PERIODIC TURBULENT STRUCTURES MODELLING IN A SYMMETRIC COMPOUND CHANNEL

Didier Bousmar and Yves Zech

Civil Engineering Dept., Université catholique de Louvain

Place du Levant, 1, B-1348 Louvain-la-Neuve, BELGIUM

E-mail: bousmar@gc.ucl.ac.be, zech@gc.ucl.ac.be

Abstract: This paper presents a theoretical and numerical study of the periodicity of the large vortices with vertical axis, located at the shear layers between the main channel and the floodplains of a flooded river. Those vortices are known to be responsible for a large momentum transfer and a subsequent conveyance reduction in compound channels. It is expected that a better knowledge of their structure could help in evaluating their effect on the channel conveyance. A hydrodynamic stability analysis of a parallel shear flow gives preliminary results on the vortices periodicity. The interaction between the two shear layers existing in a compound channel with two floodplains is briefly commented. The vortices generation is then modelled using a depth-averaged Large Eddy Simulation model. In those simulation, due to instabilities, the shear layers break up into separate vortices, whose spatial periodicity is found close to data previously published by Sellin. The vortex influence on the velocity profile and on the transversal shear stress distribution is also discussed.

Keywords: compound channels, floodplains, turbulence structures, vortex, shear layer, large eddy simulation, depth-averaged, numerical modelling

1    INTRODUCTION

Strong turbulence effects make challenging the flow modelling in a compound channel. One of the main feature of such a flow is the shear layer that appears at the interface between the main channel and the floodplains due to the velocity gradient. This shear layer generates large-scale turbulence structures, primarily in the form of vortices with vertical axes, first described by Sellin (1964). Due to the vortices, a strong momentum transfer occurs from the main channel to the floodplains, resulting in a channel-conveyance decrease that has to be evaluated for flood modelling.

These vortices are investigated using two different approaches. The first one is based on a classical hydrodynamic stability analysis. The second one uses a depth-averaged Large Eddy Simulation (LES), founded on a recent model by Nadaoka and Yagi (1998). The authors previously analysed the vortices influence on the velocity profile in a asymmetric compound channel (Bousmar and Zech, 2000; Bousmar and Zech, 2001). Here, the periodical aspect of the vortices is emphasised, for a symmetric compound channel, and is compared to previously published Sellin's data. The vortices influence on the momentum transfer is also analysed.

2    AVAILABLE EXPERIMENTAL DATA

Many velocity profiles in various compound channels geometry are available in the literature and were exploited in a former part of the authors’ study (Bousmar and Zech, 2000; Bousmar and Zech, 2001). Unfortunately, while many researchers pointed out the periodical aspect of the coherent turbulence structures in compound channel, few of them published quantitative data on this periodicity. One of the only usable data sets concerns the early experiments of Sellin (1964). Indeed, Sellin estimated the approximate wave length of the vortices, using photographs of the streamlines highlighted by aluminium powder scattered on the water surface.

Sellin's flume was 456 mm wide, with a main channel 44.5 mm depth and 114 mm wide, flanked by two symmetrical floodplains 171 mm wide (figure 1a). The bottom slope was 0.00085. From discharge measurements for the flow confined to the main channel, the estimated Manning roughness coefficient was n = 0.0064 s/m^1/3. In compound-channel experiments, the water depth was taken in the range H = 49 .. 59 mm, with discharges around Q = 2.5 l/s. The mean vortices wave-length was found equal to approximately twice the main-channel width, i.e. l » 228 mm.

As no depth-averaged velocity profile is available in Sellin's paper, the authors obtained a first approximation of this profile from the depth-averaged numerical simulation described below, before the vortex generation (figure 1b). For further use in the hydrodynamic-stability analysis, the half shear-layer width l_s is estimated by fitting a hyperbolic-tangent function U = U_m + U_s tanh [(y-y_bank)/l_s] to the left-side of the numerical profile. A value of l_s = 24.7 mm is found, i.e. less than a quarter of the main-channel width.

Fig. 1    (a) Sellin's flume cross-section, (b) numerical velocity profile.

3    HYDRODYNAMIC STABILITY ANALYSIS

Some first results on the shear flow can be gathered from a hydrodynamic linear stability analysis. This analysis aims to predict, for a given wave number, whether a small perturbation will amplify or disappear The wave number producing the highest growth rate of a perturbation can also be identified. It is expected that, at least in the first stage of their development, the vortices observed in compound channels should have a similar wave number as pointed out by Tamai et al. (1986).

Such an analysis is performed for variables written in non-dimensional form. For a general two-dimensional flow, the velocity component u in the x-direction is assumed to be the addition of a basic steady flow U, only depending on y, and a perturbation u'.

                    (1)

The perturbation u' is assumed periodic. It is supposed stable spatially and evolving only temporally. Therefore, its wave number a = 2p/l* is a positive real, where l* = l/l_s is the non-dimensional wave length; while the wave celerity is a complex c = c_r + i c_i . The growth of the perturbation is thus given by exp (a c_i t). Using (1), the flow equations for the perturbations can be linearized, to reduce either to the Rayleigh equation for inviscid flow, or to the Orr-Sommerfeld equation for viscous flow (Drazin and Reid, 1981).

For a single shear layer, the basic flow can be modelled by a parallel flow where the velocity profile is fitted by a hyperbolic-tangent function tanh (y*). Solving the Rayleigh equation shows that c_i will be positive (growing perturbation) only for wave number 0 < a < 1. The maximum growth rate is observed for a wave number a = 0.445 (Drazin and Reid, 1981). The analysis can be extended to viscous fluid, and take into account the channel roughness and its depth variation. For cases corresponding to actual channel conditions, the wave numbers corresponding to the maximum growth rate are found larger, in the range a = 0.65 (Bousmar and Zech, 2001).

In symmetrical compound channel, there is a double shear layer, that can be compared with a jet flow. The basic-steady-flow velocity profile U (y) is fitted with hyperbolic-secant function sech² (y*). Two instability modes are observed (figure 2). The first one, with maximum growth rate observed for a = 0.902, corresponds to alternated vortices (sinuous mode) and the second one, with the maximum growth rate observed at a = 0.518, corresponds to symmetrical vortices (varicose mode). The sinuous mode is found more unstable than the varicose mode.

Fig. 2    Vortices patterns for the two instability modes of a jet flow :

(a) sinuous mode, (b) varicose mode.

For usual compound channel flows, the two shear layers appear as more distant than in the sech² (y*) function. This will reduce the interaction between both shear layers. As a result, the sinuous mode will become more stable, while the varicose mode will become more unstable. For a wider main channel, the two shear layers will eventually be independent and present the same instability pattern as isolated shear layers, with maximum growth observed for the same wave number.

In the velocity profile for Sellin's cross-section, as the half shear-layer width l_s is found less than a quarter of the main-channel width, the two shear layers are expected to behave independently. Taking into account the roughness and depth variation effects, the wave number a = 0.65 found for a single shear layer can thus be used. This lead to a predicted non- dimensional wave length l* = 2p/a = 9.66, corresponding to a dimensional wave length l = l* ´ l_s = 239 mm. This value is noticeably close to the experimental one.

4    NUMERICAL MODEL

The depth-averaged LES model used is similar to the one developed by Nadaoka and Yagi (1998). In this model, the large 2D vortices are reproduced explicitly, while the smaller scale 3D turbulence is modelled through a Reynolds-averaged sub-grid model. The scale l_d of this sub-grid turbulence is assumed to be proportional to the local water depth h (H or h_fp):

l_d = c h                                    (2)

The complete 2D Saint-Venant equations include a diffusion term with an eddy viscosity u_d for the small-scale turbulence. The eddy viscosity is computed from the kinetic energy k_d and the turbulence scale :

                                   (3)

where C'_d is a constant. The equations are then made complete with a transport equation for this turbulent kinetic energy k_d that allows transport of the small-scale turbulence into the domain.

The set of equations is solved using a Mac-Cormack scheme, on a staggered mesh. A periodic boundary condition is used, assuming the same values for the variables on the upstream and on the downstream boundaries, in such a way that a uniform flow can be obtained. For the walls, the free-slip condition is used as their influence is negligible in the shear layer area. The computation starts with uniform flow conditions. The computer rounding errors are then allowed to grow in such a way that vortices may appear and develop for the most amplifying wave number. In a typical run, this vortex generation process takes about 50000 computing steps.

The cross-section of Sellin's flume is modelled using a 6 mm square grid, for a 3 m long reach, resulting in a 500 ´ 76 grid domain. One difficulty is linked with the vertical bank sides of the main-channel : these banks are modelled as oblique banks, on a width equal to one grid square. The roughness coefficient is increased on that section, in order to model also the friction on the quasi-vertical wall. The simulation is performed for a water depth H = 52 mm. The time step is chosen equal to Dt = 0.002 s.

5    NUMERICAL RESULTS

Figure 4 shows the velocity- and vorticity-field evolution with the time. The plotted velocities are relative to a reference frame moving at the same celerity as the flow mean velocity. It can be seen that perturbations in the two shear layers appears around t = 90 s. This instability grows up and the shear layer breaks up into individual vortices between t = 100 s and t = 110 s. Then, due to non-linear effects, these vortices merge into larger one, as already seen at t = 120 s. The same process is depicted on figure 3, giving the evolution in time of the transverse velocity component at the right-hand side interface. As predicted by the hydrodynamic stability analysis, the first stage of the perturbation growth (60 s < t < 105 s) can be fitted with an exponential envelop (dotted curve). The vortices expansion is then limited, due to bottom friction. Lastly, the merging process result in a lower perturbation frequency (t > 115 s).

Fig. 3    Evolution of the transverse velocity component at the interface

Fig. 4    Sellin's cross-section, computed velocity- (arrows) and vorticity field (grey scale)

Figure 5a shows the velocity profile before and after the vortices development. The main-channel velocity decreases while the velocity increases on the floodplains, at least near the interfaces. This effect is highlighted by figure 5b where the shear stress due to the eddy-viscosity and to the vortices is depicted. The vortices generate clearly a large part of the additional shear stress that is responsible for the momentum transfer between main channel and floodplain.

Fig. 5    (a) Velocity and (b) shear-stress profiles after the vortices growth

6    CONCLUSION

The periodicity of the large-scale vortices observed in compound channel is studied both theoretically and numerically, using a depth-averaged Large Eddy Simulation. The vortices wave-length and width is captured quite accurately by both techniques : this may help to characterise the momentum transfer between main channel and floodplains.

References

Bousmar, D. and Zech, Y. (2000), "Depth-averaged large eddy simulation of the turbulence structures in a compound-channel flow", Proc. Hydroinformatics'2000, Iowa City, Iowa, USA, CD-Rom proceedings.

Bousmar, D. and Zech, Y. (2001), "Periodic turbulent structures in compound open channel flows", to be submitted for publication in the Journal of Fluid Mechanics.

Drazin, P.G. and Reid, W.H. (1981), Hydrodynamic stability, Cambridge University Press, Cambridge, U.K.

Nadaoka, K. and Yagi, H. (1998), "Shallow-water turbulence modeling and horizontal large-eddy computation of river flow", Journal of Hydraulic Engineering, ASCE, 124 (5), 493-500.

Sellin, R.H.J (1964), "A laboratory investigation into the interaction between the flow in the channel of a river and that over its flood plain", La Houille Blanche, 7, 793-802.

Tamai, N., Asaeda, T. and Ikeda, Y. (1986), "Study on generation of periodical large surface eddies in a composite channel flow", Water Resources Research, 22 (7), 1129-1138.