PREDICTION OF SECONDARY FLOW IN CURVED CHANNELS OF COMPOUND CROSS-SECTION USING REYNOLDS STRESS MODELS

 

 

Shao Xuejun,  Wang Hong and Chen Zhi

Department of Hydraulic Engineering, Tsinghua University

 

Department of Hydraulic Engineering, Tsinghua University,

Beijing 100084, China

Phone: (86) 10 62788543  Fax: (86) 10 62772463

E-mail: shaoxj@mail.tsinghua.edu.cn

 

 

Abstract: In curved open channels with a compound cross-section, secondary motion is driven by both the centrifugal force and turbulent stresses. In this study two Reynolds-stress models (RSM), i.e., models by Launder-Yang (LY model) and by Speziale-Yoshizawa (SY model), are used to calculate turbulence-driven secondary motion. The governing equation system is transformed into an orthogonal curvilinear coordinate system, so that the resulting equation set takes into account the effects of centrifugal forces. The numerical procedure is based on the SIMPLER scheme in an orthogonal curvilinear coordinate system. Various cross-section configurations and channel curvatures are used in the simulation using these models, and the impacts of different cross-sections and channel curvatures on the secondary motion are compared with measurements. The predicted secondary currents have been compared with published experimental measurements on a qualitative basis. Simulation results show that the LY and the SY models gives satisfactory reproduces the secondary flow features in a curved channel of compound cross-section. The longitudinal vortex pair were predicted at the region where the flood plain connects the main channel, and when the height of the flood plain changes, predicted the position and intensity of the vortex pair also varies, which agrees with experimental observations.

 

   Keywords: secondary flow, compound channel, reynolds stress model

1    INTRODUCTION

Curved channel flows are encountered in many engineering problems such as meandering rivers, and so are flows in compound cross-sections. Most natural rivers have flood plain that extends laterally away from the mean river channel, and typically have two stages. In a curved channel, centrifugal forces act at right angles to the main flow direction, and secondary motion is pressure induced. This type of secondary motion is called Prandtl’s first kind (Demuren and Rodi, 1984). Secondary motion of Prandtl’s second kind is commonly found in straight non-circular ducts, which is caused by turbulence and is called “turbulence driven”, i.e., it cannot result from a laminar flow. Secondary motion in a curved compound channel is a combination of Prandtl’s first and second kind, and is caused by both the centrifugal forces and turbulence stresses.

Numerical simulations of turbulence driven secondary motion in straight ducts were carried out by applying algebraic stress model (e.g., Launder and Ying, 1973) or full stress equations. Reece (1976) first applied turbulence models using full stress equations to open channel flow, and successfully simulated the depression of the velocity maximum below the surface in a narrow channel, which agrees with the well-known measurements of Nikurades (1926).

Based on the understanding that a model with an isotropic eddy viscosity cannot produce any secondary motion, Launder and Ying (1973) derived a model for the stresses and , by neglecting the convection and diffusion terms in the transport equations for the stresses (assumption of local equilibrium), while the primary stresses  and  were calculated from a standard eddy-viscosity model. This algebraic stress model denoted LY has been favored especially in calculations of the flow in channels with complex geometrical cross-sections. These so-called second-order closures also have some disadvantages, e.g., more computation effort in solving transport equations, and closure models for the higher order turbulence correlations have uncertain physical foundations. To take advantage of the k-e model and improve its performance in predicting nonlinear effects, Speziale (1987) developed a nonlinear k-e model (referred to as the SY model) based on an eddy viscosity model by Yoshizawa (1984), which was applied in a curved simple cross-section duct flow calculation by Hur et al (1990) generally agrees with measured secondary flow.  

It should be noted that the above two models were developed based on experiments of duct flow with a simple cross-section (e.g., square). A model well tested for duct flow prediction may therefore need further investigation before being applied to predict flows, when either the channel curvature or the cross-section varies. For instance, both the NR model by Naot and Rodi (1982) and SY model were used in the prediction of secondary flow in straight compound channels, and calculations agree well with measurements (Lin and Shiono 1995), yet in this study, however, the NR model is found to be sensitive to channel curvature and compound cross-section configuration, and it is difficult to achieve stable solutions for all the configurations. This paper presents calculations of curved compound open-channel flows by using the LY and SY models for various channel curvature and compound cross-section configurations. The NR model is not used because of the difficulty in achieving a stable solution, which takes further study to resolve.

2    GOVERNING EQUATIONS

2.1  Mean flow equations

The three-dimensional equations governing the distribution of the mean velocity components in a curved channel may be expressed as (In the coordinate system shown in Figure 1):

Continuity equation     

Fig. 1  Coordinate system in a straight channel flow

 

streamwise(longitudinal)momentumequation

momentum equation governing the secondary velocity U2 and U3

where g1, g2 and g3=orthogonal curvilinear coordinate transformation relationship; W gravitational potential, velocity components are defined in Figure 1.

2.2    Turbulence model

In the present study the following three turbulence models are used:

LY Model  This algebraic model includes the following equations:

,

             , ,

where l is the turbulence length scale, and turbulent kinetic energy k is solved from the following transport equation

where cV=0.22, c D=0.39, sK=1.5, c =0.01.

SY model (nonlinear k-e model) The SY model has the following expressions for the Reynolds stresses

   

,

  , CD=CE=1.6，c=constant .

3    SIMULATION RESULTS

Similar to the conceptualization by Hur et al (1990), the problem to be considered consists of an incompressible viscous fluid in a curved, helically coiled channel of compound cross-section. A staggered grid system is adopted, and the source terms are discretized using a second-order accurate central difference formula, for advection and diffusion terms a power function approximation is used, and a discrete equation system is obtained using the well-known SIMPLER scheme. The discrete equations are integrated using in time using the ADI method. Flow configuration is shown in Figure 2, where RR, RL is curvature radius of the right and left boundary. The free surface is assumed to have no transverse slope and can be considered a horizontal surface, which is an accurate approximation for gentle curvatures.

Fig. 2    Flow Configuration (flow vertically into the page at the cross-section)

3.1    Secondary flow results

Experimental measurements indicate strong inclined secondary currents between a longitudinal vortex pair (a floodplain vortex and a main channel vortex) near the junction edge of the flood plain and the main channel (Figure 3). Tominaga and Nezu (1991) found that the maximum magnitude of (U22+U32)1/2 is about 4% of U1max. This study use a fixed slope J=1/1500 and B=0.3m, H=0.2m. To predict the longitudinal vortex pair, flow in a curved channel with b/B=H1/H=0.5 and RL/B=5.0 is simulated. For such an condition the both the LY and the SY models give adequate prediction of the vortex pair, as can be seen from Figure 4. The maximum magnitude of (U22+U32)1/2 is about 5~6% of U1max, due to channel curvature.

From Figure 3 and Figure4 it can be seen that the measured floodplain vortex in the pair does not extend to the wall of the flume, which is correctly predicted by the SY model. The SY model also predicted the existence of another longitudinal vortex on the flood plain, which agrees with the measurement, therefore it gives better prediction in terms of the secondary flow pattern in the simulated case. The LY model predicts a single floodplain vortex which stretches to the flume wall, and a single main channel vortex which disagree with experimental measurements, due to channel curvature and smaller width. The calculated intensity of the vortex motion is stronger compared to the prediction of the SY model. It should be noted that the comparison between Figure 3 and Figure 4 is of a qualitative nature, as the experiment and the simulation use different channel configurations.

Fig. 3    Longitudinal vortex pair in a straight compound channel(Tominaga and   Nezu 1991)

Fig. 4    Predicted longitudinal vortex pair using SY and LY models in curved channel

3.2    Primary flow prediction

Figures 5 and 6 show isovels of measured and predicted primary mean velocity, in straight channels and curved channels, respectively. It can be seen that the prediction of primary mean velocity is consistent with that for the secondary current intensity due to the longitudinal vortex pair. The LY model correctly reproduces the low velocity zone that extends from the junction edge to the free surface, which is obviously resulted from the stronger secondary current calculated. The SY model predicts a less strong intensity of the secondary current, which corresponds to a shorter extension of the low velocity zone. From a qualitative view point, the LY model gives better agreement in terms of the mean primary velocity prediction.

Fig. 5    Isovels of measured primary mean velocity

Fig. 6    Isovels of predicted primary mean velocity in curved channel

3.3    Effects of variation in cross-section

In Figures 7 and 8 the measured and predicted secondary currents for various cross-section forms are shown, where h is the height of floodplain above main channel bottom. The SY model is used for the prediction because it gives better simulation of the vortex pattern. For floodplain with a smaller h, measured result indicates a strong vortex on the floodplain that dominates the entire floodplain, and a weak main channel vortex. The SY model prediction for a small flood plain agrees with this observation, except that the floodplain vortex is clockwise, rather than anti-clockwise as measured. As floodplain becomes higher (h value gets larger), a main channel vortex at the junction edge is observed, while the floodplain vortex is not observed, which is also correctly predicted by the SY model, except that only one main channel vortex is predicted, probably because of the channel curvature in the simulated case. Again it should be noted that due to differences in channel curvature and cross-section form, comparisons between calculated and measured cases are of a qualitative nature. It is obvious that channel curvature (or, the centrifugal force) has a big impact on the secondary motion, as the predominate vortexes simulated remain clockwise and only one predominate vortex, on the floodplain or in the main channel, exist for the curved channel in simulated cases, in contrast to the two measured longitudinal vortexes in the main channel in experiments.

Fig. 7    Measured secondary current for various cross-section forms

Fig. 8    Predicted secondary current for various cross-section forms in curved channel (SY model)

4    CONCLUSIONS

Secondary motion driven by both the centrifugal force and turbulence stresses in curved compound channels is simulated by using the LY model and the SY model, in a governing equation system transformed into an orthogonal curvilinear coordinate system. Various cross-section forms and channel curvatures are calculated in the simulation using the two models, to examine the impacts of different cross-section and channel curvature on the secondary motion. The predicted secondary currents have been compared with published experimental measurements. Due to differences in channel curvature and cross-section form, comparisons between calculated and measured cases are of a qualitative nature.

The SY model gives better prediction of the secondary motion pattern, i.e., the position and number of longitudinal vortexes, while the LY model predicts a stronger secondary current intensity that agrees better with experimental observations. For primary flow velocity, the LY model correctly reproduces the low velocity zone that extends from the junction edge to the free surface, which is obviously resulted from the stronger secondary current calculated, whole the SY model predicts a less strong intensity of the secondary current, which corresponds to a shorter extension of the low velocity zone. From a qualitative view point, the LY model gives better agreement in terms of the mean primary velocity prediction.

In the simulated cases, the predicted secondary motion in response to changes in cross-section configuration has a striking similarity to experimental measurements. Nonetheless, due to impact of channel curvature on the secondary motion, only one predominate longitudinal vortex is predicted in each case, which has the same direction of rotation regardless of its position (on the floodplain or in the main channel). This gives rise to further needs of experimental verifications.

 

Acknowledgements

The authors greatly appreciate the financial support for this study by the National Natural Science Foundation of China (NSFC) through contract 59879009-E090601.

 

References

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