(1)
R.A. Eley1 and R. J. Keller2
Department of Civil Engineering, Monash University, Wellington Rd, Clayton, 3800, Victoria, Australia
E-mail: rachel.eley@eng.monash.edu.au
Abstract: Monflo is a two-dimensional numerical model that was developed to calculate the flow characteristics of channels with compound cross sections. That is, it includes the effect of momentum transfer between the main channel and the flood plains. The model was adapted to run both with and without the inclusion of momentum transfer. A series of numerical tests was then run to compare the water level and flow distribution in a compound channel predicted with both versions of the model. It was found that the effect of momentum transfer on the water level was minimal. The impact on the flow distribution varied depending on the cross-sectional shape and the surface roughness.
Keywords: open channel flow, momentum transfer, interaction phenomenon, rating curves
Flow in channels of complex cross-section, comprising a deep main channel and adjacent shallow flood plains, is a complex, three-dimensional phenomenon. An important aspect of this flow situation is the transverse transfer of momentum between the fast-moving parts of the stream (main channel) and adjacent slower moving parts. This transfer phenomenon is especially apparent in the interaction region between the main channel and the flood plain and results from the strong lateral gradient of longitudinal velocity in this region. The resultant momentum transfer may be envisaged to occur through the action of turbulent shear stresses, which act to transfer streamwise momentum from fast-moving parts of the stream to adjacent slow-moving parts.
The phenomenon has been the subject of intensive research over the last four decades. Sellin (1964 ) appears to have been the first to recognise the existence of the phenomenon and photographed the vertical vortex bank in the interaction region. Subsequent intensive studies have been undertaken by Zheleznyakov (1971 ), Rajaratnam and Ahmadi (1979 ), and Knight and co-workers (1983a; 1983b; 1984b ) among others.
At least in part, the reason for the strong interest in the phenomenon has been the potential for a significant influence on predicted water surface elevations in flood studies. Conventional backwater models, such as HEC-RAS, take no explicit account of the phenomenon and allow for it implicitly through a modified Manning¡¯s n value.
Over the last two decades, numerical open channel flow models have been developed which include the turbulent shear stresses and, hence, specifically allow for the interaction phenomenon. Such a model is Monflo, developed by Perera and Keller [1987] from earlier work by Keller and Rodi [1984a] . The model solves the basic depth-averaged equations of continuity and motion, including the depth-averaged turbulent shear stress terms. The latter are modeled with the aid of the Bousinesq eddy-viscosity concept:
(1)
where the k-e turbulence model is used to
calculate the distribution of the eddy viscosity,
. Full details of the model are presented by Keller and Rodi (1989
).
In the present paper, the program Monflo is adapted and used to investigate the influence of the interaction phenomenon on the predicted water surface elevation and flow distribution in a compound channel for a range of conditions. The adaptation of the model is discussed first. Results from the numerical experiments are then presented and discussed and conclusions drawn.
The original version of Monflo was adapted to allow the user the option of running it without considering the effect of momentum transfer. The model then transformed into a simple longitudinal model, with the bed resistance being the only shear stress calculated at each of the computational nodes.
A numerical modelling experiment was then carried out to examine the effect that momentum transfer has on the rating curve of a river with a compound cross section, and how this effect depends on the variation of certain parameters. The parameters considered were the longitudinal slope of the channel, the surface roughness, and the geometric dimensions of the depth of the main channel, D, and the relative widths of the main channel, WMC, to the floodplains, WFP, given by Wr. For simplicity, a trapezoidal cross-section shape was used for the numerical model, as illustrated by Figure 1.
Fig. 1 Cross section used in the numerical model
Each of the four above-mentioned parameters was assigned three values, as listed in Table 1. Each combination of the parameters in this table then formed one trial. Within each trial, Monflo was used to predict the water level for a range of flows, both with consideration of the interaction phenomenon (IP case), and without (NIP case).
Table 1 Parameter values
|
Parameter |
Value 1 |
Value 2 |
Value 3 |
|
D (m) |
4 |
5 |
7 |
|
Wr |
0.32 |
0.21 |
0.48 |
|
Slope |
0.002 |
0.0005 |
0.005 |
|
Roughness, n |
0.03 |
0.05 |
0.02 |
The aim of this work was to determine the effect of the interaction phenomenon on the stream rating curve. Accordingly, for each of the trials, the water levels predicted for each flow rate were plotted as stream rating curves, for both the NIP and the IP models. An example of this is shown below for Trial 1, which had the following parameters: D=7, Wr=0.32, slope=0.002, n=0.03.
Fig. 2 Stream rating curve for NIP and IP models of Trial One
As can be seen, there is a difference between the two curves. The magnitude of this difference between the IP and NIP predictions, or in other words the effect of ignoring the interaction phenomenon, varied among the trials. In order to quantify this variation, two factors were introduced:
(1)
(2)
where WL is the water level, QFP is the flow on the floodplains, QMC is the flow in the main +channel, and subscripts ¡°NIP¡± and ¡°IP¡± refer to ¡°no interaction phenomenon¡± and ¡°interaction phenomenon¡± respectively.
These factors quantify the difference between the two stream rating curves, IP and NIP, for each flow, for each trial. Both factors represent important issues for engineers dealing with flooding of a specific river. A higher Q-factor or d-factor will indicate a high similarity between the NIP and IP models.
The Q-factors and d-factors for each flow rate were calculated for each trial, and plotted versus Q/Qbf . Qbf is the bankfull flow and was used to non-dimensionalise the horizontal axis, providing a more suitable basis for comparison between the curves.
Figures 3 - 6 below show a selection of these curves. Figure 3 shows the variation of the d-factor and Figures 4-6 show the variation of the Q-factor. Each figure illustrates the effect of varying just one of the four parameters investigated, with each of the other parameters remaining fixed.
Figure 3 indicates that as the depth of the main channel was varied, the effect of the momentum transfer on the actual water level was minimal, with the lowest d-factors being around 0.978. That is, the water level predicted using Monflo without consideration of the interaction phenomenon was not lower than 97.8% of the level predicted with consideration of the interaction phenomenon. The d-factors for the remainder of the trials were found to be of a similar magnitude.
Fig. 3 Variation of d-factor with flow for various main channel depths
Fig. 4 Variation of Q-factor with flow for various main channel depths
Fig. 5 Variation of Q-factor with flow for various ratios of main channel to floodplain widths
Fig. 6 Variation of Q-factor with flow for various values of roughness
Figures 4~6 present a different result for the effect of momentum transfer on the distribution of flow. In all cases, the effect could be seen to become more significant at lower values of discharge. Thus, for a given cross section, as the discharge and thus, the water level decreased, the difference between the flow distribution as predicted by the NIP model and the IP model increased.
The effect of deepening the main channel is apparent from Figure 4. As the depth of the main channel increases, with all else remaining constant, so too does the effect of the momentum transfer on the distribution of flow. To a lesser degree, varying the width of the floodplain also has an effect on the extent to which momentum transfer alters the distribution of flow, as evident from Figure 5. It can be seen that the narrower cross sections (larger Wr), corresponded to lower Q-values, or lower similarity between the predictions of the NIP and IP models.
Figure 6 illustrates that variation of the roughness of a channel, whilst all other parameters remain fixed, results in a variation in the effect of the momentum transfer on the distribution of flow in the channel. As the roughness increase, the effect became less significant.
It was found that varying the slope of a channel had little effect on the degree of similarity between use of the NIP model and the IP model. The Q-factors and d-factors for each of the trials with differing slopes fell along the same curve.
As evident from the results of the Monflo trials, the most significant difference between the NIP and the IP models is the prediction of the distribution of flow across the floodplains and main channel. The actual water depths predicted using the two models were so close for all of the trials, that it can be concluded that considering the interaction phenomenon makes no significant difference.
As detailed in Section 2, the horizontal
momentum transfer is described by the depth averaged turbulent shear stress, txy,
given by equation (1). Thus, the extent of the momentum transfer will depend on
this velocity gradient,
, which in a compound channel is most apparent at the interface of the rapidly
moving main channel flow and the slower floodplain flow. The larger the
difference between these flows, the greater the expected degree of momentum
transfer, and consequently, the greater the difference between the predictions
made by the NIP and IP models. This is an explanation for the decrease in the
effect of momentum transfer for a given cross section, as the flow increases. As
the flow and, thus, water level, of a given channel increases, the relative
difference between the velocities in the main channel and floodplains will
decrease.
The
geometrical shape of the cross section will also have a noticeable effect on
, and consequently, on the extent of the momentum transfer. This was confirmed
through the two sets of trials undertaken which varied a component of the
channel cross section. The first involved the variation of the main channel
depth, D, the results of which are
shown in Figures 3 and 4. Deepening of the main channel had a similar effect to
decreasing the flow, in that the difference between the main channel velocity
and the floodplain velocity was increased. Thus, for the same total flow, a
deeper main channel resulted in a greater effect of the momentum transfer on the
flow distribution.
The second set of trials that dealt with the effect of cross section shape was the variation of the ratio of the main channel width to the total floodplain width, Wr. The results of these trials are shown in Figure 5. For a given flow, a cross section with a wider floodplain would have a lower water level, and thus a greater relative difference between main channel and floodplain velocities. Thus, an instinctive assumption would be that this would result in a greater effect of momentum transfer. Yet it can be seen from Figure 5 that those cross sections with the narrower floodplain predicted the greater difference between the NIP and IP models. However, further investigation revealed that for a given total flow, the wider floodplain in fact displayed the greater transfer of flow from the main channel to the floodplain. Since the ratio of flow on the floodplain to flow in the main channel was higher for this trial than for that with the narrower floodplain, this transfer of flow, although greater, had a smaller effect. In the same way, due to the greater width of the floodplain, the flow transfer did not correspond to as great an increase of the water level as was experienced by the narrower flood plain.
Thus, the difference in the effect of momentum transfer resulting from changes in the floodplain width is simply a matter of the relative impact of the momentum transfer rather that the actual magnitude of the transfer. Take for example two cross sections that are exactly the same in every respect, apart from the width of the floodplain. In the NIP model, if both the channels have the same depth of flow then the velocity distribution between the main channel and on the floodplains will be the same. Thus, the actual transfer of momentum predicted by the IP model would also be the same for the two cases. However, the resulting transfer of flow from main channel to floodplain will not be the same for the two cross sections, due to the difference in floodplain area. For the wider floodplain, the transfer of momentum will equate to a smaller increase in water level than for the narrower floodplain, and thus a smaller flow area in the main channel. Since the velocity profiles in the main channel will still be equal for the two cross sections, this corresponds to a smaller main channel flow, and thus a greater flow transfer. However, again, this greater flow transfer for the wider floodplain trial will not have as great a relative effect as does the flow transfer for the narrower floodplain trial.
The surface roughness of the channel bed will also have an effect on the velocity of the flow, and thus the degree of momentum transfer. This was illustrated by the Monflo results of the trials where Manning¡¯s n was varied, shown in Figure 6. As n was lowered, the magnitude of the difference between he predictions of the NIP and the IP models increased. This was due to an increase in the maximum channel velocity, and thus, a larger difference in velocity between the main channel and the floodplains.
Monflo was used to investigate the effect of ignoring the momentum transfer when calculating both the water level of a compound cross section and the distribution of flow across the floodplains and main channel. In both cases the effect was characterised as the similarity between the predictions of Monflo with the interaction phenomenon considered, and Monflo without the interaction phenomenon. It was found that ignoring momentum transfer did not have a noticeable effect on the former of these factors, yet had an effect of the latter.
In general, for a channel of given surface roughness and dimensions, the magnitude of the effect of momentum transfer on the distribution of flow increases as the flow decreases. The magnitude of this effect depends on the magnitude of the difference between the longitudinal velocities in the main channel and the floodplain, which in turn is influenced by both the channel dimensions and surface roughness.
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