3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow...

14
Int. J. Environ. Res., 5(2):381-394, Spring 2011 ISSN: 1735-6865 Received 10 March 2010; Revised 12 Aug. 2010; Accepted 25 Aug. 2010 *Corresponding author E-mail: [email protected] 381 3D Open Channel Flow Modeling by Applying 1D Adjustment Abedini, A. A. 1* , Ghiassi, R. 1 and Ardestani, M. 2 1 Faculty of Civil Engineering, University College of Engineering, University of Tehran, Tehran, Iran 2 Department of Environmental Engineering, Graduate Faculty of Environment, University of Tehran,Tehran, Iran ABSTRACT: A three-dimensional (3D) finite volume model with a novel adjustment scheme was developed to solve shallow water equations in open channels. An explicit finite volume method was used to discretize the governing equations in a boundary-fitted structured and collocated grid system. Because a simple second-order central scheme was used for spatial discretization and due to the occurrence of high Peclet numbers in open channel flows, some treatments were needed to reduce oscillation. Thus, a special adjustment scheme designed to minimize differences in the averaged free surface elevation and flow discharge in a 3D model and 1D flow data was applied to some cross-sections. The model was applied to simulate shallow water flow in a backward- facing step, a meandering channel with 90° bends and a 180° bend channel. A comparison of the model results with available experimental and numerical data illustrated that the proposed numerical procedure decreases the numerical oscillations and increases the stability of the 3D numerical model in open channel flow modeling. Key words: Finite volume, Open channel flow, 3D model, Adjustment scheme, Central scheme INTRODUCTION The prediction of flow patterns and characteristics of fluid flows in rivers and open channels have been the subjects of research for many years. A remarkable number of different models have been used in simulating water bodies regarding both hydraulics and contaminants fate and transport (Rowshan et al., 2007; Nakane and Haidary, 2010; Monazzam, 2009; Ardestani and Sabahi, 2009; Etemad-shahidi et al., 2009; Rajasimman et al., 2009; Sadashiva Murthy et al., 2009; Praveena et al., 2010; Etemad-shahidi et al., 2010; Nabi Bidhendi et al., 2010). Because it is expensive and time- consuming to conduct physical model tests and field measurements, many numerical models have been developed and applied to simulate river and open channel flow problems. Most of the models are two- dimensional and depth- or width-averaged. Depth- averaged models provide satisfactory results for many practical purposes. However, they give no information about the depth variation of longitudinal velocity or secondary flow; therefore, 3D models are necessary for computing all velocity components in three directions. Lane et al. (1999) compared the predictive ability and accuracy of both 3D and 2D models by using high-quality field data of a gravel-bed river confluence. They showed that the 3D model had a higher predictive ability, particularly when the 2D model was not corrected for the effects of secondary circulation on flow structure. The finite difference method has been used extensively to solve the basic governing equations in 3D hydrodynamic models. Shankar et al. (2001) investigated flow characteristics in a channel constriction using a 3D finite difference multi-level model. Chen (2003) proposed a 3D, hydrodynamic model for free surface flow using a finite difference scheme that involved two predictor–corrector steps. Due to the rectangular grid, the classic finite difference method (FDM) requires a rather fine mesh size to give a satisfactory representation for complicated and curved boundary conditions (Abbott, 1979; Chaudhry, 1979), which leads to a high computational effort. Although the finite element method (FEM) does not always require a regular mesh, it leads to dense matrices and, therefore, involves lengthy matrix inversion procedures that cause difficulties in recursive calculations of unsteady flow problems (Bauer and Schmidt, 1983). The finite volume method (FVM),

Transcript of 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow...

Page 1: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

Int. J. Environ. Res., 5(2):381-394, Spring 2011ISSN: 1735-6865

Received 10 March 2010; Revised 12 Aug. 2010; Accepted 25 Aug. 2010

*Corresponding author E-mail: [email protected]

381

3D Open Channel Flow Modeling by Applying 1D Adjustment

Abedini, A. A.1*, Ghiassi, R.1 and Ardestani, M.2

1 Faculty of Civil Engineering, University College of Engineering, University of Tehran,Tehran, Iran

2 Department of Environmental Engineering, Graduate Faculty of Environment, Universityof Tehran,Tehran, Iran

ABSTRACT: A three-dimensional (3D) finite volume model with a novel adjustment scheme was developedto solve shallow water equations in open channels. An explicit finite volume method was used to discretize thegoverning equations in a boundary-fitted structured and collocated grid system. Because a simple second-ordercentral scheme was used for spatial discretization and due to the occurrence of high Peclet numbers in openchannel flows, some treatments were needed to reduce oscillation. Thus, a special adjustment scheme designedto minimize differences in the averaged free surface elevation and flow discharge in a 3D model and 1D flowdata was applied to some cross-sections. The model was applied to simulate shallow water flow in a backward-facing step, a meandering channel with 90° bends and a 180° bend channel. A comparison of the model resultswith available experimental and numerical data illustrated that the proposed numerical procedure decreases thenumerical oscillations and increases the stability of the 3D numerical model in open channel flow modeling.

Key words: Finite volume, Open channel flow, 3D model, Adjustment scheme, Central scheme

INTRODUCTIONThe prediction of flow patterns and characteristics

of fluid flows in rivers and open channels have beenthe subjects of research for many years. A remarkablenumber of different models have been used insimulating water bodies regarding both hydraulics andcontaminants fate and transport (Rowshan et al., 2007;Nakane and Haidary, 2010; Monazzam, 2009; Ardestaniand Sabahi, 2009; Etemad-shahidi et al., 2009;Rajasimman et al., 2009; Sadashiva Murthy et al., 2009;Praveena et al., 2010; Etemad-shahidi et al., 2010; NabiBidhendi et al., 2010). Because it is expensive and time-consuming to conduct physical model tests and fieldmeasurements, many numerical models have beendeveloped and applied to simulate river and openchannel flow problems. Most of the models are two-dimensional and depth- or width-averaged. Depth-averaged models provide satisfactory results for manypractical purposes. However, they give no informationabout the depth variation of longitudinal velocity orsecondary flow; therefore, 3D models are necessaryfor computing all velocity components in threedirections. Lane et al. (1999) compared the predictiveability and accuracy of both 3D and 2D models by using

high-quality field data of a gravel-bed river confluence.They showed that the 3D model had a higherpredictive ability, particularly when the 2D model wasnot corrected for the effects of secondary circulationon flow structure.

The finite difference method has been usedextensively to solve the basic governing equations in3D hydrodynamic models. Shankar et al. (2001)investigated flow characteristics in a channelconstriction using a 3D finite difference multi-levelmodel. Chen (2003) proposed a 3D, hydrodynamicmodel for free surface flow using a finite differencescheme that involved two predictor–corrector steps.Due to the rectangular grid, the classic finite differencemethod (FDM) requires a rather fine mesh size to givea satisfactory representation for complicated andcurved boundary conditions (Abbott, 1979; Chaudhry,1979), which leads to a high computational effort.Although the finite element method (FEM) does notalways require a regular mesh, it leads to dense matricesand, therefore, involves lengthy matrix inversionprocedures that cause difficulties in recursivecalculations of unsteady flow problems (Bauer andSchmidt, 1983). The finite volume method (FVM),

Page 2: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

382

Abedini, A. A. et al.

which has the merits of both the FDM and FEM, wasfirst introduced into the field of numerical fluiddynamics (independently by McDonald (1971) andMacCormack and Paullay (1972)) for the solution oftwo-dimensional, time-dependent Euler equations(Hirsch, 1988). Recently, Ye et al. (1998) developed athree-dimensional hydrodynamic model for free surfaceturbulent flow using an implicit finite volume methodwith a standard k-ε turbulence model in a collocatedgrid. Chiavassa et al. (2003) investigated the effects ofdiscretization schemes on a 3D finite volume model ofthe Rhône River plume. Zarrati and Jin (2004)developed a three-dimensional finite volume multi-level model to simulate free surface flows.

In general, the width and length of a river are muchlarger than the depth, and the vertical velocity of waterflow is much smaller than the horizontal velocities. Thisassumption reduces the momentum equation in thevertical direction to the hydrostatic pressure law, andthe 3D Navier-Stokes equations are simplified to thetwo momentum equations in the x-y plane. Theresulting equations are referred to as the 3D shallowwater equations. In this study, an explicit 3D finitevolume model was developed to simulate shallow waterflow in rivers and open channels. The central differencescheme was used to discretize the governing equationsin a collocated and structured grid.

In numerical models, when convection is strongerthan diffusion (i.e., Peclet number > 2), a central schememay produce numerical oscillations and physicallyunrealistic results. An upwind scheme does not exhibitoscillation, but it has first-order accuracy and mayoverestimate diffusion at large Peclet numbers(Patankar, 1980). To decrease the numerical oscillation,there are several high-order upwind-type schemes suchas flux-vector splitting, flux-difference splitting, totalvariation diminishing (TVD) and fluctuation splittingas well as central-type schemes such as the Nessyahu-Tadmor method for the discretization of convectivefluxes (Blazek, 2001). These high-order schemes on astaggered (or collocated) grid usually reduce thenumerical oscillation appropriately. In addition, forcollocated grids, the well-known checkerboardinstability may appear (Rhie and Chow, 1983).Therefore, for non-staggered grids, the Rhie and Chowinterpolation should be applied.

In this study, for spatial discretization, a simplesecond-order central difference scheme on a collocatedgrid without the Rhie and Chow interpolation was used.Thus, the model may confront numerical oscillationand checkerboard instability. To solve these problems,a special adjustment scheme based on adjusting a 3Dmodel with 1D flow data was applied.

The presented model was verified and validated withthree different case studies of flow in open channels:(1) flow over a backward-facing step, (2) flow in ameandering channel with 90° bends and (3) flow in achannel with a 180° bend. The numerical results arecompared to available experimental and numerical datain terms of free surface profiles and velocities.

Governing equationsFree surface flow can be studied through mass

and momentum conservation equations (Streeter andWylie, 1958; Shames, 1962). Under the assumption of aconstant density and hydrostatic pressure distributionand neglecting wind and Coriolis forces, the resulting3D shallow water equations can be expressed in integralforms as follows (Ghiassi, 1995):

where v is the velocity vector; u and v are the velocitycomponents in the horizontal x- and y-directions,respectively; t is time; η is the water surface elevationmeasured from the undisturbed water surface; g is

gravitational acceleration; and τ x and τ y are the

components of the shear stress tensor in the x- and y-directions, respectively.Because the main idea of the research was to establisha 3D-1D adjustment scheme for shallow water flowmodeling, a basic algorithm for shear stress evaluationwas applied. Thus, the components of the shear stresstensor were calculated based on the Boussinesqapproximation, using Prandtl’s mixing length model todetermine the eddy viscosity (Cea et al., 2007). Theresults show that this model, which is categorized inzero-equation turbulence models, is generallysufficiently accurate for shallow water flows. However,for investigating a local full 3D flow, more appropriateturbulence models such as two-equation k-ε or k-ω oranisotropic turbulence models (Obi and Peric, 1994;Wallin and Johansson, 2000) can be applied.

Integrating the continuity equation over the watercolumn together while using the kinematic conditionat the free surface leads to the following free surfaceequation (Casulli, 1999):

0. =∫s

sdv (1)

∫∫∫∫ +−=+Ω∂∂

Ω sx

sx

ssdsdgsdvuud

t i .1.).( τρη

(2)

∫∫∫∫ +−=+Ω∂∂

Ω sy

sy

ssdsdgsdvvvd

t i .1.).( τρη

(3)

Page 3: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

Int. J. Environ. Res., 5(2):381-394, Spring 2011

383

where ∫−= ηh udzU ; ∫−= η

h vdzV ; and h is the

water depth from the bed to the undisturbed freesurface. This means that H(x, y, t) = h(x, y) + η(x, y, t)is the total depth of the water column.

Numerical MethodTo solve the governing 3D equations, we used a

cell-centered finite volume method on a boundary-fittedstructured and collocated grid. The boundary-fittedstructured grid has the following advantages: it isrelatively straightforward to impose proper boundaryconditions, the solution is sufficiently accurate nearthe boundary, and the conservation property isobserved. However, depending on the shape of theboundary, the quality of the structured boundary-fittedgrid can be a concern (Blazek, 2001). The computationaldomains were divided into a number of hexahedral cells,and all geometrical characteristics of the cells, such asthe coordinates, surface area and volume, werecomputed.The governing 3D equations were discretized by usinga finite volume method. The surface integrals inEquations (1), (2), (3) and (4) were approximated bysumming over the six sides of each cell as follows:

where Ω is the volume of the cell; Sx, Sy and Sz are thex-, y- and z-component areas of the cell faces,respectively; and Szt is the z-component area of thetop surface of the top cell in each water column.All ofthe variables at the cell surfaces were calculated usinga second-order accuracy central difference scheme asfollows:

where φ = u, v, w, η and l i is the distance between

the center of the ith cell and the center of the surface(i+1/2) in the ith cell. The same equations are applicableto the other surfaces of the cell.The overall numerical procedure can be summarizedby the following steps:

1.Read the physical domain geometry.2.Read the initial flow and boundary conditions.3.Generate the 3D boundary-fitted structured grid usingthe algebraic grid generation method.4.Read the 1D flow data.5.Improve the initial flow conditions based on the 1Dflow data.6.Determine the wet and dry cells in thecomputational domain based on the latest values ofthe water surface elevation (ηn) (refer to section 3-5).7.Solve the free surface equation explicitly to computeηn+1 by applying a modified form of Equation (8) asfollows:

The values of u and v are taken from the last time step(n).1.Adjust the calculated free surface elevations (ηn+1)based on the difference between the average free

surface elevation (η i )3D and the 1D free surface

elevation (η i )1D (refer to section 3-3).

2.Solve the x-direction momentum equation explicitlyto compute un+1 by using the latest values of the watersurface elevation (ηn+1).From Equation (6), we have the following:

∑=

=++6

10... )(

sszyx swsvsu (5)

∑=

=+++Ω∆

−+

6

1)...(.

1

s sszuws yuvs xuut

un

un

(6)

∑=

+++∑=

−61 )...(

161 ).( s sszxzsyxysxxxs ssxg τττρη

∑=

=+++Ω∆

−+

6

1)...(.

1

s ss zvws yvvs xvut

vn

vn (7)

∑=

+++∑=

−6

1 )...(16

1 ).( s sszyzs yyysxyxs ss yg τττρη

0=∂∂

+∂∂

+∂∂

yV

xU

tη (4)

04

1 )..(max

1.

1

=∑=

+∑=

+∆

−+

s ssyvsxuk

kS ztt

nnηη

(8)

llll

ii

kjiikjiikji

1

,,1,,1,,2/1

+

+++ +

+=

φφφ (9)

∑=

+∑=

∆−=

+ 41 )..(

max

1

1s ss yvs xu

k

kS zt

tnnηη

(10)

∑=

++Ω

∆−=

+ 6

1 )...(1

s ss zuws yuvs xuut

un

un (11)

∑=

++

Ω

∆+∑

∆−

6

1 )...(

16

1).(

s sszxzsyxysxxx

t

s ssxt

g

τττ

ρη

Page 4: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

384

Open Channel Flow Modeling

All of the variables on the right hand side of theequation, except η, are taken from the last time step (n),while η is taken from the current time step (n+1).1.Solve the y-direction momentum equation explicitlyto compute vn+1 by using the latest values of the watersurface elevation (ηn+1).From Equation (7), we have the following:

All of the variables on the right hand side of theequation, except η, are taken from the last time step (n),while η is taken from the current time step (n+1).1.Solve the continuity equation explicitly to computewn+1 by using the latest values of the horizontalvelocities (un+1, vn+1).In each water column (i, j), the vertical velocitycomponent was computed from the bottom to the topcell, so for the first cell (i, j, 1) that is next to the bedwith the vertical velocity at the bottom surface equal

to zero ( 0)1,,( =w jibottom ), Equation (5) can be

written as follows:

( ww jitopjibottom )1,,()2,,( = ), so the value of

w jitop )2,,( can be calculated. These calculations

are continued until reaching the top cell, and then the

vertical velocities at the center of the cells ( w kji ,, )

are equal to 2/)( ),,(),,( ww kjitopkjibottom + .

1.Calculate the 3D flow discharge in each section andadjust the calculated velocities based on the difference

between the 3D flow discharge (Qi )3D and the 1D

flow discharge (Q1D) (refer to section 3-3).2.Repeat the computation procedure from step 6 tostep 12 for the next time step.

In this study, for spatial discretization, a simplesecond-order central difference scheme on a collocatedgrid without the Rhie and Chow interpolation was used.To treat the numerical oscillation and preventinstability, a special adjustment scheme based onadjusting the 3D model with 1D flow data was applied.Two parameters in the 3D model, including the averagefree surface elevation and the total flow discharge ineach cross-section, were selected for adjustments with1D flow data, which were computed as follows:

∑=

++Ω

∆−=

+ 6

1 )...(1

s sszvws yvvsxvut

vn

vn

(12)

∑=

++

Ω

∆+∑

∆−

6

1 )...(

16

1 ).(

s sszyzsyyysxyx

ts ssy

tg

τττ

ρη

0.).(

6

1 )..(

=+∑

+∑=

+

s topzwtopfaceslateral szw

s ssyvsxu

(13)

The values of u and v are taken from the current timestep (n+1), and if the values of Sz#0 for the lateralfaces, then the values of w on the right hand side ofthe equation are taken from the last time step (n).However, in most of the cells (except for the cells thatare next to the lateral boundaries), all of the lateral facesare vertical, which means Sz=0. For the second cell,the vertical velocity at the bottom surface is known

∑=

∑=

−=max

1

max

1 cossin,,cos ,cos,,(3)(j

j

k

k iv kjijiiu kjiDQi αβα

∑=

∑=

−−=max

1

max

1*)sin ,,,cos ,sin,,cos ,cos,,(3)(

j

j

k

k jiw kjijiiv kjijiiu kjiDQi ββαβα

∑=

∑== max

1 ,

max

1),,(

3)( j

j L ji

j

j L jijiDi

ηη

(15)

βcos ,2/1,,

22 )(* jikjiyx ss + (16)

The value of wtop can be calculated as follows:

∑=

+

−∑−=

6

1 )..(

1).(

1

s ssyvsxu

s topzfaceslateral szws topz

wtop

(14)

where L is the width of the cell (distance between thecenter of surface (j-1/2) and surface (j+1/2)), and αand β are the horizontal and vertical angles of the cell,respectively.In each cross-section, to prevent increases in numericalerror, the 3D results were adjusted with 1D flow data.The amount of mass residual in each cross-section,which is equal to the summation of the absolute valueof the difference between the outflow and inflow of

Page 5: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

385

Int. J. Environ. Res., 5(2):381-394, Spring 2011

where v and w are adjusted in the same manner as u.All 1D flow data can be obtained by using existingnumerical 1D model such as HEC-RAS, MIKE-11 orCECAD-REM.

According to the theory of characteristics, whenthe flow regime is subcritical, two physical boundaryconditions, one at the upstream and another at thedownstream, are needed. The remaining unknownparameters at each boundary can be computed usingRiemann invariants (Yoon and Kang, 2004). Forsupercritical flow, these two boundary conditions areneeded at the upstream.In the model, four types of boundary conditions wereconsidered, including a known velocity; a known watersurface level; an open boundary (zero-gradientcondition) and a closed boundary. These boundaryconditions can be applied for upstream, downstream,and lateral boundaries in the computational domain,depending on the flow regime and boundary type. Atthe free surface, the water flux across the surface wasassumed to be zero, and the wind stress was ignored.On the bed, the water flux was also assumed to bezero, and a quadratic friction law was used to calculatethe bottom shear stress (Henderson, 1966).For free surface tracking, in each water column, the k-index for the top layer “Kmax” was allowed to varywith time. This technique eliminates the use of a thicktop layer to cover the variation in free surface elevationand allows the surface to move from one layer toanother. The “Kmax” index for each water column wascalculated and saved at each time step. When the freesurface at the (n+1)th time step drops below one thirdof the top cell height, the “Kmax” index is reduced by1. The geometrical characteristics of the top cells,including the cell volume and cell lateral faces, werecalculated again based on the water surface elevationand were updated at each time step.

The above solution procedure is a combination ofsecond-order central spatial discretization, an explicitmultistage temporal discretization and a 3D-1Dadjustment scheme. Each explicit scheme remains stableonly up to a certain value of the time step (∆t). To bestable, an explicit scheme has to fulfill the so-calledCourant-Friedrichs-Lewy (CFL) condition. On a 3Dstructured grid, the CFL condition for a control volume

Ω i may be expressed as follows (Blazek, 2001):

where, Cr is the CFL number and is specified in the

range 0<Cr ≤ 1.With Equation (19), a local time step that is valid forone control volume is obtained. For a global time stepfor all volumes, a minimum over all volumes is taken.

each water column, was considered as an indicator ofthe numerical error. Water surface elevations wereadjusted based on the difference between the 1D free

surface elevation (η i )1D and the 3D average free

surface elevation (η i )3D, and velocities were adjusted

based on the ratio between the 1D and 3D flowdischarges as follows:

)3)(1)((,, DiDijiadjusted

ji ηηηη −+= (17)

)3)/(1)((*,,,, DQiDQiu kjiuadjusted

kji = (18)

s zhigwisyhigvisxhigui

iCrt i )()()( +++++

Ω=∆

(19)

)(min tt ii

∆∆ = (20)

In this study, to choose an appropriate time step thatincreases the stability, the CFL number was specifiedbetween approximately 0.1~0.5.

Model verification and validationThe numerical model was verified and validated

with available laboratory experimental data andnumerical results.This case is commonly used to validate models thatcalculate flows with large recirculation regions. Thegeometry is depicted in Fig. 1. The dimensions of thecomputational domain were l = 8×s, L = 22×s, h = 2×sand H = 3×s, in which s is the step width, equal to 1 min this test case.

A total of 9585 cells were used, with the followingdistribution: 27×30×1 in the entrance region and195×45×1 in the region after the step. The meshgeneration was non-uniform with a greaterconcentration of points in the region of the expansion.The grid dimensions in the expansion area were equalto 5×5 cm. The Cartesian coordinates x and y weremeasured from the lower step corner, and the Reynoldsnumber based on the step width was considered as7860.

A comparison of velocity profiles at the point x/s= 5.3, which shows good agreement between ourresults and the available experimental data (Kim et al.,1980) and numerical references (Mansour et al., 1983;Zdanski et al., 2003), is presented in Fig. 2.

Page 6: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

386

The velocity vectors are shown in Fig. 3. Basedon the achieved results, the reattachment point waspredicted as xr / s = 6.7. Experimental data from Eatonand Johnston (1981) show xr / s = 7.0±1.0, whichconfirms the accuracy of the model results.

Fig. 2. Velocity profiles at x / s = 5.3 of thebackward-facing step model

Fig. 1. Backward-facing step geometry plan

To investigate the convergence and discretizationerrors, the grid convergence index (GCI) proposed byRoache (1994) was applied. Hence, flow computationswere performed for two additional mesh systems with10×10-cm and 20×20-cm grid dimensions in theexpansion area (designated as medium (2) and coarse(3) meshes, respectively). The GCI corresponding to

the fine grid solution f fine is defined as follows

(Cadafalch et al., 2002):

in which:

1

1/)21(

−=

rp

fffFsGCI fine

(21)

r

ffffp

ln

)12/()23(ln −−= (22)

where Fs = 3 is a safety factor; r is the grid refinementratio, equal to two in this test case; p is the order ofaccuracy, which is bounded to two because a second-order central spatial discretization was used; and f1, f2and f3 are the evaluated quantities of the parameters ofinterest for fine (1), medium (2) and coarse (3) grids,respectively.In this example, the longitudinal velocities (U) at severalsections were selected as evaluated quantities.Information on the calculated GCIfine is summarized inTable 1. It can be seen that the maximum numericaluncertainty in the fine grid solution for the longitudinalvelocities (U) is 5.83%. The position of the maximumGCI in all sections is near the expansion area, whichshows the need for mesh refinement in the expansionarea.

In this test, the absolute value of the Peclet numbervaried from 0.0009 to 414.5, with an average value of21.0. For 23.0% of the cells, the absolute value of thePeclet number was less than two (|Pe| < 2), and for77.0% of the cells, |Pe| > 2. Thus, for most of the cells,the central scheme without adjustment might producenumerical oscillation. To evaluate the effects of theadjustment scheme, a convergence test in terms of themass residual history was performed for fine andmedium grids with three different CFL numbers. Thetotal mass residual was computed by summing theabsolute value of the difference between the outflowand inflow of each water column, scaled by the totalvolumes of the cells. The results are presented in Fig.4. As shown, when the CFL number was equal to 0.5,

Abedini, A. A. et al.

Page 7: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

387

Int. J. Environ. Res., 5(2):381-394, Spring 2011

Fig. 3. Velocity vectors in the backward-facing step model

the model without the adjustment scheme did notconverge and, especially for the medium mesh, wasunstable. However, by applying the adjustmentscheme, the model was stable and converged to thefinal results. For lower CFL numbers, while the modelwithout the adjustment scheme did not convergewithin the specified iteration numbers, applicationof the adjustment scheme resulted in more stabilityand decreased the required number of iterations aswell.

Tamai et al. (1983) studied the flow characteristicsof a meandering channel. The experimental channelconsisted of ten consecutive bends with a rectangularcross-section. Each curve was composed of a circularchannel and a straight reach so that the centerline ofthe channel closely followed a sine-generated wave.The radius of the channel centerline was 0.6 m with 90°bends. The length of the straight reach was 0.3 m, andthe channel width was 0.3 m. The Manning roughnesscoefficient of the channel was estimated to be equal to0.013. For a flow discharge of 0.002 m3/s with alongitudinal bed slope of 1/1000, the average flow depthand Froude number were 0.03 m and 0.42, respectively.The ratio of the channel-width to water-depth wasequal to ten, and the ratio of the curvature-radius tochannel-width was equal to two, similar to commonmeandering channels in nature (Zarrati et al., 2005).

Table 1. Calculation of the Grid Convergence Index in the backward-facing step test

GCI fine (%) Posit ion Section ( x / s ) Average Maximum Minimum ( y / s )max ( y / s )min

5.3 2.10 4.76 0.14 0.525 1.700 4.3 1.12 3.42 0.10 0.525 1.500 3.3 1.26 2.44 0.27 0.275 1.500 2.3 2.01 5.83 0.27 0.475 1.900

According to Jin and Steffler (1993), when the water-depth to curvature-radius ratio is small in a curvedchannel, the pressure distribution is hydrostatic.Because the water-depth to curvature-radius ratio was0.05 in this test, which is very small, the assumption ofa hydrostatic pressure distribution was valid. A 3Dmesh with 94×10×6 cells was generated for twoconsecutive bends as shown in Fig. 5.

Water surface profiles at three sections acrossthe bend (as illustrated in Fig. 5) were compared to theexperimental data (Fig. 6). The agreement between theresults and the laboratory measurement data is good.The maximum coefficient of variation (CV=standarddeviation/mean value) between the model results andexperimental data was 3.1%.

Longitudinal and transverse velocity profiles atthree sections along the bend and in each section atfive different locations, denoted by 1 to 5, are comparedwith experimental data in Figs 7 and 8, respectively.Location 3 is at the centerline, and the other sectionsare at 5-cm intervals on the right and left sides of thecenterline, with location 1 near the outer bank. Despitethe complex flow pattern and the use of a simpleturbulence model, the agreement between thecalculated longitudinal velocities and experimentaldata is very good. However, some differences can be

Page 8: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

388

Open Channel Flow Modeling

seen between the computed transverse velocities andthe experimental data. It seems that the use of a moreadvanced turbulence model can improve the transversevelocities.

For all of the cells, the absolute value of the Pecletnumber was more than two (2.1<|Pe|<48905.0, with anaverage value of 115.1) so, as in the previous test, the

central scheme without adjustment might producenumerical oscillation. A convergence test wasperformed for two different CFL numbers. As shown inFig. 9. for both CFL numbers, application of theadjustment scheme decreased the required number ofiterations and stabilized the model in converging tothe final results.

Fig. 4. Mass residual history in the backward-facing step model

Page 9: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

389

Int. J. Environ. Res., 5(2):381-394, Spring 2011

Experiments in a laboratory open channel with a180° bend were conducted by De Vriend (1977), anddepth-averaged velocity distributions and watersurface profiles were reported. The channel cross-section was rectangular with a 1.7 m width, thecenterline radius was 4.25 m, and the inlet and outlet ofthe bend were connected to 6-m-long straight reaches.The channel boundaries were hydraulically smooth.The flow discharge (Q) was 0.19 m3/s with a constantdownstream water depth (H0) of 0.18 m and a Froudenumber of 0.47.

Fig. 5. Layout of the calculated reach and measurement sections of the 90° bends

Fig. 6. Comparison of calculated transversal water surface with experimental data of 90° bends( • Experimental data, – – – Present work )

Computations were carried out for the sameconditions used in the experiment, and the simulationresults were compared with the available experimentaldata of De Vriend (1977) and the numerical results ofYe et al. (1998). A 3D mesh with 56×26×10 cells wasapplied for the numerical computations. Fig. 10 givesthe depth-averaged velocity distributions at severalcross-sections of the channel. The agreement betweenour results and the above-mentioned laboratorymeasurement data and numerical results is very good.

Page 10: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

390

Fig. 7. Longitudinal velocity profiles at different locations along the 90° bends(circles: experimental data; solid line: Zarrati and Jin; dashed line: present work)

Fig. 8. Transverse velocity profiles at different locations along the 90° bends(circles: experimental data; solid line: Zarrati and Jin; dashed line: present work)

Fig. 9. Mass residual history for the model of a meandering channel with 90° bends

Abedini, A. A. et al.

Page 11: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

Fig. 11 shows velocity vectors in the bed and in thetop layers of the flow model. As shown in Figures 10and 11, the maximum velocity at the beginning of abend (0°) occurs near the inner bank, and at the end ofa bend (180°), it occurs near the outer bank.

Fig. 12 shows the secondary currents at variouscross-sections. As can be seen, the maximum velocityof secondary currents at the beginning of a bendoccurs near the inner bank and shifts towards the outerbank at the end of the bend, related to the location ofthe maximum longitudinal velocity, which causesmaximum centrifugal forces. This figure also showswetting and drying of the cells at the water surface.Near the outer bank, due to super-elevation of the watersurface, a new layer of cells was added to thecomputational domain (for example, zone 0.0-0.4 at97.5°).

As with the previous test, for all of the cells, theabsolute value of the Peclet number was greater thantwo (2.6<|Pe|<619007.0, with an average value of 572.2),so the central scheme without adjustment mightproduce numerical oscillation. In this test, due to veryhigh Peclet numbers, the adjustment scheme did notallow for the use of higher CFL numbers but, of course,decreased the required number of iterations and

Fig. 10. Depth-averaged velocity distribution at different locations of the 180° bend channel( • De Vriend, — Ye et al., - - Present work )

stabilized the model in converging to the final resultsfor a CFL number equal to 0.1 (Fig. 13).

CONCLUSIONIn the present study, a three-dimensional finite

volume model is developed to predict water surfaceelevation and 3D velocity profiles in open channelflows. The solution procedure is a combination of asecond-order central spatial discretization on acollocated grid, an explicit multistage temporaldiscretization and a novel adjustment scheme. Themodel is validated, by using three different cases ofopen channel flows. In all the test cases, the modelresults were in good agreement with experimental dataand numerical references. Some overestimation wasseen in longitudinal velocity profiles near the outerbank of meandering channel with 90° bends and alsoin depth-averaged velocities near the inner bank ofchannel with 180° bend that can be investigated infuture studies.The developed model, despite using azero-equation turbulence model and hydrostaticassumptions, leads to good results in the modeling of3D shallow water flow in open channels. Based on thepresented results, the proposed numerical proceduredecreases the numerical oscillations and increases thestability of the 3D numerical model. Thus, it can be

Int. J. Environ. Res., 5(2):381-394, Spring 2011

391

Page 12: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

Fig. 11. Velocity vectors at (A) the bed layer and (B) the top layer of the 180° bend channel

Fig. 12. Secondary currents at various cross-sections of the 180° bend channel

Open Channel Flow Modeling

392

Page 13: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

Fig.13. Mass residual history in the model of the 180° bend channel

used to predict the major characteristics of shallowwater flows in open channels.

REFERENCESAbbott, M. B. (1979). Computational hydraulics: elementsof the theory of free-surface flows, Pitman: London.

Ardestani, M. and Sabahi, M. S. (2009). Inverse Method toEstimate the Mass of Contamination Source by ComparingAnalytical with Numerical Results. Int. J. Environ. Res., 3(2), 317-326.

Bauer, S. W. and Schmidt, K. D. (1983). Irregular-grid finite-difference simulation of lake Geneva surge. J. Hydraul. Eng.,109 (10), 1285-1296.

Blazek, J. (2001). Computational fluid dynamics: principlesand applications, Elsevier Science Ltd, UK.

Cadafalch, J., Perez-Segarra, C. D., Consul, R. and Oliva, A.(2002). Verification of finite volume computations on steadystate fluid flow and heat transfer. J. of Fluids Eng., 124, 11-21.

Casulli, V. (1999). A semi-implicit finite difference methodfor non-hydrostatic free-surface flows, Int. J. Numer. Meth.Fluids, 30, 425-440.

Cea, L., Puertas, J. and Vazquez-Cendon, M. E. (2007). Depthaveraged modelling of turbulent shallow water flow with wet-dry fronts. Arch. Comput. Methods Eng., 14, 303-341.

Chaudhry, M. H. (1979). Applied Hydraulic Transient, VanNostrand Reinhold: New York.

Chen X. (2003). A fully hydrodynamic model for three-dimensional free-surface flows, Int. J. Numer. Meth. Fluids,42, 929-952.

Chiavassa, S. A., Rey, V. and Fraunié, P. (2003). Modeling3D Rhône river plume using a higher order advection scheme,Oceanologica Acta, 26, 299-309.

De Vriend, H. J. (1977). A mathematical model of steady flowin curved shallow channels, J. Hydraul. Res., 15, 37-53.

Eaton, J. K. and Johnston, J. P. (1981). A review of research onsubsonic turbulent flow reattachment. AIAA J., 19, 1093-100.

Etemad-Shahidi, A, Afshar, A. , Alikia, H. and Moshfeghi,H. (2009). Total Dissolved Solid Modeling; KarkhehReservoir Case Example, Int. J. Environ. Res., 3 (4), 671-680.

Etemad-Shahidi, A ., Faghihi, M . and Imberger, J. (2010).Modelling Thermal Stratification and Artificial De-stratification using DYRESM; Case study: 15-KhordadReservoir, Int. J. Environ. Res., 4 (3), 395-406.

Ghiassi, R. (1995). Three dimensional coastal flow modelingusing the finite volume method, Phd thesis, University ofBradford, UK.

Henderson, F. M. (1966). Open channel flow, MacmillanCo. Press, New York.

Hirsch, C. (1988). Numerical computation of internal andexternal flows, Vol. 1, Fundamentals of numericaldiscretization, John Wiley & sons Ltd.

Jin, Y. C. and Steffler, P. M. (1993). Predicting flow in curvedopen channels by depth-averaged method, J. Hydraul. Eng.,119 (1), 109-124.

Kim, J., Kline, S. J. and Johnston, J. P. (1980). Investigationof a reattaching turbulent shear layer: flow over a backward-facing step. J. Fluid Eng., 102, 302-308.

Lane, S. N., Bradbrook, K. F., Richards, K. S., Biron, P. A.and Roy, A. G. (1999). The application of computationalfluid dynamics to natural river channels: three-dimensionalversus two-dimensional approaches. Geomorphology, 29,1-20.

MacCormack, R. W. and Paullay, A. J. (1972). ComputationalEfficiency Achieved by Time Splitting of Finite Difference

Int. J. Environ. Res., 5(2):381-394, Spring 2011

393

Page 14: 3D Open Channel Flow Modeling by Applying 1D Adjustment ·  · 2013-06-093D Open Channel Flow Modeling by Applying 1D Adjustment ... investigated flow characteristics in a channel

Operators, American Institute of Aeronautics andAstronautics, Paper, 72-154. American Institute ofAeronautics and Astronautics: San Diego, Calif.

Mansour, N. M., Kim, J. and Moin, P. (1983). Computationof turbulent flows over a backward-facing step, NasaTechnical Memorandum, 85851.

McDonald, P. W. (1971). The computation of transonicflow through two-dimensional gas turbine cascades, AmericanSociety of Mechanical Engineers, Paper, 71-GT-89. AmericanSociety of Mechanical Engineers: New York.

Monazzam, M. R. (2009). Optimization of Profiled DiffuserBarrier Using the New Multiimpedance DiscontinuitiesModel. Int. J. Environ. Res., 3 (3), 327-334.

Nabi Bidhendi, Gh. R., Mehrdadi, N. and Mohammadnejad,S. (2010). Water and Wastewater Minimization in TehranOil Refinery using Water Pinch Analysis. Int. J. Environ.Res., 4 (4), 583-594.

Nakane, K. and Haidary, A. (2010). Sensitivity Analysis ofStream Water Quality and Land Cover Linkage Models UsingMonte Carlo Method. Int. J. Environ. Res., 4 (1), 121-130.

Obi, S. and Peric, M. (1994). Second-Moment CalculationProcedure for Turbulent Flows with Collocated VariableArrangement. AIAA Journal, 29, 585-590.

Patankar, S. V. (1980). Numerical Heat Transfer and FluidFlow, McGraw-Hill, New York.

Praveena, S. M., Abdullah, M. H. and Aris, A. Z. (2010).Modeling for Equitable Groundwater Management. Int. J.Environ. Res., 4 (3), 415-426.

Rajasimman, M., Govindarajan, L. and Karthikeyan, C.(2009). Artificial Neural Network Modeling of an InverseFluidized Bed Bioreactor. Int. J. Environ. Res., 3 (4), 575-580.

Rhie, C. M. and Chow, W. L. (1983). Numerical study ofthe turbulent flow past an airfoil with trailing edgeseparation, American Institute of Aeronautics andAstronautics Journal. 21 (11), 1525-1532.

Rowshan, G. R., Mohammadi, H., Nasrabadi, T., Hoveidi,H. and Baghvand, A. (2007). The Role of Climate Study inAnalyzing Flood Forming Potential of Water Basins.International Journal of Environmental Research, 1 (3), 231-236.

Sadashiva, Murthy, B. M., Ramesh, H. S. andMahadevaswamy, M. (2009). Pollution Migration Study inSubsurface Environment. Int. J. Environ. Res., 3 (4), 545-556.

Shames, I. H. (1962). Mechanics of fluids, McGraw–Hill,New York.

Shankar, N. J., Chan, E. S., and Zhang, Q. Y. (2001). Three-dimensional numerical simulation for an open channel flowwith a constriction. J. Hydraul. Res., 39 (2), 187-201.

Streeter, V. L. and Wylie, E. B. (1958). Fluid mechanics,McGraw–Hill, New York.

Tamai, N., Ikeuchi, K., Yamazaki, A. and Mohamed, A. A.(1983). Experimental analysis on the open channel flow inrectangular continuous bends. J. of Hydroscience andHydraul. Eng., 1 (2), 17-31.

Wallin, S. and Johansson, A. V. (2000). An explicit algebraicReynolds stress model for incompressible and compressibleturbulent flows, Journal of Fluid Mechanics. 403, 89-132.

Ye, J., Mccorquodale, J. A. and Barron, R. M. (1998). Athree-dimensional hydrodynamic model in curvilinearcoordinates with collocated grid, Int. J. Numer. Meth. Fluids,28, 1109-1134.

Yoon, T. H. and Kang, S. K. (2004). Finite volume model fortwo dimensional shallow water flows on unstructured grids.J. Hydraul. Eng., 130 (7), 678-688.

Zarrati, A. R. and Jin, Y. C. (2004). Development of ageneralized multi-layer model for 3-D simulation of freesurface fows. Int. J. Numer. Meth. Fluids, 46, 1049-1067.

Zarrati, A. R., Tamai, N. and Jin, Y. C. (2005). Mathematicalmodeling of meandering channels with a generalized depthaveraged model, J. Hydraul. Eng., 131(6), 467-475.

Zdanski, P. S. B., Ortega, M. A., Nide, G. C. R. and Fico, Jr.(2003). Numerical study of the flow over shallow cavities,J. Computers & Fluids, 32, 953-974.

Abedini, A. A. et al.

394