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J. Hydrol. Hydromech., 54, 2006, 3, 269–279

NUMERICAL MODELLING OF THE VELOCITY DISTRIBUTION IN A COMPOUND CHANNEL 1)

F. OTHMAN and 2)E.M. VALENTINE

1) Department of Civil Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia; mailto: [email protected] 2) School of Engineering and Logistics, Charles Darwin University, Darwin NT 0909, Australia; mailto: [email protected]

Uniform flow in compound channels has been studied in terms of a numerical model, called the NKE model. The model uses the three dimensional Navier-Stokes equations in conjunction with the non-linear k-ε turbulence model. The latter is used for the calculation of the Reynolds stress components responsible for the generation of the secondary currents. This model is based on the SIMPLE technique, and computes the six parameters U, V, W, P, k, and ε using wall functions on a Cartesian grid. The NKE model was used to simulate the compound open channel flows of the UK Flood Channel Facility run 080301 (Shiono and Knight, 1989). The Reynolds Stress Model (RSM) of FLUENT was also used as a comparison. The results obtained have shown that the NKE and RSM models can reasonably predict the primary mean velocity and secondary currents. Although agreement is certainly not perfect in every detail, the main features of the flow are reproduced. The bulging of the contours at the bottom corner of the main channel, the inclination of the contours near the free surface towards the channel centre, and the depression of the maximum velocity below the free surface can be seen. These are consistent with the pattern of the secondary flows, which are mainly formed by two vortices, namely the main channel vortex and flood plain vortex. These vortices, which originate near the main channel-flood plain junction, can be reproduced by the NKE and RSM models. KEY WORDS: Compound Channel, Navier-Stokes Equation, Non-linear k-ε Turbulence Model, Reynolds Stress Model (RSM), Velocity Distribution, Secondary Currents. F. Othman, E. M. Valentine: NUMERICKÉ MODELOVANIE ROZDELENIA RÝCHLOSTI V ZLOŽENOM KORYTE. Vodohosp. Čas., 54, 2006, 3; 25 lit., 7 obr. Štúdia pomocou numerického modelu NKF analyzuje ustálený rovnomerný prúd vody v koryte zloženom z kinety a dvoch symetrických beriem so zvislými stranami. Model využíva tri rovnice Naviera– Stokesa a nelineárny k-ε model turbulencie, ktorý simuluje Reynoldsove napätia, zodpovedné za druhotné prúdy. Tento model, založený na tzv. SIMPLE technike, počíta šesť parametrov U, V, W, P, k a ε pri použití stenových funkcií a karteziánskej siete. NKE model simuloval prúdenie, experimentálne pozorované na zariadení UK Flood Channel Facility ako séria č. 080301 (Shiono a Knight, 1989). Model pre Reynoldsove napätia (RSM) z balíka FLUENT bol tiež využitý na porovnanie. Výsledky ukázali, že modely NKE a RSM sú schopné predpovedať ako základné rýchlostné pole, tak aj vyvolané druhotné prúdenia. Aj keď zhoda s experimentom nie je v každom detaile úplná, hlavné znaky rýchlostného poľa sú zobrazené. Na simulácii možno vidieť zaoblenie rýchlostného poľa v rohoch dna hlavného kanála, sklon poľa v blízkosti hladiny smerom do stredu, ako aj pokles maxima rýchlosti pod voľnú hladinu. Tieto efekty sú v súlade s obrazom druhotných prúdov, ktoré sú tvorené hlavne dvoma vírmi – vírom kinety a vírom bermy. Tieto víry, vznikajúce pri spojení kinety s bermou, môžu byť reprodukované modelmi NKE a RSM. KĽÚČOVÉ SLOVÁ: zložený profil, rovnice Naviera–Stokesa, nelineárny k-ε model turbulencie, model Reynoldsových napätí (RSM), rozdelenie rýchlostí, druhotné prúdy.

Introduction Many natural rivers have cross-sections which can be characterised as having a compound section

that consists of a deeper, narrower main channel and adjacent shallow flood plains. The river flow is contained within the main channel sections most of the time. However, during flood events the river 269

F. Othman, E.M. Valentine

spills over onto the flood plains, resulting in a compound channel flow. The extent of danger caused by floods to both life and property depends on the stage reached for a given flow rate. Some of the main hydraulic features of compound channel flows are shown in Fig. 1 for a symmetric compound channel. It can be seen from the figure that there is momentum transfer between the main channel and the flood plain. This figure also shows the circulation of secondary currents. The limitations of the availability of good quality data, and the difficulty in obtaining the measurements of secondary currents, have prompted the computational studies of flow characteristics in compound open channels. However, the complex cross-section and heterogeneous nature of the boundary roughness, on top of other parameters, make the modelling of such flows particularly difficult and challenging.

The concept of apparent shear force has been used extensively in one dimensional flood routing and estuarial modelling. Two-dimensional numerical models based on depth-averaged parameters have also been developed to give the lateral distributions of both velocity and boundary shear stress (Keller and Rodi, 1988; and Wormleaton, 1988). Shiono and Knight (1988) employed the analytical solution to obtain the lateral variation of depthaveraged velocity and boundary shear stress. However, the lateral momentum transfer and the secondary circulation in these compound channels mean that the flows are highly three-dimensional in structure. Three-dimensional modelling for predicting flow characteristics (such as primary mean velocity, turbulent parameters, and secondary currents) in

Fig. 1. Hydraulic aspects of overbank flow (after Shiono and Knight, 1991). Obr.1 Hydraulická schéma prúdenia v zloženom profile koryta (podľa Shiono, Knight,1991).

compound channels has also been attempted using the k-ε and the algebraic stress model. To calculate the turbulent stresses, Krishnappan and Lau (1986) employed the algebraic stress relation of Thatchell (1975), while Kawahara and Tamai (1988) used the algebraic stress model of Launder and Ying (1973). More recently, Naot et al. (1993a,b) combined the 270

energy dissipation model with the algebraic stress model of Naot and Rodi (1982) to represent the turbulence. The non-linear k-ε model of Speziale (1987) was employed by Pezzinga (1994) and Lin and Shiono (1995). A full Reynolds stress transport model of turbulence has also been applied for computing flows in compound channels (Cokljat and

Numerical modelling of the velocity distribution in a compound channel

Younis, 1995). Predictions of the model were compared with some available experimental data and have shown satisfactory agreement. However, difficulties in producing the whole flow field especially near the junction region of the main channel and flood plain (M/F) and free surface have been reported (Kawahara and Tamai, 1989; Shiono and Lin, 1992; Pezzinga, 1994; Lin and Shiono, 1995). Three-dimensional modelling provides a promising and reliable means of simulating the flow in compound channels. There exist a number of numerical models for predicting turbulence characteristics in open channels. Some have been used by researchers mentioned above. The difficulties in producing the exact flow field near the junction of the main channel and flood plain and free surface due to the underestimation of the secondary flow vector have been demonstrated. This study used a different non-linear k-ε model (Baker and Orzechowski, 1983), along with the continuity and

momentum equations, which will be referred to as the NKE model. It has been reported by Pender and Manson (1994) that this version of the non-linear k-ε model is numerically robust. Governing equations Three-dimensional, steady, incompressible, turbulent flow in compound channels can be described by the Reynolds-averaged Navier-Stokes equations. In deriving the governing equations, a uniform flow in the streamwise x-direction is assumed. Therefore, all x-gradient terms are neglected. The continuity and momentum equations for steady uniform turbulent flow are written in terms of mean velocity and assume the form: Continuity equation:

∂V ∂W + =0. ∂y ∂z

(1.1)

Momentum equations:

V

∂U ∂U ∂ 2U ∂ 2U ∂u ' v ' ∂u ' w ' +W = gSo + ν +ν − − ∂y ∂z ∂y ∂z ∂y 2 ∂z 2

(1.2a)

V

∂V ∂V 1 ∂P ∂ 2V ∂ 2V ∂ v ' v ' ∂ v ' w ' +W =− +ν +ν − − ∂y ∂z ∂y ∂z ρ ∂y ∂y 2 ∂z 2

(1.2b)

V

∂W ∂W 1 ∂P ∂ 2W ∂ 2W ∂ v ' w ' ∂ w ' w ' +W =− +ν +ν − − . ∂y ∂z ∂y ∂z ρ ∂z ∂y 2 ∂z 2

(1.2c)

In the above equations, U, V, and W represent the velocity in the x, y, and z directions respectively. Note that the momentum equations are the simplified version of the Reynolds equations. It is also assumed that events at a particular cross-section do not propagate upstream of it to preserve the parabolic nature of the governing equations. This would enable a marching integration from upstream to downstream to be employed. Before Eq. (1.1) to (1.2c) can be solved, a turbulence model must be introduced for determining the Reynolds stresses u’u’, u’v’, u’w’, v’v’, v’w’, w’w’ appearing in the momentum equations. The choice of turbulence models to characterize the turbulence always posed a problem in solving the above equations. Turbulent flows are three-dimensional and time-dependent. The standard k-ε model is one of the most widely used and validated turbulence models. However, this model is not capable of rep-

resenting turbulent secondary flow due to the isotropic eddy viscosity assumptions. To overcome these defects, the non-linear k-ε has been developed. In this model, a set of extra terms is included in the Reynolds stresses to account for the nonlinearity of the turbulence. This model is characterized by an expression of the Reynolds stresses that are quadratic in the mean velocity gradients, which allows reproduction of the secondary currents due to turbulence anisotropy. The turbulent stresses are modelled using the non-linear k-ε model of Baker and Orzechowski (1983), which has been modified to make it applicable to compound open channel flows. The stresses can be expressed in the following relationships:

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F. Othman, E.M. Valentine

k2 ⎛⎛ ∂U ⎞ ⎛ ∂U ⎞ −u ' u ' − c1k + c2c4 ⎜ ⎜ ⎟ + ε ⎜ ⎝ ∂ y ⎠ ⎜⎝ ∂ z ⎟⎠ ⎝

2⎞

2

⎟ ⎟ ⎠

v' w' = c4

(1.3a)

k 2 ⎛ ∂V ∂W ⎞ k ⎜ + ⎟ + c2c4 ε ⎝∂z ∂y ⎠ ε

⎛ ∂U ⎞ ⎛ ∂U ⎞ ⎜ ⎟⎜ ⎟ . ⎝ ∂ y ⎠⎝ ∂ z ⎠

(1.3f)

The eddy viscosity (νt) assumes the form

k2 ⎛ ∂U ⎞ −u ' v ' = c4 ⎜ ⎟ ε ⎝∂y⎠

(1.3b)

k2 ⎛ ∂U ⎞ ⎜ ⎟ ε ⎝ ∂z ⎠

(1.3c)

The turbulent energy (k) and dissipation rate (ε) are computed using the following equations

(1.3d)

V

−u ' w ' = c4

k2 ⎛ ∂V ⎞ k2 ⎛ ∂U ⎞ + v ' v ' = −c3k + 2c4 ⎜ c c ⎟ 2 4 ⎜ ⎟ ε ⎝∂ y⎠ ε ⎝∂y⎠ w' w' = −c3k + 2c4

V

2

k2

ν t = Cµ . ε

(1.4)

∂k ∂k ∂ ⎛ ν ∂k ⎞ ∂ ⎛ ν ∂k ⎞ +W = ⎜ t ⎟ + ⎜ t ⎟ + P − ε , ∂y ∂z ∂y ⎝ σk ∂y ⎠ ∂z ⎝ σk ∂z ⎠

(1.5)

2

k2 ⎛ ∂W ⎞ k2 ⎛ ∂U ⎞ + c c ⎜ ⎟ 24 ⎜ ⎟ ε ⎝ ∂z ⎠ ε ⎝ ∂z ⎠

(1.3e)

ε ε2 ∂ε ∂ε ∂ ⎛ ν t ∂ε ⎞ ∂ ⎛ ν t ∂ε ⎞ C P C +W = ⎜ + + − , ⎟ ⎜ ⎟ 1ε 2ε k k ∂y ∂z ∂y ⎝ σ ε ∂y ⎠ ∂z ⎝ σ ε ∂z ⎠

(1.6)

where the Production (P) is defined in the following form P = u 'i u ' j

∂U i , ∂x j

which can be written in the x, y, and z directions as follows

⎛ ∂U ∂U ∂V ∂V ∂W ∂W ⎞ P = −⎜ u 'v ' + u ' w' + v 'v ' + v 'w' + w'v ' + w' w' ⎟. ∂y ∂z ∂y ∂z ∂y ∂z ⎠ ⎝

(1.7)

The values of the coefficients in the above equations are given below. These values are taken from Rodi (1980), and Baker and Orzechowski (1983). c1 0.94

c2 0.67

c3 0.56

c4 0.09

Cµ 0.09

Boundary conditions

The boundary conditions include the prevailing conditions at the solid/rigid wall, centerline and free surface. The conditions at the solid boundaries are specified using the wall function technique proposed by Launder & Spalding (1974). According to this technique, the conditions are specified at a grid point which lies outside the laminar sublayer. It is assumed that the shear stress and the velocity at this grid satisfy the logarithmic portion of the universal law of the wall given by U 1 ⎛ U* y ⎞ , (1.8) = ln E U* κ ⎜⎝ ν ⎟⎠ 272

C1ε 1.44

C2ε 1.92

σk 1.0

σε 1.3

where U – resultant velocity along the boundaries, U* – friction velocity, κ – von Karman constant (= 0.42), y – distance from the wall, E – parameter representing wall roughness. Introducing the dimensionless variables y y + = w U* , the above equation can be written as

ν

( )

U 1 = ln Ey + . U* κ U* is considered to be variable along the crosssection since the value of shear stress varies across the wall. U* is also linked to the turbulent energy k

Numerical modelling of the velocity distribution in a compound channel

by Eq. (1.11). The procedure involve in determining U* is therefore iterative. The value of E depends on the roughness of the boundary. For hydraulically smooth boundaries, E ≅ 9.0, whereas in the more general case of a rough wall, E is evaluated using the following expressions;

E = exp

(κBs ) ⎛ k sU * ⎞ ⎜ ⎟ ⎝ ν ⎠

,

(1.9)

2 ⎛ ⎛ ⎛ ksU* ⎞⎞ ⎞⎟ ⎛ ksU* ⎞⎞ ⎛⎜ Bs = ⎜⎜5.50+ 2.50ln⎜ ⎟⎟⎟exp − 0.217⎜⎜ln⎜ ⎟⎟⎟ ⎝ ν ⎠⎠ ⎜⎝ ⎝ ⎝ ⎝ ν ⎠⎠ ⎟⎠ 2 ⎛ ⎛ ⎛ ⎛ ksU* ⎞⎞ ⎞⎟⎞⎟ ⎜ ⎜ + 8.5 1− exp − 0.217⎜⎜ln⎜ ⎟⎟ . ⎜ ⎜ ν ⎠⎟⎠ ⎟⎠⎟ ⎝ ⎝ ⎝ ⎝ ⎠

(1.10)

Normal velocity components at the solid boundaries are set to zero. To specify conditions for k and ε at the grid point near the wall, the hypothesis of equilibrium of production and dissipation (P = ε) is assumed. This yields the following expressions for k and ε;

U *2 Cµ

3/ 2

,

(1.13)

where kf is the kinetic energy at the free surface; and yf is the distance between nearest grid point and the free surface. Cf is an empirical constant, and the value used is 0.164. Numerical procedure

where ks is the equivalent sand roughness parameter of the wall, and Bs – the roughness parameter obtained by the curve fitted expression (Krishnappan and Lau, 1986);

kw =

⎛ kf ⎞ ⎟ Cf ⎜ ⎜ Cµ ⎟ ⎝ ⎠ εf = κ yf

,

(1.11) 3

34 U 3 Cµ k 2 εw = * = . κ yw κ yw

(1.12)

At the plane of symmetry, no cross flux or diffusion is assumed. The velocity component normal to the plane of symmetry is zero, while for all other variables the gradients normal to + the plane are set to zero. The boundary condition at the free surface are specified following the approach of Rastogi and Rodi (1978) which considers the free surface to act as a plane of symmetry for all variables except ε. At the free surface ε is calculated using the expression of Krishnappan (1984);

The mean flow and turbulence model equations were solved using the 3-dimensional finite volume method. The method is parabolic in the streamwise direction but elliptic in the cross-stream planes. The numerical scheme proposed by Patankar and Spalding (1972) for three-dimensional parabolic flow is applied to solve the equations simultaneously. Calculations are carried out on a nonuniform and staggered grid. The power-law scheme is used for the formulation of the convectivediffusive terms, the Semi-Implicit Method for Pressure-Link Equations (SIMPLE) method of Patankar and Spalding is used to solve the partial equations, and the Triadiagonal Matrix Algorithm (TDMA) is used for solving the resulting algebraic equations. Since flows in open channels are driven by gravity, the pressure gradient can be represented by the term gSo. Starting with an assumed pressure field (except for the streamwise direction where the pressure gradient is known), the velocities were solved using the momentum equations. The SIMPLE algorithm was applied for the pressure correction at the cross section. The Reynolds stresses were determined and finally the transport equations (k and ε) were solved. The entire procedure is repeated at the same streamwise location until a fully converged solution is obtained. In the shallow water theory, where the hydrostatic pressure distribution can be assumed for all direction, this results in velocity–depth coupled equations. However, by using the control volume method to discretise the equations, the velocity field has been found to be proportional to the square of depth field and most of the coefficients will be dependent on both depth and velocity (Jian Guo Zho, 1995). Consequently the SIMPLE algorithm cannot be used to solve the equations. Modification of the SIMPLE algorithm is therefore needed before the equations can be solved.

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Due to the highly non-linear systems of the equations, there is a possibility of more than one ‘close’ solution. However, it is not being considered here as it may be dependent on the use of different methods and approaches in solving the equations. Model application

The model was applied to simulate the experimental runs performed at the UK Engineering and Physical Sciences Research Council Flood Channel Facility (EPSRC FCF) at Hydraulics Research Ltd., Wallingford, UK. The EPSRC Flood Channel Facility is a large laboratory flume 56m long and 10m wide. Compound channels with semi base width, b = 0.75m, and height of the flood plain above the main channel bed, h = 0.15m were laid on a constant bed slope of 1.027x10-3 m m-1. The FCF run 080301 (Shiono and Knight, 1989) was used to validate the primary mean velocity and secondary flows obtained from the model. This run was carried out in a symmetric channel with smooth walls. In all numeric runs, the bed slope is specified equal to the measured one, while the prescribed flow depths follow very closely the measured depths. The flow field is solved and then the discharge is deduced by integration. Starting with certain arbitrary assumptions for U, V, W, P, k, and ε, the solution is found with an iterative procedure until convergence criteria are satisfied. Due to the symmetric condition, the calculation is carried out in only half of the channel crosssection, which greatly reduces the computational effort. A staggered and non-uniform grid is employed. The grid is composed of meshes with variable size both in y (vertical) and z (lateral) directions. Near the wall, the mesh size is smaller, to accommodate the higher velocity gradient. The grid size increases further from the wall. In addition to the numerical model, the RSM model FLUENT (a computational fluid dynamic software package) was also used to simulate the experimental data. In employing the RSM model FLUENT, some modifications to the boundary have been made. FLUENT models the free surface approximately by assuming it to act as a plane of symmetry. This is not appropriate for a free surface flow. At the free surface, the dissipation increases with a resultant decrease of the eddy viscosity. Thus, the boundary condition of epsilon at the free surface has to be changed. This is done by inserting a small routine into FLUENT via the User Defined Subroutine option. 274

Primary velocities and secondary currents

The longitudinal velocity is normalised by the computed mean velocity (U/Uavg) to obtain dimensionless isovels. A comparison between the measured primary mean velocity and the corresponding isovels computed by the numerical models (NKE and RSM) for FCF run 080301 is shown in Figs 2 to 4. Fig. 2 shows the isovels obtained experimentally, while Figs 3 and 4 illustrate the computed isovels by the NKE model, and RSM of FLUENT respectively. It can be seen from the figures that the agreement between the measured isovels with NKE and RSM models is acceptable. The bulging of the contours at the corner and the inclination of the contours near the free surface towards the channel centre can be seen clearly, although it is predicted to a lower degree by the RSM model. The velocity in this region is decelerated due to the transport of lowmomentum fluid by the secondary currents. The effects of secondary currents in suppressing the maximum velocity below the free surface, called the velocity-dip phenomenon, can be produced very well by the NKE and RSM models, as shown in Fig 3 and 4. These are similar to the findings of Tominaga et. al (1989a), Tominaga and Nezu (1991), and Nezu and Naot (1995). It appears that the NKE model overestimates the secondary currents, leading to excess bulging of the contours towards the channel centre and depressing the maximum velocity much below the free surface. Furthermore, the maximum isovel predicted by the NKE and RSM models is less than the one obtained experimentally. The maximum isovels predicted are 1.25 and 1.22 by the NKE and RSM models respectively compared to 1.5 experimentally. Results obtained by other models have also reported difficulties in reproducing well the whole flow field (Larsson, 1989; Naot et al, 1993a; Pezzinga, 1994). Secondary currents

Figs 5 – 7 show the vector description of the secondary currents. The measured secondary currents are illustrated in Fig. 5, while those computed by the NKE and RSM models are shown in Figs 6 and 7 respectively. It can be seen that both the NKE and RSM models can produce the general pattern of the secondary flow in the main channel reasonably well, although some differences can be seen on the flood plain.

Numerical modelling of the velocity distribution in a compound channel

Depth (m)

Lateral Distance (m)

Fig. 2. Isovels of mean velocity (after Shiono and Knight, 1989). Obr. 2 Izočiary zostrednenej rýchlosti (podľa Shiono, Knight, 1989).

U/Uavg

Depth (m)

Lateral Distance (m)

Fig. 3. Isovels of the computed mean velocity (NKE model). Obr. 3 Izočiary zostrednenej rýchlosti počítanej podľa NKE modelu.

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U/Uavg

Depth (m)

Lateral distance (m) Fig. 4. Isovels of the computed mean velocity (RSM of FLUENT). Obr.4 Izočiary zostrednenej rýchlosti počítanej podľa modelu RSM (FLUENT).

The distinctive feature is the strong inclined secondary currents generated from the junction edge towards the free surface. This secondary circulation is formed mainly by two vortices that originate near the corner between the main channel and flood plain. One of the vortices is directed towards the centre of the main channel, namely the main chan-

nel vortex, while the other is directed towards the flood plain, namely the flood plain vortex. Both of the vortices are strong enough to reach the free surface. The computed main channel vortex extends towards the channel centre, which causes the velocity-dip phenomenon.

Depth (m)

Lateral distance (m)

Fig. 5. Secondary flow vectors (after Shiono and Knight, 1989). Obr.5 Vektory druhotných prúdov (vírov) podľa Shiono, Knight, 1989.

276

Numerical modelling of the velocity distribution in a compound channel

Depth (m)

Lateral distance (m)

Fig. 6. Secondary flow vectors (NKE model). Obr.6 Vektory druhotných prúdov podľa NKE modelu.

Depth (m)

Lateral distance (m)

Fig. 7. Secondary flow vectors (RSM of FLUENT). Obr.7 Vektory druhotných prúdov podľa modelu RSM (FLUENT).

Conclusions

Turbulent flows in open channels are very complicated in nature. The limitation of the data available and the difficulty in obtaining some of the measurements have prompted the numerical studies of flow in compound channels. This paper has employed the numerical computation in order to understand the behaviour of turbulent flows in compound open channels. In this study, the non-linear k-ε model of Baker and Orzechowski (1983) has been chosen to model the turbulence. In addition, the suitability of FLUENT, one of the most popular

Computational Fluid Dynamic packages, to model free surface flows has also been studied. It can be deduced from this study that secondary currents do have an influence on the primary mean velocity. The bulging of the contours at the corner and the inclination of the contour near the free surface towards the channel centre demonstrate the effects of secondary currents on the primary velocity. The effect of secondary currents is further demonstrated by the depression of maximum velocity below the free surface. These phenomena have been observed experimentally by Tominaga and Nezu (1991) and Nezu and Naot (1995). 277

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The NKE and RSM model can reasonably predict the primary mean velocity qualitatively. Difficulties for the model to predict the average velocities have been encountered. The average velocity obtained by the NKE model is 0.486 m s-1, while RSM model gives a value of 0.633 m s-1. The actual value is 0.619 m s-1. The NKE model seems to underestimate the velocities whereas the RSM model overestimates it. The general pattern of the secondary currents can be reproduced fairly well by both the NKE and RSM model, especially in the main channel. However, the NKE model overestimates the secondary flows at the corner of the main channel-flood plain region towards the free surface which lead to an excess bulging of the contours towards the free surface and the depression of the maximum velocity well below the free surface. The RSM model, on the other hand, produced this effect to a lesser extent. Difficulties with the model in reproducing exactly the shear stress at the bottom and corner banks may contribute to the discrepancies of the results. Further refinement to the numerical model may be required to improve the results. Acknowledgements. The authors acknowledge Prof. D.W. Knight from the University of Birmingham for providing the EPSRC FCF data, Dr. I. Potts from the Department of Mechanical Engineering, Dr. D.C. Swailes from the Department of Engineering Mathematics, University of Newcastle Upon Tyne, UK, and Assoc. Prof. G. Pezzinga of Catania University, Italy, for their help and invaluable advice and comments throughout the study. List of symbols b c1, c2, c4,cµ CD, Cµ, Cµ‘ E f h H k ks NKE P R Re RSM SIMPLE

– – – – – – – – – – – – – – –

TDMA U, V, W

– –

u, v, w

–

278

semi width of the main channel, constant in turbulent model, constant in turbulent model, roughness parameter, friction factor, height of flood plain above main channel bed, flow depth of channel, turbulent kinetic energy, Nikuradse equivalent roughness, non-linear k–ε, stress production of k, pressure, hydraulic radius, Reynolds number, Reynolds Stress Model, Semi-Implicit Method for Pressure-Link equations, Triadiagonal Matrix Algorithm, mean velocity in the x, y, and z directions respectively, instantaneous velocity in the x, y, z directions respectively,

u' , v' , w ' – U* y ε ρ µ ν κ

– – – – – – –

turbulent fluctuation of the velocity in the x, y, z directions respectively, resultant friction velocity, distance from the wall, dissipation rate of k, fluid density, dynamic viscosity, kinematic viscosity, von Karman constant.

REFERENCES BAKER A.J. and ORZECHOWSKI J.A., 1983: An Interaction Algorithm for Three-Dimensional Turbulent Subsonic Aerodynamic Junction Region Flow. AIAA J. Vol. 21, 4, 524. COKLJAT D. and YOUNIS B.A., 1995: Second Order Closure Study of Open Channel Flows. J. Hydraulic Engng, Vol. 12, 2, 94. KAWAHARA Y. and TAMAI N., 1988: Numerical calculation of turbulent flows in compound channel with an algebraic stress turbulence model. 3rd International Symposium on Refined Flow Modelling and Turbulence Measurements, Tokyo, Japan. KELLER R.J. and RODI W., 1988: Prediction of Flow Characteristics in Main Channel/Flood Plain Flows. J. Hydraulic Research, Vol. 26, No. 4, 425. KRISHNAPPAN B.G., 1984: Laboratory Verification of Turbulent Flow Model. J. Hydraulic Engng, Vol. 110, No. 4, 500. KRISHNAPPAN B.G. and LAU Y.L., 1986: Turbulence Modelling of Flood Plain Flows. J. Hydraulic Engng, ASCE, Vol. 112, No. 4, 251. LARSSON R., 1988: Numerical simulation of flow in compound channels. 3rd International Symposium on Refined Flow Modelling and Turbulence Measurements. Tokyo, Japan, 633. LAUNDER, B.E. and YING Y.L., 1973: Prediction of flow and heat transfer in ducts of square cross section. Proceedings of Inst. Mechanical Engineers, Vol. 187, 455. LIN, B. and SHIONO K., 1995: Numerical modelling of solute transport in compound channel flows. J. Hydraulic Res., Vol. 33, No. 6, 773. NAOT D., NEZU I. et al., 1993a: Hydrodynamic behaviour of compound rectangular open channels. J. Hydraulic Engng, ASCE, Vol. 119, 3, 390. NAOT D., NEZU I. et al., 1993b: Calculation of compound open channel flow. J. Hydraulic Engng, ASCE, Vol. 119, 12, 1418. NAOT D. and RODI W., 1982: Calculation of secondary currents in channel flow. J. Hydraulic Div., ASCE, Vol. 108, 948. NEZU I. and NAOT D., 1995: Turbulence structure and secondary currents in compound open channel flows with variable depth flood plains. Proc. 10th Symposium on Turbulent Shear Flow, Pennsylvania. PATANKAR S.V. and SPALDING D.B., 1972: A Calculation Procedure for Heat, Mass and Momentum Transfer in Three Dimensional Parabolic Flows. Int. J. Heat and Mass Transfer, Vol. 15, 1787. PENDER G. and MANSON J.R., 1994: Developments in Three Dimensional Numerical Modelling of River Flows. Proceedings of the Second International Conference on River and Fluid Hydraulics, York, England, John Wiley.

Numerical modelling of the velocity distribution in a compound channel PEZZINGA G., 1994: Velocity distribution in compound channels. J. Hydraulic Engng, ASCE, Vol. 120, 1176. RASTOGI and RODI W., 1978: Predictions of heat and mass transfer in open channels. J. Hydraulic Div., ASCE, Vol. 104, 397. SHIONO K. and KNIGHT D.W., 1989: Transverse and vertical measurements of Reynolds stress in a shear region of a compound channel. Proceeding of 7th International Symposium on Turbulent Shear Flows., Stanford, USA. SHIONO K. and KNIGHT D.W., 1991: Turbulent openchannel flows with variable depth across the channel. J. Fluid Mechanics, Vol. 222, 617. SPEZIALE C.G., 1987: On Non-Linear k-l and k-ε models of turbulence. J. Fluid Mechanics, Vol. 178, 459. TOMINAGA A. and NEZU I., 1991: Turbulent structure in compound open channel flows. J. Hydraulic Engng, ASCE, Vol. 117, 1, 21. TOMINAGA A., NEZU, I., EZAKI H., NAKAGAWA, 1989a: Experimental study on secondary currents in compound open channel flows. Proc. of the 23rd IAHR Congress, Ottawa, Canada, A-15. TATCHELL D.G., 1975: Convection Processes in Confined Three-Dimensional Boundary Layers. PhD Thesis. Imperial College of London, England. WORMLEATON P.R., 1988: Determination of discharge in compound channels using the dynamic equation for lateral velocity distribution. International Conference on Fluvial Hydraulics, Budapest. ZHOU J.G., 1995: Velocity-Depth Coupling in Shallow-Water Flows. J. Hydraulic Engng., Vol. 121, 717. Received 9. November 2005 Scientific paper accepted 14. June 2006

NUMERICKÉ MODELOVANIE ROZDELENIA RÝCHLOSTI V ZLOŽENOM KORYTE F. Othman, E.M. Valentine Turbulentné prúdenia v otvorených korytách sú veľmi komplikované. Nedostatok vhodných pozorovaných údajov a ťažkostí pri ich získavaní provokujú využívanie numerických štúdií prúdenia v zložených korytách. Táto štúdia využíva numerické výpočty k porozumeniu správania sa turbulentných prúdov v takýchto korytách. V štúdii bol využitý nelineárny k-ε model Bakera a Orzechovského (1983) na modelovanie turbulencie. Navyše sme tiež študovali vhodnosť FLUENTu, jedného z najobľúbenejších balíkov výpočtovej hydrodynamiky. Zo štúdie možno usúdiť, že druhotné prúdy ovplyvňujú rozloženie hlavného rýchlostného poľa. Tento efekt je demonštrovaný zaoblením obrazu rýchlosti v rohoch

koryta a jej priebehom v blízkosti hladiny. Vplyv druhotných prúdov je ukázaný aj poklesom maxima rýchlosti pod voľnú hladinu. Tieto efekty boli pozorované experimentálne v prácach Tominaga a Nezu (1991) a Nezu a Naot (1995). Modely NKE a RSM môžu kvantitatívne simulovať aj hlavné rýchlostné pole. Ťažkosti, ktoré sa pri tom vyskytujú, boli zhodnotené. Priemerná rýchlosť podľa NKE modelu bola 0,486 m s-1, kým podľa RSM modelu 0,633 m s-1. Aktuálna experimentálna hodnota pri tom bola 0,619 m s-1. Zdá sa teda, že NKE ju podhodnocuje a RSM nadhodnocuje. Všeobecný obraz druhotných prúdov môže byť reprodukovaný pomerne dobre oboma modelmi, najmä pre hlavný prúd. Avšak NKE model nadhodnocuje aj druhotné prúdy v rohoch koryta, čo vedie k prehnane vypuklým obrazom rýchlosti smerom k voľnej hladine a poklesu maxima pod hladinu. RSM model tieto efekty zvýrazňuje v menšom rozsahu. Ťažkosti s presnejšou simuláciou šmykových napätí na dne a v rohoch prispievajú k rozpornostiam vo výsledkoch. Zlepšenie výsledkov môže priniesť iba zlepšenie samotného matematického modelu turbulencie. Zoznam symbolov b c1, c2, c3, c4 CD, Cµ, Cµ´ E f h H k ks NKE P R Re RSM SIMPLE TDMA U, V, W u, v, w u´, v´, w´ U* y ε ρ µ ν κ

– polovičná šírka hlavného koryta, – konštanty turbulentého modelu, – konštanty turbulentého modelu, – parameter drsnosti, – koeficient trenia, – hĺbka vody v berme, – hĺbka vody nad dnom kinety, – turbulentná kinetická energia, – Nikuradzeho náhradná drsnosť, – nelineárne k-ε. – produkcia napätia k, tlak, – hydraulický polomer, – Reynoldsovo číslo, – Reynoldsov tlakový model, – poloimplicitná metóda pre tlakové rovnice, – trojdiagonálny maticový algoritmus, – priemerná rýchlosť v smeroch x, y, z, – aktuálna rýchlosťv smeroch x, y, z, – fluktuácie rýchlosti v smeroch x, y, z, – rýchlosť trenia, – vzdialenosť od steny, – rýchlosť disipácie k, – hustota kvapaliny, – dynamická viskozita, – kinematická viskozita, – Karmanova konštanta.

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