Assessment of water depth change patterns in 120°sharp bend using numerical model

Azadeh Gholami,Hossein Bonakdari*,Ali Akbar Akhtari

Department of Civil Engineering,Razi University,Kermanshah 67149,Iran

Assessment of water depth change patterns in 120°sharp bend using numerical model

Azadeh Gholami,Hossein Bonakdari*,Ali Akbar Akhtari

Department of Civil Engineering,Razi University,Kermanshah 67149,Iran

In this study,FLUENT software was employed to simulate the fl ow pattern and water depth changes in a 120°sharp bend at four discharge rates.To verify the numerical model,a 90°sharp bend was first modeled with a three-dimensional numerical model,and the results were compared with available experimental results.Based on the numerical model validation,a 120°bend was simulated.The results show that the rate of increase of the waterdepth atthe cross-section located 40 cm before the bend,compared with the cross-sections located 40 cm and 80 cm after the bend,decreases with the increase of the normalwater depth in the 120°curved channel.Moreover,with increasing normalwater depth, the dimensionless water depth change decreases atallcross-sections.At the interior cross-sections of the bend,the transverse water depth slope of the inner half-width is always greater than thatof the outer half-width of the channel.Hence,the water depth slope is nonlinear ateach crosssection in sharp bends.Two equations reflecting the relationships between the maximum and minimum dimensionless water depths and the normal water depth throughout the channel were obtained.

Water depth change;120°sharp bend;Experimental model;Numerical model;Discharge rate;Normal water depth;FLUENT software

1.Introduction

As flow enters a bend,a transverse gradientis created atthe water surface due to a centrifugal force.Consequently,the water surface elevation increases along the outer bend wall and decreases along the inner wall(Lien et al.,1999; Baghlani,2012).The difference between these two water surface elevations is called superelevation or billow.Various factors are involved in the bend fl ow pattern,leading to a change in the velocity component distribution and water depth profiles.Studying these factors is necessary for comprehensive understanding of channel and naturalriver behavior(Ferguson et al.,2003;Ottevanger et al.,2011).One of the most important factors affecting the fl ow pattern is the ratio of thecurvature radius to channel width(Rc/b),based on which the bend type is identifi ed.According to Leschziner and Rodi (1979),if Rc/b＜3,the bend is sharp.Otherwise,the bend is mild.With increasing Rc/b,the secondary current power decreases along the bend.In a curved channel,the location of the maximum longitudinal velocity is a function of Rc/b,and the longitudinal pressure gradient and secondary currents are both involved in flow pattern formation.In sharp bends,the longitudinal flow power is so high that it prevails over the secondary current,and the maximum velocity occurs near the inner channel wall along the bend.In mild bends,the secondary current prevails over the longitudinal pressure gradient and the location of the maximum longitudinal velocity is transferred to the outer channel wall(DeMarchis and Napoli, 2006;Naji et al.,2010).The fl ow behavior in sharp bends is much more complex than that in mild bends(Ye and McCorquodale,1998;Gholami et al.,2016).Therefore,a sharp channel bend was selected for this study.

Many researchers have conducted experimental and fi eld studies on the fl ow patterns in curved channels(Rozovskii, 1961;Anwar,1986;Bergs,1990;Sui et al.,2006;Jing et al.,2014).Steffler et al.(1985)studied the transverse water depth profiles in a 270°mild bend.Based on laboratory fi ndings,they reported that the distribution of the transverse water depth profi le along the bend is linear,and they also presented relationships to predict this profile.However,they did not assess this profi le at the bend inlet and outlet. Blanckaert and Graf(2001)conducted extensive experimental research on a 120°sharp bend with a movable bed.They studied the secondary current cells and turbulent fl ow parameters,such as the turbulence intensity,average kinetic energy,and Reynolds shear stress.However,they did notfocus on links with water depth changes.Yan et al.(2011)investigated the influence of the distance to the sidewall and the aspectratio on the maximum velocity location in open channel fl ow experimentally.

In addition to experimental studies,many researchers have employed numericalmodeling to study water depth changes in curved channels.Ye and McCorquodale(1998)conducted three-dimensional flow pattern modeling in curved channels. They found that secondary currents and the water depth superelevation start at the upstream bend section and gradually reach the interior bend sections.Numerical investigations of the free surface flow in a 90°sharp bend with a nonlinear water surface gradient were conducted by Bodnˊar and Pˇríhoda (2006).DeMarchis and Napoli(2006)examined the depth distribution of velocity components,secondary currents,and water depth changes in a 270°mild bend.Their numerical results indicated thatthe effects of the water depth change due to the channel curvature were modeled well,and the water depth was high near the outer channel wall and low near the inner wall.Zhang and Shen(2008)presented a threedimensional numerical model to investigate water depth changes,longitudinal and transverse velocity profiles,and the fl ow separation phenomenon in curved channels.Naji et al. (2010)numerically examined the depth distribution of velocity components,secondary currents,and streamlines in a 90°mild bend.They found that secondary currents were the main factor leading to the changes in velocity components.However,their study did not report results on water depth distribution and changes.Ramamurthy et al.(2013)used two numerical turbulence models to simulate three-dimensional fl ow patterns in a 90°sharp bend.They employed different simulation methods,the rigid lid method(Liu and García, 2008)and volume of fl uid(VOF)method,to simulate free water surfaces and showed that the numerical results from Reynolds stress modeling(RSM)for turbulence and the VOF method were in acceptable agreement with the experimental results.However,water depth change results were not reported.Gholamietal.(2014)used experimentaland numerical models to study the fl ow patterns in a 90°sharp open channel bend.According to their results,the maximum velocity in a sharp bend remained at the inner channel wall until the fi nal section,and at the sections after the bend the maximum velocity was transmitted to the outer channel wall.They studied the transverse and longitudinal water depth profi les butdid not calculate water depth changes in relation to the normal water depth.Gholami et al.(2015a)investigated the application of a soft-computing method,a gene expression programming (GEP)model,to predictvelocity fi elds in a 90°bend with five different discharge rates.The presented model predicted the velocity fi elds accurately.Although the model could not present an equation for velocity prediction,it could be utilized to predict velocity at different discharge rates for more rapid prediction.Moreover,flow depth changes were not reported.

Few prior studies have addressed water depth changes in a 120°bend.The aim of this study was to carry out a numerical investigation of water depth changes upstream,at the beginning,at interior cross-sections,at the exit,and downstream of a 120°sharp open-channel bend with differentdischarge rates. First,using the available experimental results and FLUENT software,a 90°bend model was checked in terms of threedimensional modeling,and the numerical model was validated with the experimental results from Akhtari et al.(2009) and Gholami et al.(2014,2015b).The model accuracy was examined using the statistical indices,the relative root mean square error(RRMSE)and mean absolute error(MAE).Then, a numerical model of the 120°bend was simulated in accordance with the experimental geometry and four hydraulic conditions.Moreover,the water depth change patterns at different cross-sections and in the longitudinaldirection of the channel in different hydraulic conditions were examined and compared.The water depth changes of both inner and outer mid-widths of each cross-section were also assessed numerically.

2.Method

2.1.Experimental model

Experiments were conducted in a curved flume with a 90°bend to measure fl ow variables at the University of Mashhad, Iran(Akhtari et al.,2009;Gholami et al.,2014).The curved channel consisted of a 90°bend,an upstream channel 3.6 m long,and a downstream channel 1.8 m long,as shown in Fig.1.The square cross-section was 40.3 cm × 40.3 cm (channel width× height),and the curvature radius of the channel was Rc=60.45 cm.Thus,the bend was sharp with Rc/b＜3.The bed and channel walls were fixed and made of Plexiglas with a roughness coeffi cientof n=0.008.The bend was therefore hydraulically smooth.A sharp-crested rectangular weir,with an adjustable height,was setat the end of the flume to adjust the water depth in the channel.In the main reservoir,a sharp-crested triangle weir was used to measure the discharge through the channel.Velocity measurements were taken after adjusting the discharge rate and channelwater depth.A one-dimensional velocity meter(propeller)was used to measure the axial velocity in the fl ume with an accuracy of 2 cm/s(Armfi eld Limited,Co.,1995).The velocity meter was mounted with a vernier ruler in the transverse direction with 0.5-mm accuracy and an analog caliper in the depth direction with 0.1-mm accuracy. A micrometer (mechanicalbathometer)was used to measure the water depth with 0.1-mm accuracy.The analog caliper with a needle pointer was placed at an appropriate position so thatthe needle was tangent to the water surface.To remove the unbalanced effect of the vernier ruler,the caliper was submerged in water atthe same position. The water depth was calculated from the differences between the measured bed height and water surface height.Experimentaltests were conducted for four discharge rates in the 90°bend.The hydraulic characteristics of flow used in the laboratory test are shown in Table 1.

Fig.1.Experimental model.

In this study,the fl ow velocity and water depth were measured at different points in different cross-sections in the channel.Eight cross-sections,i.e.,the cross-section located 40 cm before the bend(S1);the cross-sections at0°(S2),22.5°(S3),45°(S4),67.5°(S5),and 90°(S6)in the bend;and the cross-sections located 40 cm(S7)and 80 cm(S8)after the bend were selected,as shown in Fig.2.At each cross-section (S1 through S8),four positions along the depth direction were determined for flow velocity and water depth measurements, i.e.,z=3,6,9,and 12 cm from the channel bed.At each depth,13 points were set in each transverse profi le,with the middle point at the center of each profi le,and the interval between adjacent points was 3 cm.

The longitudinal and transverse water depth profiles were evaluated numerically in a 120°bend model because an experimental study on a 120°bend was not available.First,a numerical model corresponding to the experimental model of a 90°bend was designed using FLUENT software.The numerical model was validated by comparing the numerical results with available experimental results of the 90°bend. Subsequently,a numericalmodelof a 120°bend was designedin accordance with the geometric and hydraulic characteristics of the experimental model of the 90°bend,except that the central angle was 120°.Moreover,to study the influences of various hydraulic conditions,four hydraulic conditions employed in laboratory tests,as shown in Table 1,were considered in the 120°bend model.

Table 1 Laboratory hydraulic conditions for 90°channel bend.

Fig.2.Eight cross-sections in 90°bend.

2.2.Numerical modeling,grid,and boundary conditions

The governing equations for the motion of a viscous incompressible fl uid in a turbulent state are the continuity and momentum equations(Navier-Stokes equations)(Versteeg and Malalaskera,2007).In this study,FLUENT software,based on the finite volume method,was used to solve the equations.

Different grids with various sizes were tested based on sensitivity analysis of meshing to simulate the 90°and 120°bends.To adjustthe meshing in the 90°and 120°bend models, the grid near the bed,at the walls,inside the bend,and at the interface between two phases was finer,while the rest of the grid was coarser.Moreover,in the longitudinal direction,the grid was chosen in line with the flow direction to save computational time.The grid became finer and coarser as it approached and exited the bend,respectively.Overall,the grid employed had 67500 nodes for a 90°bend(30,25,and 90 nodes for the width,height,and length,respectively)and 75000 nodes(30,25,and 100 nodes for the width,height,and length,respectively)for a 120°bend.Figs.3 and 4 show grids for both bend models.

The VOF multiphase modelwas used to simulate the water depth change.The Reynolds stresses were modeled with the k-ε(RNG)turbulence model(Yakhotand Orszag,1986).The model integrated the governing equations of fluid flow over each volume control and discretized algebraic equations using various discretization schemes(FLUENT Manual,2005).In the development of the numerical model,the Presto plan wasused to expand the pressure,the Piso plan was used for velocity-pressure coupling,the Quick plan was used for the momentum and volume fraction,and the second-order upwind (SOU)scheme was used to separate the displacement item.

Fig.3.Mesh for 90°bend.

Fig.4.Mesh for 120°bend.

Also,relaxation coeffi cients less than 1 were used for pressure,momentum,turbulence kinetic energy(k),and the turbulence kinetic energy dissipation rate(ε)in order to prevent solution divergence.The essential time step for solving the equations was considered equal to 0.001 s for the divergence process of this simulation.

The velocity inlet boundary condition was used separately for water and air at the inlet.The velocity at the water inlet was based on the entrance velocity in each hydraulic condition listed in Table 1,and the velocity at the air inlet was considered a smallvalue,such as 0.0001 m/s,in allconditions.In the FLUENT setting,the pressure outlet boundary condition was considered for the outlet,the pressure inletboundary condition for the free surface,and the channel bed and walls relied on the wall boundary condition.

2.3.Model performance evaluation

The validation errors are calculated as follows:

where N is the number of measurement points,yexpiis the measured data,and ymodiis predicted by the model.RRMSE and MAE show the difference between the modeled and experimental results.

3.Numerical model validation

To validate the numerical model,the experimental results of a 90°bend were used.Fig.5 compares the longitudinal velocity distributions for different transverse profi les predicted by the numerical model with experimental data for z=3 cm and z=12 cm from the channel bed and depthaveraged velocity(DAV)distribution at cross-sections S1 through S8,with a discharge rate of 25.3 L/s.Table 2 shows the performance of the numerical model of the 90°bend.In general,the results of the numerical model are in acceptable agreementwith those of the experimentalmodelfor all crosssections,with average RRMSE values of 4.95%,5.00%,and 4.62%for z=3 cm,z=12 cm,and the DAV distribution, respectively.Fig.5 shows that,at cross-sections S7 and S8, the numerical model simulates the reversible axial velocity profile well at the inner channel wall for z=12 cm.From Table 2,it can be found that for z=3 cm(near the channelbed),the numerical model has relatively low errors in predicting the flow velocity at cross-sections S6 through S8,and relatively high errors at the interior bend cross-sections;for z=12 cm(near the water surface),the numerical model has high prediction errors at cross-sections S6 through S8,with MAE values of 1.56 and 2.76 cm/s at cross-sections S6 and S8,respectively;and,regarding the DAV distribution,the highestand lowesterrors occurred atcross-sections S6 and S5 based on RRMSE,with values of 6.19%and 3.33%,respectively.However,the maximum and minimum values of MAE for the DAV distribution occurred atcross-sections S2 and S7, respectively.

Fig.5.Comparisons of flow velocity(u)distributions from numerical and experimental models at different cross-sections in 90°bend.

Table 2 Comparisons of RRMSE and MAE in velocity prediction between numerical and experimental models at different cross-sections in 90°bend.

Fig.6.Comparison of longitudinalprofiles of simulated water depths along 90°bend with experimental results.

In Fig.6,the longitudinal profi les of the water depth near the inner and outer walls and along the longitudinalaxis of the channel predicted by the numerical model are compared with the experimental results for the normal water depth of 15.0 cm.Divided by the normal water depth(15.0 cm),the water depth was nondimensionalized.Evidently,there is acceptable agreement between the water depth profiles predicted by the numerical model and the experimental results, especially along the channel axis,with mean RRMSE and MAE values of0.98%and 0.125 along the bend.This indicates that the numerical model was quite capable of simulating the water depth distribution.However,the numerical model did not predict the water depth at the inner wall well,as the RRMSE value was high,with a value of 2%.This was induced by the greater effect of centrifugal force on water depth changes at the inner wall.Subsequently,the water depth changes in a 120°sharp bend and the infl uence of hydraulic conditions on these changes were investigated and discussed.

4.Results and discussion

4.1.Water depth changes at cross-sections before and after 120°bend

In the study of the water depth changes in the 120°bend, eight cross-sections,i.e.,the cross-section located 40 cm before the bend(T1);the cross-sections at 0°(T2),30°(T3), 60°(T4),90°(T5),and 120°(T6)in the bend;and the crosssections located 40 cm(T7)and 80 cm(T8)after the bend were selected.Table 3 shows the dimensionless average waterdepth at cross-sections T1,T7,and T8,respectively,in four experimental conditions(Table 1).For each discharge rate,the water depth was nondimensionalized through division by the corresponding normal water depth,and the water depth at the three cross-sections was almost invariable for each normal water depth.According to Table 3,the water depth at crosssection T1 increases for all normal water depths due to the energy required by flow to enter the bend and the energy loss of flow as it advances through downstream cross-sections in the bend.At the exit of the bend,the presence of a secondary current does notallow the water surface to be fixed(to achieve the normal water depth).In a sharp bend,a secondary current always exists along the bend and the rotational secondary current remains in the channel even after the bend.By advancing along the channel and with the flow development after the bend,and the secondary current diminishes.Ye and McCorquodale(1998)addressed the flow pattern in sharp curved channels and pointed out thatthe water depth increases at cross-sections before the bend.

Table 3 Dimensionless water depths at cross-sections before and after 120°bend and rate of increase of water depth at cross-section before 120°bend,compared with water depths atcross-sections after 120°bend,fordifferentnormalwater depths.

Table 3 also exhibits the rate of increase of the water depth at cross-section T1,compared with the water depths at crosssections T7 and T8.It is evident that,with increasing normal water depth,the rate of increase of the water depth at crosssection T1 descends.In other words,at a low discharge rate, there is a greater increase in the water depth before the bend.

4.2.Investigation of water depth changes at beginning, interior cross-sections,and end of 120°bend

The water depth profi les at the beginning,interior crosssections,and end of the 120°bend were checked.The difference between water depths near the inner and outer walls was considered the water depth changes at all cross-sections. In order to assess the effects of different discharge rates,the measured water depth changes were divided by the normal water depth to achieve dimensionless quantities.Fig.7 shows the dimensionless water depth changes for differentdischarge rates at cross-sections T2 and T6.It is clear that there is a correlation between the water depth change and discharge rate,meaning that the water depth change in the channel is dependent on the hydraulic conditions.Moreover,with the increase of the discharge rate,the dimensionless water depth change decreases at any cross-section.Fig.8 shows thecorrelations between the water depth change and normal water depth at cross-sections T2 and T6.

Fig.7.Effect of discharge rate on water depth changes at cross-sections T2 and T6.

Fig.8.Effect of normal water depth on water depth changes at cross-sections T2 and T6.

Fig.9.Transverse distributions of water depth at beginning,interior cross-sections,and end of 120°bend for different normal water depths.

Fig.9 shows transverse profi les of the water depth distribution atthe beginning,interior cross-sections(T3 through T5), and end of the 120°bend for different discharge rates.Through division by the channel width(40.3 cm),the distance from the channel axis in different cross-sections is nondimensionalized, with negative values towards the outer wall,and positive values towards the inter wall.Evidently,as flow enters the bend,the centrifugalforce causes a transverse water depth slope.Thus,at the interior cross-sections,the water depth increases towards the outerchannelwalland decreases towards the inner wall.By advancing along the bend and in the presence of a secondary current,the maximum water depth at the outer channel wall gradually increases.With the elimination of the secondary current,the transverse water depth slope is destroyed,and the water depth reaches the initial value at the exit of the bend.

Table 4 shows the dimensionless water depth changes at cross-sections T2 through T6.This table and Fig.10 indicate that,in each case,the maximum dimensionless water depth change occurs atcross-section T4(mid-pointof the bend),andthe minimum dimensionless water depth change occurs at cross-section T2.Also,the water depth change at each crosssection decreases with the increase of the discharge rate.

Table 4 Dimensionless water depth changes at cross-sections T2 through T6 of 120°bend for different normal water depths.

Fig.10 shows the water depth changes in the vicinity of the inner wall,channel axis,and outer wall of the channelfor four normal water depths along the 120°bend.Clearly,the water depth at cross-section T1 is higher than the normal water depth.The water depth near the outer wall increases at first, and then gradually decreases after it reaches the maximum value throughout the channel,while the situation near the inner wall is just the opposite.The maximum and minimum water depths occur at the outer and inner walls,respectively. The longitudinal profiles of the water depth change are the same for different discharge rates.

Table 5 shows the dimensionless water depth changes of the inner and outer mid-widths at cross-sections T2 through T6 of the 120°bend calculated with the numerical model for differentnormalwater depths.For allthe normalwater depths, the water depth change of the inner mid-width is greater than that of the outer mid-width at all cross-sections.For each normal water depth,the maximum and minimum water depth changes are,in mostcases,the values of the innermid-width at cross-section T4 and the outer mid-width at cross-section T2, respectively,exceptthatfor the normal water depth of 9.0 cm, the maximum value occurs at cross-section T5,and for the normal water depth of 12.0 cm,the minimum value occurs at cross-section T6.Furthermore,with increasing normal water depth,the dimensionless water depth changes of the inner and outer mid-widths decrease at each cross-section.Therefore, the transverse water depth distribution is nonlinear.In Fig.10, the water depth changes of the inner and outer mid-widths areindicated with blue and red arrows,respectively.Itis observed that for all different discharge rates,the length of blue arrows is larger than that of red arrows.The results are the same as those in Leschziner and Rodi(1979).Bodnˊar and Pˇríhoda (2006)also conducted numerical studies on a 90°sharp bend and concluded that the transverse water depth distribution is nonlinear.

Fig.10.Water depth changes in vicinity of inner wall,channelaxis,and outer wallof 120°bend atdifferent cross-sections for different normal water depths.

Table 5 Dimensionless water depth changes of inner and outer mid-widths of different cross-sections in 120°bend.

Fig.11.Maximum and minimum dimensionless water depths in 120°bend for different normal water depths.

Fig.11 shows the maximum and minimum dimensionless water depths throughoutthe channelfor differentnormalwater depths.From Fig.11,two equations refl ecting the relationships between the maximum and minimum dimensionless water depths and the normal water depth throughout the channelare obtained.With increasing normal water depth,the maximum dimensionless water depth decreases,while the minimum dimensionless water depth increases throughout the channel,respectively,conforming that,the dimensionless water depth changes decrease with the increase of the discharge rate.

5.Conclusions

In this study,the changes in water surface geometry at various cross-sections of a sharp 120°bend were assessed numerically using different normal water depths.The results indicate that,for different normal water depths,the dimensionless water depth at the cross-section located 40 cm before the bend is higher than those at the cross-sections located 40 cm and 80 cm after the bend,and the rate of increase of the water depth atthe cross-section located 40 cm before the bend, compared with the other two cross-sections,decreases with theincrease of the normalwater depth.For different normal water depths,the maximum and minimum dimensionless water depth changes mostly occur atthe 60°cross-section and atthe beginning of the bend,respectively.With increasing normal water depth,the dimensionless water depth changes of the inner and outer mid-widths decrease at each cross-section.

Two equations reflecting the relationships between the maximum and minimum dimensionless water depths and the normal water depth throughoutthe channelwere obtained.According to the presented relations,with increasing normalwater depth,the minimum dimensionless water depth increases,while the maximum dimensionless water depth decreases throughout the channel.Despite the applicability of the equations to conditions with different normal water depths,these equations have some limitations and can only be used for 120°sharp bends. Thus,the equations should be improved through consideration of the bend angle.The currentstudy results can be utilized to study flow behaviors in sharp bends,especially to design channelwalls and side structures(e.g.,spur dikes,channel branches,inlet channels,etc.)thatmay require exactwater depths.

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Received 29 November 2015;accepted 26 April 2016

Available online 6 January 2017

*Corresponding author.

E-mail address:bonakdari@yahoo.com(Hossein Bonakdari).

Peer review under responsibility of Hohai University.

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