SECONDARY-CURRENT IMPACT ON MOMENTUM TRANSFER FOR COMPOUND CHANNELS FLOW MODELLING

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: Coherent turbulence structures are responsible for large momentum transfer in compound channels. Large vortices, with vertical axis, located at the shear layer between the main channel and the floodplains generate the main momentum transfer and can be adequately captured through depth-averaged Large Eddy Simulation. This momentum is supposed to be transported on the floodplains through secondary currents. Due to those secondary currents, the velocities computed with the actual friction factor of the channel are generally underestimated when compared with the measured ones. This paper presents an attempt to take into account the secondary-current effects in a depth-averaged numerical simulation, through adequately modelled dispersion terms. The velocity profile prediction can be greatly improved, but the physical meaning of the necessary dispersion coefficients has to be deepened.

Keywords: compound channels, floodplains, secondary currents, dispersion, depth-averaged, numerical modelling

1    INTRODUCTION

Obviously, flow in compound channels is affected by the shear layer between the main channel and the floodplains. Due to the momentum transfer occurring through this shear layer, velocities are decelerated in the main channel and accelerated on the floodplains, making the conveyance estimation difficult for such compound channels. The main feature of this turbulent flow is the vortices with vertical axis developing across the shear layer. Those vortices were identified by Sellin (1964) as the instrument for the momentum transfer between main channel and floodplains. The flow structure in compound channel is also characterised by helical secondary currents, with horizontal axis (see figure in table 1), developing in both main channel and floodplains (Knight and Shiono, 1996).

In a previous work, Bousmar and Zech (2000, 2001a, 2001b) applied a depth-averaged Large Eddy Simulation (2DH-LES) to the compound-channel flow. This 2DH-LES reproduces well the development of the vortices with vertical axis at the interface between main channel and floodplains. The effect of these vortices on the momentum transfer is evidenced by a corresponding additional shear stress and its influence on the transverse profile of the depth-averaged longitudinal velocity component. Nevertheless, if the velocity predictions around the interface is clearly improved, the velocity on the floodplains away from the interface remains underestimated (see figure 2 in further section).

The easiest way to overcome this underestimation would be to decrease the roughness coefficient on the floodplains. Unfortunately this leads to an underestimation of the bottom shear stress, an accurate prediction of which is also of interest for the river engineer. In fact, the discrepancy between measured and computed velocities has to be explained by the presence of helical secondary currents (Shiono and Knight, 1996). These currents transfer a part of the momentum transversally and can increase the velocity on the floodplains if taken into account in the numerical simulations. Of course, such secondary currents can be modelled thoroughly only by 3D modelling. But their effect can also be taken into account in depth-averaged modelling, through the addition of a dispersion term. Such an example of a secondary current term is given by the Shiono and Knight method (1991).

In this paper, after a short review of the present knowledge on secondary currents, the dispersion terms are derived from the Navier-Stokes equations and a dispersion model is proposed. Then the influence of dispersion on the velocity profile is explored, in conjunction with the large vortices effect, as obtained from the 2DH-LES simulations.

2    ORIGIN OF SECONDARY CURRENTS

Several sources of secondary currents can be identified, depending on the flow conditions (Table 1) : centrifugal forces in curved channels (see e.g. Chang, 1988), turbulence anisotropy in prismatic channels (Nezu and Nakagawa, 1993). For compound channel, turbulence anisotropy due to the presence of the re-entering corner generates also Reynolds stresses and the corresponding secondary currents (Tominaga and Nezu, 1991). For all the considered cases, it can be observed that the secondary-current generation processes are controlled by the main flow. As a consequence, the transversal velocity component v will be proportional to the longitudinal component u. Another observation of interest is that the secondary-current cell width is generally of the same range as the water depth.

    Table 1    Sources of secondary currents

For narrow channels (width B = 2 ´ water depth h), Ikeda (1981) developed a simplified model of secondary currents, corresponding to a pair of counter-rotating cells, and thus neglecting the corner effect. Ikeda obtained the following expression for the transversal velocity v variation :

                (1)

where y and z are the transverse and vertical positions relative to the bottom left corner, U_* is the mean shear velocity, k is the Karman constant and d is a calibration factor. Such a profile is depicted on figure 1. This profile, developed for narrow channels, can also approximate the secondary cells in wide channels. It will be used to calibrate the proposed dispersion model.

3    DERIVATION OF DISPERSION TERMS

The dispersion terms arise from the integration of the Navier-Stokes equations to the Saint-Venant equations. All the terms are integrated along the water depth and the local variables are replaced by their depth-averaged forms. In the convective terms of the Navier-Stokes equations, several velocities products have to be integrated. As those velocities are not constant along the water depth (figure 1), the depth-averaged value of their product should not be equal to the product of their depth-averaged values. Several authors introduce a Boussinesq factor b in order to take this effect into account (Yen, 1973), but most of them assume then than its value is equal to b = 1. Here, the dispersion terms will be explicitly developed, in order to clarify their influence.

Fig.1    Typical vertical profiles of local longitudinal and transverse velocity components, For a wide channel, 0.2 m depth, with U_* = 0.045 m/s.

The depth integration of the product of the local velocities u and v, between the limits z_b (bed level) and z_w (water surface) can be written as a function of the depth-averaged velocity components U and V (Yulistianto, 1997) :

    (2)

On the second line, the second and third terms disappear as U and V are exactly the integrated value of u and v. The second term on the last line of this equation is the so-called dispersion term. Similar terms can be deduced for the velocities squares u² and v². All those terms are then to be added in the momentum equations of Saint-Venant : Yulistianto (1997) gives the complete derivation and Rodi (1980) mentions the final result, unfortunately with a sign error for the dispersion terms. For example, the depth-averaged x-momentum equation is :

      (3)

where S_fx is the friction slope in the longitudinal (x) direction and t_xx and t_xy are the depth-averaged turbulent shear stress. The last two terms are the dispersion terms.

4    PROPOSED DISPERSION MODEL

In order to analyse the dispersion terms influence on the predicted velocity profiles, these terms need to be estimated. Dealing with depth-averaged modelling, it is necessary to express their values as a function of the depth-averaged variables whose values are known. It is suggested to assume a proportionality between each dispersion term and the square of the depth-averaged longitudinal velocity U. This proportionality is evident for the u values, it can also be deduced from the secondary currents generation process for the v value. The dispersion terms are now :

  , ,    (4)

where c_uu ,c_uv and c_vv are defined as the dispersion coefficients.

The value of the so-defined dispersion coefficients should ideally be deduced either from theoretical considerations or from experimental data. A theoretical estimation of them can be obtained by assuming the logarithmic profile for the longitudinal velocity component u and the Ikeda profile (1) for the transversal component v (figure 1), with the abscissa y and the d parameter values chosen to get the maximum velocity approximately equal to v_max » 0.04 U (see table 1). Indeed, the absolute value of the coefficients c_uv and c_vv is expected to have a maximum value at the secondary-current cell centres, as v is maximum there, and to be null between two adjacent cells. The dispersion terms are found to be rather constant with water depth. Their estimation for h = 0.20 m is given in table 2.

An second estimation of the dispersion term can be obtained from experimental data, for example from the data set of the large scale Flood Channel Facility (FCF) at Wallingford, UK (Knight, 1992). This data set provides LDA measurements of u and v values at several location in the main channel and on the floodplains, for about 10 different geometries or water depths. Unfortunately, few points are available on each vertical on the floodplains, and the measurements have to be used carefully as the experimenters faced probe orientation problems that could lead to inaccurate values of v (Shiono and Knight, 1991). Nevertheless, the direct integration of the available velocity data gives dispersion coefficient surprisingly close to the ones expected from the theoretical estimation (Table 2). The c_uu value is nearly constant for the whole channel width and through all the tested geometries. The c_uv and c_vv values present a linear growth from the main channel central axis to the interface with the floodplain, with equal maximum values for all the tested geometries. These coefficients then present a linear decrease through the floodplain, but with more scattering, probably due to the fewer number of points available. A unique secondary-current cell, extending on the whole channel width, can be expected from such a coefficient profile, although it seems not realistic in regard to the width to depth ratio. Table 2 gives the coefficient maximum values.

    Table 2    Estimated maximum values for the dispersion coefficients

5    NUMERICAL RESULTS

Numerical simulations were performed for the asymmetrical geometry of the FCF 060501 test (Knight, 1992) that was formerly studied using the 2DH-LES without dispersion (Bousmar and Zech, 2000). The main channel is 1.50 m width and 0.15 m depth, and its banks slope is 1:1. The unique floodplain is 2.25 m width. Channel bottom slope is 1.027 10^-3. For the considered test, the water depth was h = 0.198 m and the discharge Q = 0.343 m³/s. The Manning roughness coefficient is estimated as n = 0.011 s/m^1/3. The numerical model solves the Saint-Venant equations using a Mac-Cormack scheme. Cyclic boundary conditions are imposed in order to let the vortices develop themselves with time. The grid size is 0.05 ´ 0.05 m² and the time step is 0.0025 s.

Figure 2 shows the computed velocities compared with the experimental data for the flow without vortices (common simulation) and with vortices, thanks to the 2DH-LES model. Taking into account the vortices improves clearly the velocities prediction around the shear layer. But, as already pointed out, the transverse extension of their influence is not enough developed for increasing sufficiently the computed velocities. Moreover, as already said, changing the roughness coefficient value to make the velocities more exact would damage the bottom-shear-stress prediction accuracy.

Fig. 2    Velocity profiles : for common simulation, with LES only, and with LES and dispersion (theoretical values)

On figure 2, the results of a similar simulation are also depicted, now including the dispersion terms, with dispersion coefficients as deduced previously : constant c_uu = 0.0080, linearly variable c_uv and c_vv, with maximum values c_uv = 0.0015 and c_vv = 0.0005. The dispersion terms effect is perceptible, mainly in the shear layer area, but is not sufficient.

Assuming a uniform flow, the x-momentum equation (3) reduces to the balance of four terms : the momentum available through the bottom slope (S₀ = －¶z_w/¶x) is mainly dissipated by the friction (S_fx), while the shear stress due to turbulence (t_xy) and the dispersion can either dissipate or produce momentum :

                      (5)

In the main-channel area, for a velocity quite constant along y, the positive gradient of c_uv lead to a positive value of the dispersion term. There is an additional dissipation and, as shown by figure 2, the computed velocity will be lowered. On the floodplain, a negative gradient of c_uv allows to add momentum and increase the velocity. Nevertheless, it is clear that the dispersion coefficient values estimated a priori are to small to improve significantly the velocity prediction.

According to these comments, figure 3 shows, For both velocity and bottom shear stress, the results of a trial and error fitting of the dispersion coefficient. To obtain the optimised value, it was necessary to use values of the coefficient dispersion varying between c_uv = –0.0100 and c_uv = 0.0020, thus up to 6 times greater than the theoretical one. Such great values are in fact needed in order to get a sufficient gradient and a sufficient value of the dispersion term in (5). This result of a linear gradient is similar to the calibration of the secondary current term of Shiono and Knight (1991). Nevertheless, even if these dispersion coefficients give an accurate modelling of the velocity, they are difficult to explain in term of physical process. Indeed, they would correspond to a unique secondary current cell on the floodplain, which is unrealistic when considering its width to depth ratio; as for such a width to depth ratio, numerous cells with a width comparable to the water depth (h = 0.05 m) would have been expected.

Fig. 3    Velocity and bottom shear stress profiles ,with LES only, And with LES and dispersion (optimised values)

6    CONCLUSION

The influence of the helical secondary currents on the floodplain velocity in a compound channel flow is simulated by the use of dispersion terms. These dispersion terms are developed from the Navier-Stokes equations integration and are modelled as proportional to the square of the depth averaged longitudinal velocity. An appropriate calibration of the dispersion coefficient allows an accurate modelling of both velocity and bottom shear stress profiles, but remains difficult to explain in term of turbulence structures.

References

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