1991), and measurements from river Severn at. Montford, UK (Knight et al 1989). 3.1 Laboratory data. The SERC Flood Channel Facility is a laboratory flume 56 ...
Performance of lateral velocity distribution models for compound channel sections J.F. Weber Universidad Nacional de Córdoba and Universidad Tecnológica Nacional, Córdoba, Argentina.
A.N. Menéndez INA (National Institute for Water) and Universidad de Buenos Aires, Buenos Aires, Argentina
ABSTRACT: In this paper, the scope of 2D-Horizontal and 1D-Lateral models for lateral velocity distribution is addressed, and their relative and absolute performances are tested by making comparisons between their predictions, on the one side, and experimental and field velocity data, on the other side. The models considered are: the Divided Channel Method (DCM), the Lateral Distribution Method (LDM) and a 2D Horizontal hydrodynamic finite-element model (RMA2). 1 INTRODUCTION Hydraulic engineering studies related to floods usually require the lateral distribution of flow velocity across the compound channel section, constituted by the main channel and the floodplain. 2D-Horizontal numerical models (such as the public domain software RMA2 of USACE) are now broadly accepted as an appropriate theoretical model to solve this problem. However, as in the majority of applications the longitudinal scale of flow variation is much larger than the lateral scale of variation (given by the flow width), the longitudinal and lateral flow variations can be accounted for separately (Menéndez 2003); the first one, through well-established 1DLongitudinal models based on de Saint Venant equations (Cunge et al. 1980), while the second one, using 1D-Lateral models. 1D-Lateral models, which solve the lateral distribution of the longitudinal depth-averaged flow velocity, have become the subject of analysis and application in recent years. They run between simple and relatively old empirical and heuristic formulations - such as the methods of Lotter (1933), or Divided Channel Method (DCM, used in software HEC-RAS, 2001), Horton (1933), and Pavlovski (1931) - to physically-based equations, like the Lateral Distribution Method (LDM) proposed by Wark et al. (1990). 2 MODEL DESCRIPTIONS In the following paragraphs, the three models used in the present work are briefly described: the DCM; the analytical solution proposed by Shiono & Knight
(1988,1991) to the LDM, and the RMA2-WES 2DHorizontal hydrodynamic finite-element model (Donnell et al. 2001). 2.1 DCM Model Lotter (1933), and later Einstein & Banks (1950) divided the cross section into subsections, and assumed that the energy grade slope was the same for any subsection, and that the interfaces between them behaved as impermeable boundaries, i.e., that there was no diffusion of lateral momentum (leaving the friction as the only energy loss), to calculate the partial discharge Qi for any such subsection:
Qi = K i S 0
1
2
(1)
where S0 = longitudinal slope; and Ki = hydraulic conveyance, given by
AR Ki = i i ni
2
3
(2)
where Ai, = flow area, Ri = hydraulic radius, and ni = Manning’s roughness coefficient for subsection i. The lateral velocity distribution, Vi, can then be obtained as
Vi =
Qi Ai
(3)
This is the method implemented in software HEC-RAS (HEC, 2001) as Flow Distribution Option.
2.2 LDM Model The LDM (Wark et al. 1988) is based on the steadystate continuity and momentum depth-averaged equations of motion. Combining both equations, and neglecting longitudinal derivatives, the following expression is obtained
ρgS 0Y −
1 d ⎡ f 2 ∂V ⎤ f ρV 2 1 + 2 + ⎢ ρλ Y V ⎥ = 0 (4) 8 8 dy ⎣ ∂y ⎦ s
where: ρ = density of water; g = gravitational acceleration; Y = local flow depth; f = Darcy’s friction coefficient; V = depth-mean velocity; s = lateral bed slope; y = lateral coordinate; and λ = nondimensional Boussinesq eddy viscosity. Equation 4 takes into account the effects of friction and diffusion as dissipative terms. The source term represents the effect of water weight. Shiono & Knight (1988) obtained the analytical solutions of Equation 4 across subsections where the depth (Y) is constant or a linear function of the lateral coordinate. These solutions are given in equations 5 (for constant depth) and 6 (for linear depth). In these expressions, η, ψ, and ω are constants given by equations 7 - 9. Coefficients a and b in equations 5 - 6 must be calculated solving a linear system of equations. This system is obtained by imposing the matching conditions expressed in equations 10 -11 at the interface between subsections. For a section composed by N subsections, 2N-2 equations result from applying 10 - 11. Although Abril (2003) showed that the boundary condition given by Equation 11 is not strictly true, it will be consider as correct for the present work.
1 s 1+ s2 ψ= 1+ 2 λ
8f −
1 2
gS 0
ω=
1+ s2 f λ − s 8 s2
V1 = V2 n
f 8
n
∂V1 ∂V = 2 ∂y ∂y n
n
(11)
The remaining two equations arise from the boundary conditions at the two extremes. For example, in the case of a cross section as shown in Figure 1, the two boundary conditions correspond to the noslip condition at point p and zero derivative in point m (symmetric section). 2.3 2D model The RMA2 model (Donnell et al. 2001) is a 2D depth-averaged finite-element hydrodynamic numerical model. It was originally developed by Norton, King and Orlob (1973) and actually supported by WES-USACE. It solves the shallow water equations, i.e. (neglecting wind shear stress and Coriolis acceleration):
⎛ ∂ 2u ∂ 2 u ⎞ ∂u ∂u ∂u Y + Yu + Yv − Y ε ⎜ 2 + 2 ⎟ + ∂t ∂x ∂y ⎝ ∂x ∂y ⎠
Y
⎛ ∂ 2v ∂ 2v ⎞ ∂v ∂v ∂v + Yu + Yv − Y ε ⎜ 2 + 2 ⎟ + ∂t ∂x ∂y ⎝ ∂x ∂y ⎠
⎛ ∂z ∂Y ⎞ gvn 2 2 2 + gY ⎜ b + ⎟ + 13 u + v = 0 ∂ ∂ y y ⎝ ⎠ Y
V( y ) = a1eηy + b1e −ηy +
8 gS 0Y f
V( y ) = a2Y ψ + b2Y −ψ −1 + ωY 24 f 1 η= λ 8Y
(9)
(10)
2 ⎛ ∂z ∂Y ⎞ gun + gY ⎜ b + + u 2 + v2 = 0 ⎟ 1 ⎝ ∂x ∂x ⎠ Y 3
⎛ ∂u ∂v ⎞ ∂Y ∂Y ∂Y + Y ⎜⎜ + ⎟⎟ + u +v =0 ∂t ∂x ∂y ⎝ ∂x ∂y ⎠
Figure 1. Scheme of a simplified compound channel section
(8)
(12)
(13)
(14)
(5)
where u and v are the x and y components of the velocity vector, respectively, and ε is the horizontal eddy-viscosity, calculated as:
(6)
ε = λYV* = λ gS0 Y
(7)
The numerical solution of equations 12 – 14 is implemented through the finite-element method. The program uses both quadrilateral and triangular elements. The weighted-residuals Galerkin method is
3
2
(15)
used for spatial integration. An explicit finitedifference scheme is used for time integration. 3 EXPERIMENTAL DATA Two sources of experimental data were used for validation: those from the SERC Flood Channel Facility experimental flume (Shiono & Knight 1988, 1991), and measurements from river Severn at Montford, UK (Knight et al 1989). 3.1 Laboratory data The SERC Flood Channel Facility is a laboratory flume 56 m long and 10 m wide, with discharges up to a maximum of 1.1 m3/s. Velocities were measured by means of mini propeller current meters (Shiono & Knight 1991). For the present work, eight experiments, distributed in two groups according to floodplain width, are considered. Taking as a reference Figure 1, the total half-width B is 3.15 m for Group A and 1.65 m for Group B; the main channel half-width is the same for both groups, and equal to 0.75 m; the longitudinal slope is fixed in 0.001027; the main channel depth relative to floodplains (h) is 0.15 m. The total depth H and the discharge Q vary for each case, as shown in Tables 1 and 2. These tables also present the non-dimensional geometric parameter Dr, defined as:
Dr =
H −h H
Table 1. Hydraulic parameters for experimental data. Group A. Case 1 Case 2 Case 3 Case 4 Case 5 H (m) 0.1690 0.1780 0.1870 0.1980 0.2879 Q (m³/s) 0.2483 0.2821 0.3237 0.3830 1.1142 Dr 0.111 0.157 0.197 0.242 0.479 Table 2. Hydraulic parameters for experimental data. Group B. Case 6 Case 7 Case 8 H (m) 0.1667 0.1987 0.3000 Q (m³/s) 0.2421 0.3325 0.8349 Dr 0.100 0.245 0.500
The relation between parameters n and f is the following:
8 gn 2 3 Y
Field data used for the present work consists of three velocity profiles measured at a cross section in river Severn at Montford, UK (Knight et al. 1989). This is the longest river of England. Its mean discharge is 43 m3/s. The longitudinal slope at the surveyed location is 0.0002. Table 3 shows the discharge and depth associated to the three cases considered. Table 3. Hydraulic parameters for River Severn, UK. Discharge Depth m³/s m Case 1 330.8 7.81 Case 2 220.6 6.92 Case 3 188.8 6.15
Roughness coefficients were taken as reported in the original work (Knight el al. 1989). For the application of DCM and RMA2 models, the corresponding values of Manning’s n were obtained. Table 4 shows all these values. Table 4. Roughness coefficients for River Severn, UK. Manning’s coefficient n y Point m f Case 1 Case 2 Case 3 1 0.00 0.038 0.0212 2 8.00 0.038 0.0230 0.019 0.0193 3 71.00 0.5 0.1037 0.1002 0.0981 4 79.00 0.04 0.0318 0.0312 0.0308 5 96.00 0.5 0.1037 0.1002 0.0981 6 105.00 0.1 0.0373 0.0309 0.0312 7 128.00 0.1 0.0345
(16)
Manning’s roughness coefficient n was used instead of Darcy’s coefficient f to characterize channel roughness. Coefficient n was assumed constant and equal to 0.0104. It was verified (Weber 2003) that results are relatively insensitive to which of the coefficients is considered as a constant.
f =
3.2 Field data
(17)
4 MODELS APPLICATION Applications of the three models to the two sets of experimental data are described in the following. 4.1 DCM model The DCM model was applied through software HEC-RAS (HEC 2001). This soft, aimed at solving gradually varied flow with a 1D-Longitudinal model, has a feature called Flow Distribution, which calculates (using the DCM) the lateral velocity distribution in specified cross sections. Users can control the points where they want to obtain the results, through the specification of a ∆y value. For the application of HEC-RAS, a rectilinear prismatic channel, with the known flow depth imposed as downstream boundary condition, was considered. The lengths of the 1D models were 56 m for the laboratory flume experiments and 10,000 m for River Severn. The results for middle-located stations were used for comparison with experimental data, in order to avoid influence from the boundary conditions.
Case 1
Case 3 2.0 field data 1D - N=10 1D - N=15 1D - N=5 ground
1.5 V (m/s)
In order to investigate the influence of lateral discretization on the accuracy of the predictions, different discretization steps were tested. Figures 2-3 show the lateral velocity profiles predicted for two cases in the laboratory flume (Case 1 with H = 0.169 m and Case 5 with H = 0.288 m). Figures 4-5 are the velocity profiles computed for two cases in River Severn (Case 1 with H = 7.81 m and Case 3 with H = 6.15 m). It is observed that a finer discretization does not necessarily means higher precision for the DCM.
1.0 0.5
40
60
80 y (m)
100
120
Figure 5. Lateral velocity distribution for Case 3 of River Severn as predicted by DCM
1.4 n=5 1.2
n = 10
4.2 LDM model
n = 15
1.0
experimental
V/Vo
0.8 0.6 0.4 0.2 0.0 0.0
0.5
1.0
1.5 2.0 y (m)
2.5
3.0
3.5
Figure 2. Lateral velocity distribution for Case 1 of SERC channel as predicted by DCM
Case 5 1.6
U/Uo
1.2
V(δy ) = δV
0.8
n=5 n = 10 n = 15 experimental
0.4 0.0 0.0
0.5
1.5 2.0 2.5 3.0 3.5 y (m) Figure 3. Lateral velocity distribution for Case 5 of SERC channel as predicted by DCM
1.0
V (m/s)
Case C1 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 -
field data 1D - N=10 1D - N=15 1D - N=5 ground
0
20
40
The analytical solution of LDM was applied using a specially-developed software, named “Lateral” (Weber & Menéndez 2003). This soft proceeds through the following steps: - It calculates the characteristic constants η, ψ, and ω (Equations 7-9) for each subsection. - It calculates the coefficients matrix and the independent-term vector. - It solves the algebraic system of equations. - It calculates the mean-depth lateral profile of the longitudinal velocity for a specified ∆y. In case of natural streams with no vertical walls, the non-slip condition leads to the trivial solution, because Y = 0. Then, this condition was slightly modified to:
60
80
100
120
(17)
where δY = minimum imposed depth considered in calculations, and δV = minimum imposed velocity corresponding to depth δY. Although δY and δV appear as new parameters, experience shows that they have only a local influence. Values of 0,001 m and 0,1 m were used for δY in laboratory flume and in River Severn, respectively. The minimum velocity δV was taken as 0,1 m/s for both cases. Instead of working with V(y), function V2(y) was used, which allows to deal with a linear system of algebraic equations. Solution to this system were obtained through Gauss Elimination Method. Figures 6-9 show the results of the application of LDM to the same cases as Figures 2-5. These figures correspond to similar results reported in Shiono & Knight (1988) and Knight et al. (1989).
140
y (m)
Figure 4. Lateral velocity distribution for Case 1 of river Severn as predicted by DCM
4.3 2D model The 2D model RMA2 was used to predict the lateral velocity distribution for the same cases as the previous models. A prismatic geometry was developed using an ad-hoc software called GEO. It constructs a structured rectangular finite-element mesh, as shown in Figure 10.
1.6 Case 1
1.4
LDM Experimental ground
1.2 U/Uo
1.0 0.8 0.6 0.4
Figure 10. Finite-element mesh generated by software GEO
0.2 0.0 0
1
2 y (m)
3
4
Figure 6. Lateral velocity distribution for Case 1 of SERC channel as predicted by LDM 1.6
Case 5
LDM
V (m/s)
1.2
Experimental
0.8 0.4 0.0 0
0.5
1
1.5 2 y (m)
2.5
3
3.5
Figure 7. Lateral velocity distribution for Case 1 of SERC channel as predicted by LDM Case 1 river Severn 2.0
V (m/s)
1.0 0.5 0
50
100 y (m)
Figure 8. Lateral velocity distribution for Case 1 of River Severn as predicted by LDM Case 3 1.6
No. of elements 4998 3213
Group no. 1 Group no. 2
No. of nodes 15317 9932
The turbulence parameter E = ρε was correlated with parameter λ used in LDM model through Equation 15. For the laboratory cases, artificially high values of E were taken, in order to assure numerical stability. Values between 14.2 to 114.4 Pa.s were used for field data and 3 to 20 Pa.s for laboratory data. The known water level was imposed as the downstream boundary condition. Figures 11-14 show the lateral velocity distribution as calculated by RMA2. 1.0
field data LDM ground
1.2 0.8 0.4
Case 1
0.8 V (m/s)
V (m/s)
Table 5. Mesh dimensions for SERC channel
Table 6. Mesh dimensions for River Severn No. of elements No. of nodes Case 1 5550 17025 Case 2 5550 17025 Case 3 4500 13861
field data LDM ground
1.5
The length of the channel was taken as the minimum one necessary to achieve a stabilized solution. It resulted lengths of 10 m for the laboratory cases and 600 m for River Severn. The ∆x step was assumed fixed at 0.1 m for laboratory cases and 4 m for River Severn. The ∆y step was different for the main channel, the banks and the floodplains. For the laboratory cases, ∆y was 0.05 m for the main channel, 0.025 m for the banks and 0.1 m for the floodplains; whereas for River Severn cases, ∆y was taken as 1.50 m for the main channel, 2 m the for banks and 4 m for the floodplains. With these selections, the resulting mesh dimensions are as shown in Tables 5-6.
RMA2 Experimental ground
0.6 0.4 0.2
40
60
80
y (m)
100
120
140
Figure 9. Lateral velocity distribution for Case 3 of River Severn as predicted by LDM
0.0 0
1
2 y (m)
3
4
Figure 11. Lateral velocity distribution for Case 1 of SERC channel as predicted by 2D model
Another consequence of this lack of diffusion is that the lateral velocity distribution depends strongly on the discretization step. Specifically, it is better to minimize the number of subsections considered, as recommended in HEC-RAS user’s manual (HEC 2001), which constitutes a way of artificially smoothing out the profile. For laboratory cases, the DCM model tends to overestimate main channel velocities and underestimate floodplain velocities. This anomaly increases as floodplain flow depth increases (see Figures 2-3).
1.4
Experimental ground RMA2
Case 5 1.2
U/Uo
1.0 0.8 0.6 0.4 0.2 0.0 0
0.5
1
1.5
2
2.5
3
3.5
y (m)
Figure 12. Lateral velocity distribution for Case 5 of SERC channel as predicted by 2D model Case 1 2.0
field data ground RMA2
V (m/s)
1.5 1.0 0.5 0
50
100 y (m) Figure 13. Lateral velocity distribution for Case 1 of River Severn as predicted by 2D model Case 3 2.0
field data V (m/s)
1.5
ground
5.2 LDM model As was presented in early works (Shiono & Knight 1988, 1991), the LDM model adequately predicts the lateral velocity distribution, particularly for laboratory cases. Dispersion in the prediction of velocity profiles for natural rivers is a frequent feature, although there are some examples of very satisfactory applications of LDM model to natural streams (Weber & Tarrab 2003). Additionally, LDM model results an adequate predictor of stage-discharge relation, as can be seen in Figure 15, related to Group A of laboratory cases. Weber (in press) shows an example of application of the LDM model to predict the stage-discharge relation for a natural stream. Predictions are relatively insensitive to nondimensional eddy viscosity values for the floodplains over a relatively large range (Weber & Del Prete 2003). On the contrary, main channel λ values are decisive to perform a good prediction.
RMA2
1.0
5.3 2D model
0.5 0
50
y (m)
100
Figure 14. Lateral velocity distribution for Case 3 of River Severn as predicted by 2D model
5 MODELS PERFORMANCE
RMA2 model shows limitations to predict the lateral velocity distribution for laboratory cases, due to excessive artificial eddy viscosity necessary to guarantee numerical stability, thus producing excessively smoothed-out profiles. This effect is more significant as floodplain depth decreases (see Figures 1112). This is not the case for field data comparisons (River Severn). Instead, the limitations are associated to assuming constant E values. 0.32
Absolute and relative performances of the used models are discussed next.
Rating Curve - group A
5.1 DCM Model The Divided Channel Method main limitation is associated to the absence of lateral momentum diffusion. As a consequence, the lateral velocity profile tends to copy the channel bottom form (see Figures 4-5). This limitation is particularly significant for skewed V-shaped sections, where diffusion plays an important role.
H (m)
0.28
0.24
0.20
LDM Experimental
0.16 0.2
0.7
1.2 Q (m³/s)
Figure 15. Rating curve for SERC channel. Group A cases
In contrast to the LDM model, RMA2 model is not a handy tool to predict stage-discharge relations in compound channels, due to the relatively large amount of time-machine necessary to perform the calculations.
1.6
LDM Experimental ground DCM RMA2
1.2
U/Uo
1.0
5.4 Relative performance of models
0.8 0.6 0.4 0.2 0.0 0
0.5
1
1.5
2
2.5
3
3.5
y (m)
Figure 16. Comparison among DCM, LDM and 2D models predictions for Case 4 of SERC channel 1.4
Case 7
1.2 1.0 U/Uo
Figures 16 and 17 show the lateral velocity distribution predicted by the three models considered in this paper, together with the experimental data, for Cases 4 and 7, respectively, of SERC channel. For common parameters, the same or equivalent values are assumed. It can be clearly noted the absence of diffusive effects in the DCM prediction, and, on the contrary, the excess of diffusion showed by the 2D model. With the DCM model, the velocities in the floodplain are usually better represented than in the main channel, where the models tend to give a more uniform profile. The calculation gets poorer as the water depth in the floodplain increases, and its use is not recommended for relative depths over about 0.4. The LDM model has the possibility of considering the energy loss due to the vertical-axis eddies which develop in the shear layer between the main channel and the floodplain. Additionally, LDM model can include a sink term to account for the longitudinal-axis eddies, known as secondary currents, which take place at the toe of the main channel bank, acting as an apparent shear stress at the main channel-floodplain interface (Wormleaton & Merrett 1990) and, hence, producing an additional and significant momentum transfer from the main channel to the floodplain. As a consequence, the LDM can provide a much better agreement, if properly tuned. Regarding 2D-Horizontal models performance, it is shown that it is better for natural than for laboratory channels. In the last case, the difficulty lies in the fact that the artificial viscosity necessary to attain numerical stability (at least for RMA2) is usually higher than the physical eddy viscosity; hence, velocity gradients are smoothed out. The agreement of this solution improves for higher floodplain water heights, and gets acceptable for relative depths over about 0.4, thus complementing with the DCM model. On the other hand, for natural channels the agreement is relatively satisfactory for the whole range of water depths. In any case, 2D-Horizontal model could not match the LDM performance, and this is due to the fact that there is no formulation yet to account automatically for the effects of secondary currents at the toe of the main channel (RMA2 already includes an implementation of a parametric formulations to represent the effects of secondary currents in curved flow).
Case 4
1.4
LDM Experimental ground DCM RMA2
0.8 0.6 0.4 0.2 0.0 0
0.5
1 y (m)
1.5
2
Figure 17. Comparisons among DCM, LDM and 2D models predictions for Case 7 of SERC channel
6 CONCLUSIONS The main conclusions from the present work are the following: - The DCM model (through software HEC-RAS), in spite of its simplicity, gives a reasonable prediction of the lateral velocity distribution from the engineering point of view. - The main limitation of DCM model is associated to the absence of momentum lateral diffusion, which lowers precision specially in the calculated main channel velocity values. - The LDM model adequately represents both the lateral velocity distribution and stage-discharge relation. - 2D-Horizontal models performs better for natural than for laboratory channels. In the last case, the difficulty lies in the fact that the artificial viscosity necessary to attain numerical stability (at least for software RMA2) is usually much higher than the physical eddy viscosity; hence, velocity gradients are smoothed out - For laboratory cases, the calculation with DCM gets poorer as the water depth in the floodplain increases, and its use is not recommended for relative depths over about 0.4. On the other hand, RMA2 results improve for higher floodplain water heights,
and gets acceptable for relative depths over about 0.4, thus complementing with the DCM model. 7 ACKNOWLEDGMENTS Authors would like to thank to Prof. Koji Shiono for providing his experimental data set. First author also acknowledges Science & Technology Secretary, University of Cordoba (Argentina), for support through its High Education Scholarship Program. REFERENCES Abril, B. 2003. Benchmark comparisons of the analytical and finite element solutions of the SKM. Technical Report EPSRC Research Grant – GR/R54880/01. University of Birmingham, UK. Cunge, J., Holly, F. & Verwey, A. 1980. Practical Aspects of Computational River Hydraulics. Boston: Pitman. Donnell, B. P. (ed.) 1996. User’s Guide to RMA2 WES version 4.3, U.S. Army Corps of Engineers, Waterways Experiment Station, Hydraulic Laboratory. Einstein, H. A. & Banks, R. B. 1950. Fluid resistance of composite roughness. Transactions, American Geophysical Union 31(4): 603-610. HEC 2001. HEC – RAS Hydraulic Reference Manual, U.S. Army Corps of Engineers, Davis, CA. Horton, R. E. 1933. Separate roughness coefficients for channel bottom and sides. Engineering News Record 111(22): 652–653. Knight D. W., Shiono, K. & Pirt, J. 1989. Prediction of depth mean velocity and discharge in natural rivers with overbank flow. Proceedings of the International Conference on Hydraulic and Environmental Modellling of Coastal, Estuarine and River Waters: 419-428. Bradford. Lotter, G. K. 1933. Soobrazheniia k gidravlicheskomu raschetu rusel s razlichnoi sherokhovatostiiu stenok (Considerations on hydraulic design of channels with different roughness of walls). Izvestiia Vsesoiuznogo NuachnoIssledovatel’skogo Instituta Gidrotekhniki (Transactions, All-Union Scientific Research Institute of Hydraulic Engineering) 9: 238-241. Leningrad. Menéndez, A. 2003. Selection of optima mathematical models for fluvial problems. Proceedings of 3rd IAHR Symposium on River, Coastal and Estuarine Morphodynamics. Barcelona, Spain. Pavlovskii, N. N. 1931. K voprosu o raschetnoi formule dlia ravnomernogo dvizheniia v vodotokahk s neodnorodnymi stenkami (On a design formula for uniform movement in channels with nonhomogeneous walls). Izvestiia Vsesoiuznogo NuachnoIssledovatel’skogo Instituta Gidrotekhniki (Transactions, All-Union Scientific Research Institute of Hydraulic Engineering). 3: 157 –164. Leningrad. Shiono, K. & Knight, D. W. 1988. Two-dimensional analytical solution for a compound channel. In Y. Iwasa, N. Tamai & A. Wada (eds.), Proceedings of 3rd International Symposium on Refined Flow Modeling and Turbulence Measurements:503-510. Tokyo, Japan. Shiono, K. & Knight, D. W. 1991. Turbulent open-channel flows with variable depth across the channel. Journal of Fluid Mechanics 222: 617-646. Great Britain. Wark, J.B., Samuels, P.G. & Ervine, D.A. 1990. A practical method of estimating velocity and discharge in a compound channel. In White, W.R. (ed.), River flood hydraulics: 163172. Chichester, UK: John Wiley & Sons.
Weber, J. F. & Del Prete, P. 2003. Calibration and uncertainty analysis of parameters of analytical model for floodplain flow distribution. Mecánica Computacional 22. Bahía Blanca, Argentina: AMCA. In Spanish Weber, J. F. & Menéndez, A. N. 2003. LATERAL model to predict lateral velocity distribution in natural streams. Mecánica Computacional 22. Bahía Blanca, Argentina: AMCA. In Spanish Weber, J. F. & Tarrab, L. 2003. Modeling lateral velocity distribution in natural streams: case Parana river. Mecánica Computacional 22. Bahía Blanca, Argentina: AMCA. In Spanish Weber, J. F. 2003. 1D-2D integrated modeling of floodplain flows. M Sc thesis, University of Cordoba, Argentina. In Spanish.
