooplains), an abrupt change will occur in epth causing momentum exchange between main channel an flooplains. Combination of higher effect of flow epth an lower influence of roughness in main channel an flooplains brings about a consierable velocity ifference in the main channel an flooplains. Such a ifference causes transverse mass an momentum exchange which reuces cross-section carrying capacity consierably while increases flow in flooplain. Velocity witnesses a suen reuction in the area between the main channel an flooplains which can be attribute to apparent shear an momentum exchange. Compoun cross-sections are a combination of a eep main channel an wie flooplains; flooplains are usualy ry an have higher roughness coefficient than the main channel (Fig. 1) [2,3]. (a)transverse cross-section of river (b)regular compoun cross-section Figure 1. a sample of flooe rivers with compoun cross-sections (along with main channel an flooplains) [2] 2. SECONDARY FLOWS Centrifugal force is exerte as the flow bens; such a force is varie in terms of ben raius an irection ue to velocity fluctuations. Centrifugal force in the ben forms a transverse slope on water surface; consequently, water surface rises in external ben while it falls in internal ben. Such a phenomenon results in lateral pressure graient insie the * Corresponing Author: E. Teymourei, Science an Research branch, Islamic Aza University, Civil Eng Kerman, Sirjan. (Phone: ; fax: ) Teymourei@yahoo.com 682

2 Teymourei et al.,2013 cross-section. Now, if the pressure graient overcomes centrifugal force, a flow (known as seconary flow) is forme in transverse irection insie the cross-section. Seconary flow makes the particles on the surface water an in the bottom move towars external an internal walls, respectively.aspecial flow moel, terme as helical flow, is forme as a result of interaction of seconary flow an irregular profile of longituinal velocity; helical flow exerts several changes in flow pattern of the ben compare with the flow in irect channel. Furthermore, flow velocity insie the channels encounters consierable variations with formation of the flow cause by transverse rotational cell in channel cross-section. Before facing to a ben, flow moves through irect channel with the maximum longituinal velocity being virtually in the center of channel cross-section; as it enters the ben, the flow changes an its maximum moves towar internal wall. Seconary flow as well as pressure longituinal graient shoul, thus, come along in orer to etermine flow pattern in the ben [2]. Determination of seconary flows is important ue to their influence on flow hyraulics. Such flows originate from shear stress ifferences between liqui layers. Owing to velocity ifference between main channel an flooplains in compoun cross-sections, shear stress is forme insie liqui layers forming seconary flows in this area. 3. VELOCITY TRANSVERSE DISTRIBUTION Velocity transverse istribution in compoun cross-sections is much more ifferent an complex than in simple cross-sections. Flow cross-section can be regare as the main channel as long as there is water in the main channel; however, as soon as water enters flooplain, flow hyraulics cannot be consiere as it is in simple cross-section anymore. Since such phenomena as ey flows, momentum exchange, seconary flows, etc. increase in the cross-sections, conventional relations are no longer responsive to the situations in compoun cross-sections; therefore, it is necessary to make use of the relations capable of moeling such flow phenomena with more complex cross-sections consiering physical status of flow hyraulics. Accuracy of etermining capacity of compoun cross-sections in esign process is questionable ue to huge ey flows in interface of main channel an flooplains. Preicting transfer capacity of velocity istribution is very ifficult. Variety of preiction moels have been recently propose for simulation of complex behavior of irregular flows in channels accoring to mean epth momentum equation. The propose metho, SKM (Shiono an Knight Metho), has been unanimously accepte because it presents proper analytic solutions for istribution of lateral velocities in mean epth on the basis of hyraulic parameters. Such ey results in transfer of momentum of main channel to flooplains, energy loss, an consequently, reuction of spee an flow rate of the river [4]. Flow rotational cells in flooplains have a irection against irection of main channel eies. So, simulation of velocity transverse istribution an obtaining rating curve are pivotal in floos. Such ey flows are mainly ue to shear stress between fast an slow flows in insignificant cross-sections which causes momentum exchange from main channel to flooplain; as a result, velocity ecreases in main channel while increases in flooplain [3]. The consierable ifference in flow velocities between flooplains an main channel accounts for formation of an interaction area in the interface between them; this brings about a significant mass exchange an movement. Complex an 3-D architecture of flow in this area an evelopment in cross-section with towar neighboring areas invaliates 1-D flow hypothesis an therefore, consierable errors occur in estimation of Maning 1-D relation from transfer capacity [6]. Influence of seconary flow on latituinal velocity ue to mutual effect of flow is evient in Figure 2 [2]. Figure 2. seconary flow between channel main channel an flooplain [2] 4. FLOW BEHAVIOR IN DIRECT COMPOUND CHANNEL Flow behavior in irect channels with flooplains along the channel has been evaluate for 50 years. In 1985, Sciences an Engineering Research Council an Hyraulics Research Institute in Wallingfor ecie to make a series of equippe facilities. These facilities involve a 56m long an 10m wie channel with a capacity of 1.1 m3/s built for research about mutual effect of flow between main channel an flooplains. The imensions of FCF-SERC channel enable it to form such a flow as the one in natural rivers where the channel was absolutely 3-D along with high momentum change between ifferent cross-sections [Zahiri, 1999]. The most important flow behavior along regular flow is isturbe shear reaction between fast flow in main channel an low-velocity flow in flooplain (Fig. 3). 683

3 J. Basic. Appl. Sci. Res., 3(2s) , 2013 Figure 3. mechanism of flow architecture in a channel with compoun cross-section [SKM, 1991] Moreover, rotational flow with vertical axes can be seen; this flow results in a significant reuction of energy an lateral momentum transfer which occurs from main channel to flooplain. 5. RATING CURVE Manning equation is useful for rating in compoun cross-sections. Hyraulic raius occurs haphazarly in shallow flooplains an therefore, if flow rate is estimate for an iniviual channel, it cannot be free from errors. The conventional metho for overcoming this problem is consiering vertical walls with friction an calculation of flow rate for each sub cross-section (Fig. 4). (a) Rectangular cross-section (b) Trapezoial cross-section Figure 4. various moels of analysis in compoun cross-sections (1:1 horizontal pattern; 1:2 incline pattern; 1:3 vertical pattern) This metho was valiate by Chow (1956) as follows: Q = A R S + A R S + A R S (1) If relative epth is very low in the above equation, (Yf an Yc stan for flooplain an main channel epths, respectively) causes isturbance reaction leaing to faulty estimation of Q. Q = K Q + K Q (2) Q, Q Occur when there is no reaction; k, k stan for coefficients of flooplain an main channel, respectively where k is an k These methos were further evelope by Wormleaton et al. (1988). These experiments were performe on a fixe be with varying transvers cross-section an two ifferent amounts of roughness in flooplain. 684

4 Teymourei et al., FLOW HYDRAULICS IN FLOODED RIVERS (COMPOUND CROSS-SECTIONS) As iscusse before, velocity transverse istribution in compoun cross-sections is much more ifferent an complicate than that in simple cross-sections. As water flows in main channel, flow cross-section may be consiere a simple cross-section; however, once entere flooplain, flow hyraulics cannot be consiere as that in simple crosssections anymore because such phenomena as ey flows, momentum flows, an seconary flows aggravate. An analytic- 2-D moel for solving velocity transverse istribution in compoun cross-sections was propose in 1988 base upon Navier-Stokes equations. Effect of seconary flows is neglecte in this mathematical moel. The results obtaine from this mathematical moel in experimental an river compoun cross-sections showe that seconary flows play important role in etermination of velocity transverse istribution. Numerical solution for this 2-D moel by use of limite subtractions [Ayubzaeh an Zahiri, 2002] an limite components is also propose. Application of these two numerical solutions in floo tren fining in regular compoun cross-sections has resulte absolutely ientical results [1]. In 2000, a 2-D metho accoring to Navier-Stokes equations was propose for irect compoun cross-sections [4] where effect of seconary flow in compoun cross-section with bens was shown to be 10 times as much as its counterpart in compoun cross-sections with irect route. In 2004, a metho was recommene for calculation of rating curve in regular an river heterogeneous compoun cross-sections. In this metho, seconary flows coefficient of main channel in heterogeneous compoun cross-sections was correcte as a thir-orer function of relative roughness. 7. EFFECTIVE FACTORS ON MUTUAL EFFECT OF FLOW When compoun cross-section involves eep (main channel) an shallow (flooplains) cross-sections, an abrupt change will occur in epth leaing to momentum exchange phenomenon between main channel an flooplains. Combination of big effect of flow epth an low effects of roughness in main channel an floo-plains results in consierably ifferent velocities in main channel an flooplains. Such a velocity ifference leas to momentum exchange which in turn, reuces carrying capacity of the cross-section. Recent investigations by Sellin (1964) an Zheleznyakov (1971) showe rotational movements an their effect on velocity an flow rate in overflows. 8. HELICAL SECONDARY FLOWS Helical flows are forme in main channel on the walls. These longituinal rotations are seen all along isturbe flows leaing to isturbance in points pattern [Perkins, 1979; Knight & Patel, 1985]. Helical seconary flows are forme in the bottom of cross-section an on walls in a clockwise manner. Nevertheless, irections of these flows on corners of the main channel are more intensive an forme in anticlockwise. Iniviual rotational flows exist on the corners of cross-section of flooplains. These flows an helical seconary flow in the bottom of the cross-section are unable to expan to free water surface ue to low energy (Figs. 5 an 6). Figure 5. Figure 5: rotational flows in the interface of main channel an flooplain Figure 6. isturbance of flow in the interface of main channel an flooplain 685

5 J. Basic. Appl. Sci. Res., 3(2s) , MAIN EFFECTIVE PARAMETER ON FLOW MUTUA EFFECT PHENOMENON IN COMPOUND CROSS- SECTION A compoun cross-section can be generally imagine as follows: b f B H h s 1 b Figure 7. compoun cross-section (trapezoial cross-section) with the parameters In figure 7, H, B, h, bf, bc, Sc, an Sf stan for total height of the cross-section (water height in the main channel), total with of water level, height ifference between bottom of flooplain an main channel (epth of main channel), with of flooplain, with of main channel with, lateral slope of main channel, an lateral slope of flooplain cross-section, respectively. Main parameters in etermination of flow in compoun cross-sections are orere in terms of importance as follows: Relative epth (D r ) (ratio of flow epth in flooplain to flow height in main channel) Relative with (W r ) (ratio of half with of cross-section in flooplain to half with of main channel) Relative roughness (n r ) (ratio of roughness in flooplain to that in main channel) Shape ratio (ASP) (the ratio of half with of main channel to epth of main overflown cross-section Number of flooplains in the consiere compoun cross-section (N f ) Lateral slope of main channel (S c ) Parametric escription of effects of above-mentione factors may vivily present variations of mutual effect D METHODS IN COMPOUND CROSS-SECTIONS 1-D methos for etermination of insignificant flow rates, velocity istribution, etc., as conventional methos in the investigations on seiment transfer, walls erosion, an sustainable channels esign, lack efficiency. 2-D moels are able to calculate velocity istribution, insignificant flow rates an consequently, total flow rate. Variety of 2-D moels have been propose such as Wark metho (LDM), Shiono-Knight [8], Lambert-Selin [2], Sponer-Shiono [9], Projen [7] D MODEL FOR VELOCITY TRANSVERSE DISTRIBUTION (SKM) Several 2-D hyraulic methos have been propose by Shiono-Knight [12], Lambert-Salin [2], Arvin et al. [11], an Projen et al. [7]. ρ v + ω = ρgsinθ + + (3) 2-D equations of SKM metho have been put forth for solving transverse istribution of flow velocity in compoun cross-sections; the following equation is the equation presente in [10] without consiering seconary flow. 1 1/ f U 0 1 gh S fu 0 2 H U 8 s y 8 y (4) where U,, g, an λ stan for mean velocity in epth, volume mass of water, gravitational acceleration, an coefficient without viscosity (isturbe flow), respectively. Furthermore, H, S, an y stan for flow epth in each point, slope of channel bottom, an transverse irection of channel, respectively [10]. Analytic solution of equation [4] is presente below for the areas with fixe epth in main channel an flooplain (Fig. 8): Stress Lear U τ y h H h Figure 8. shear layer in the interface of main channel an flooplain 686

6 Teymourei et al.,2013 U = β + C exp y + C exp ( y) (5) where a =, β = ρghs, e =, C1, an C2 are unknown constants of the equation. The function epth for linear slope range shown in Fig. 7 is as follows: H(y) = H (6) where S stans for lateral slope of main channel. Substituting equation (6) in (4), the analytic solution for mean epth velocity in linear slope amplitue can be obtaine. Equation (3) i use only if seconary flows are not consiere; otherwise, (4), which was propose by SKM in 1991, is aopte instea of the ifferential equation (3) [10]. 1 1/ f u ρghs0 ρf U 1 ρλh U H(ρ UV) Γ 2 8 s y 8 y y (7) where s stans for transverse slope of the channel. SKM metho is relate to effect of seconary flows through the right phrase of (7) an is mean velocities over time in longituinal an latituinal irections, respectively. In general, whenever vertical an horizontal components of flow velocity become important, seconary flow is forme an it affect hyraulics of compoun cross-sections consierably. Like SKM metho [12], the metho propose by Arvin et al. [11] is applicable for heterogeneous compoun cross-sections. By solving Navier-Stokes equations in regular compoun crosssections with irect route (4), Arvin et al. [11] inicate importance of seconary flows phrase to be 7-10% of that in rotational compoun cross-sections. Through experimental ata for homogeneous compoun cross-section with irect route an big imensions (FCF), SKM coul propose the effect of seconary flows as a function of be shear stress: H ( UV ) y gs s 0 H In this equation, the coefficient for homogeneous compoun cross-sections in normal an floo flows are 0.05 an 0.15 in the main channel an in flooplains. Definite coefficients have not been propose for heterogeneous compoun cross-sections where roughness coefficient of flooplain is higher than the main channel (as it is in majority of rivers). V ku (9) where k is proportion coefficient. This coefficient is a function of epth an roughness of flooplain an rotation egree of river. After analysis of the propose 2-D mathematical moel, proportion coefficient in the main channel an flooplains are 0.25 an 0.00 for experimental compoun channels with irect irection, respectively. This coefficient in wie an homogenous compoun cross-section of rivers with irect route is negligible. 13. PROPOSED MODELS OF SKM SKM moel A 2-D analytic solutions for velocities istribution has been examine in open channels. Seconary flow is a very important parameter because it is irectly engage in force balance of momentum equation. Therefore, it is of high importance for a proper etermination of velocity istribution. It can be seen in the recommene moel A through SKM metho that the istributions [ρuv] are somewhat ientical to ifferent epths Dr. The istributions [H ρuv], however, are largely epth-relate. As a result, seconary flow can be presente in a more efficient manner by [ρuv] than [H ρuv] istributions SKM moel B This moel is use for simulation of seconary flow in slope lateral wall. Analytic solution of this moel is performe for velocity lateral istribution by consiering Г=HГ^* SKM moel C This moel is aopte for simulation of seconary flow in slope lateral wall. This moel is solve for velocity lateral istributions by consiering Г=( (Hᴪ))/ y. the most common moel for etermination of seconary flow has been recommene on the basis of ᴪ consiering ᴪ=[ρUV] to have linear istribution. 14. CONCLUSION The present paper was formulate in orer to etermine 2-D flow moel an effect of seconary flow on flow velocity istribution in compoun channels by use of the mathematical metho SKM. The results obtaine from the present stuy about variation pattern of seconary flow coefficient, it can be conclue that the propose moels in compoun channels have a goo performance an negligence of seconary flows in compoun channels leas to more errors in calculations of flow rate. Therefore, abrupt change in profile of shear stress may be prevente by consiering linear istribution from seconary flow coefficient an raw acceptable results. SKM can make proper results by consiering parameter estimation of seconary flow. Consiering the propose moels an ientical status for the moels, the following conclusions can be rawn: (8) 687

7 J. Basic. Appl. Sci. Res., 3(2s) , 2013 The moels B an C set no-slip consition uring analutic solution without any numerical problem. All three moels have the same solutions for amplitue of channels with fixe epth; however, there are ifferences for varie linear flow. There are also ifferences in the results of U preicte for simple channels with the three moels where the higher U, the higher seconary flow. Such increase usually occurs in lateral walls where there is a strong shear layer. The ifferences between the propose moels are higher in the channels with rather same seconary flows. If Γ is not that big in the moels (especially in the channel with rather small apparent ratio), the ifference is very insignificant. No-slip bounary conition has a tiny effect in the moel A while bounary conition is always set in the moels B an C. also, the moel C is a suitable general moel for wie variety of applications. Comparison of analytic results shows that all the propose moels by SKM are capable of forming the velocity istributions conforming to experimental ata. It is also noteworthy that the mentione ata are valiate by comparing with experimental ata. REFERENCES 1. Yang, K., Cao, S., an Knight, D. W. Flow patterns in compoun channels with vegetate flooplains. Journal of Hyraulic Engineering 133(2), Zhang, M., Shen, Y., anwu, X. 3D numerical simulation on overbank flows in compoun channels (in Chinese). Journal of Hyroelectric Engineering 25(5), Huai, W., Chen, W., an Tong, H. 3D numerical simulation on overbank steay open channel flows in compoun channels (in Chinese). Avances in Water Science 14(1), Ervine, D. A., Babaeyan-Koopaei, K., an Sellin, R. H. J. Two-imensional solution for straight an meanering overbank flows. Journal of Hyraulic Engineering 126(9), 2000, Shiono, K. an Knight, D. W.Turbulent open-channel flows with variable epth across the channel.journal of Flui Mechanics 222, 1991, Huai, W., Xu, Z., Yang, Z., an Zeng, Y. Two imensional analytical solution for a partially vegetate compoun channel flow.2008:applie Mathematics an Mechanics (English Eition) 29(8), DOI /s y 7. Xu, W. Stuy on computational metho of overbank flow in channels with compoun cross section (in Chinese). Journal of Hyraulic Engineering (6), Yang, Z. H. an Gao, W. Two imensional analytical solution for overbank flows with interaction between the main channel an flooplain (in Chinese). Journal of Sichuan University (Engineering Science Eition) 41(5), Castaneo, S., Meina, R., an Menez, F. J. Moels for the turbulent iffusion terms of shallow water equations. Journal of Hyraulic Engineering 131(3), 2005, Abbott, M. B. an Price, W. A. Coastal, Estuarial, an Harbour Engineers Reference Book, 1994,Taylor & Francis, Lonon/New York 11. Shome, M. L. an Steffler, P. M. Lateral flow exchange in transient compoun channel flow.journal of Hyraulic Engineering 124(1), 1998, Stelling, G. S., Wiersma, A. K., an Willemse, J.Practical aspects of accurate tial computations. Journal of Hyraulic Engineering 112(9), 1986, Blumberg, A. F. an Mellor, G. L. A escription of a three-imensional coastal ocean circulation moel. Three- Dimensional Coastal Ocean Moels (e. Heaps, N. S.), American Geophysical Union, Washington, DC, Huai, W., Gao, M., Zeng, Y., an Li, D. Two-imensional analytical solution for compoun channel flows with vegetate flooplains. Applie Mathematics an Mechanics (English Eition) 30(9), 2009, DOI /s z 15. Tang, X. an Knight, D. W.A general moel of lateral epth-average velocity istributions for open channel flows. Avances in Water Resources 31(5), 2008, Knight, D. W., Omran, M., an Tang, X.Moeling epth-average velocity an bounary shear in trapezoial channels with seconary flows. Journal of Hyraulic Engineering133(1), 2007, McGahey, C.A Practical Approach to Estimating the Flow Capacity of Rivers Application an Analysis, 2006, Ph. D. issertation, Open University, Milton Keynes, UK 18. Ikea, S. (1981)Self-forme straight channels in sany bes. Journal of the Hyraulics Division 107(4), Chlebek, J. an Knight, D. W.A new perspective on siewall correction proceures, base on SKM moelling. Proceeings of the International Conference on Fluvial Hyraulics (es. Ferreira, R. M. L., Alves, E. C. T. L., Leal, J. G. A. B., an Carosa, A. H.), 2006, Taylor & Francis, Lonon/NewYork, Byja. K., M. Small. Salah, Mohamma Ayub. SA,The effect of seconary flows on steep graes, Combine with the,two-imensional moel of momentum within ", Journal of Soil an Water Research, 2010,Volume 41, Number