THREE DIMENSIONAL EXPERIMENTAL/ NUMERICAL
 RESULTS AND APPLICABILITY OF k-e MODEL
FOR MEANDERING COMPOUND CHANNELS

B.L.Jayaratne¹, N.Tamai² and Y. Kawahara³

¹Consulting Engineer, Ph.D., Pacific Consultant International, Tokyo, Japan

²Professor, Department of Civil Engineering, University of Tokyo, Tokyo, Japan

³Professor, Department of Safety Systems Construction, University of Kagawa, Kagawa, Japan

 Address for correspondence:

Pacific Consultant International, Water Resources Department,

7-5, Sekido 1-chome, Tamashi, Tokyo 206-8550, Japan.

 Telephone: +81-42-372-6030

Fax: +81-42-372-6359

E-mail: jayara@pcitokyo.co.jp   or   bljayar@mc.kcom.ne.jp

Abstract: Flow structures in a compound meandering channel where flood plain levees are shifted ahead by 30°, were investigated experimentally for shallow and deep water over bank flow conditions in order to understand the effect of phase shift as well as water stage on flow behaviour. Flow in compound meandering channels with parallel levees has also been studied. Flow structures in compound meandering channels strongly depend on several factors including meandering wave length, curvature, phase shift of levees and water stage etc.  The resultant influence caused by these factors on inertia force (main driving force), centrifugal force and flow mixing etc, makes the flow complex and three dimensional. Three-dimensional mathematical model with standard k-e method was applied to simulate the flow. It reveals that standard k-e model can be adopted to regenerate the compound meandering channel flow with sufficient accuracy except the flow separation reaches.

Keywords: compound meandering channel, phase shift, over bank flow, water stage, k-e model

1  INTRODUCTION

Many researchers are concerned about the flow process and bank erosion in compound meandering channels as downstream of many rivers consists with compound cross sections. It has increased the demand for mathematical models that can describe the flow process in compound curved channels. Velocity and depth distributions should be predicted with an adequate accuracy as the basis for rational river planning and design or maintenance by selecting appropriate locations and structures for river improvement work.

Tamai et al. (1983) reported the mean velocity distribution and water surface elevation in a single meandering channel. Ikeda et al. (1984) performed detailed measurement in a meandering duct and also carried out a numerical simulation with the k-e model. A comparison with experimental data confirmed that the turbulence model reproduces the mean velocity field and turbulent field adequately. Imamoto et al. (1983) explained the edges of flood plains coexist with the helicoidal flow. The flow on the flood plain moves deep scouring position from near the outer bank of main channel towards the center of channel (Mori and Kishi (1985), Kinoshita (1988)). Asida et al. (1990) also investigated flow characteristics in meandering compound channels. Willets et al. (1990, 1993), Ervine et al. (1993) and Sellin et al. (1993) indicated that main channel sinuosity has a major influence on the water level and discharge. Muto and Shino et al. (1995, 1996) studied the turbulent flow structure including the main flow velocity distribution and secondary currents using three experimental flumes of different sinuosities. Fukuoka et al. (1995,1996) examined the flow in compound meandering channels and riverbed changes. Jayaratne (1998) discussed the 3-D flow structure in meandering compound channels using experimental and numerical results.

This paper discusses the flow fields of shallow and deep water over bank flow in a 30°phase shifted compound meandering channel with explanations of the effect of phase shift on flow mechanism. In addition, applicability of standard k-e model to generate the meandering compound channel flow is also illustrated.

2  EXPERIMENTAL SETUP

The experiments were conducted in a 30m long compound meandering channel in which flood plain levees are 30⁰ ahead to the main channel levees. The width of main channel is 30cm and height of flood plain is 5cm where as width of flood plain at each side is varying along the channel. Details of the experimental flume are given in the Table 1. In order to minimize the upstream and downstream effects, one of the meandering waves at the middle of the flume was selected. Seven measuring sections selected for depth and velocity measurements are as shown in Fig.1. The details of two series of experiments conducted for shallow water and deep water flood flow conditions are as shown in the Table 2.

                           Table 1  Channel dimensions

                        Table 2  Experimental conditions

Three velocity components were measured in a fine grid mesh at each cross section. The sampling time was 60s at the rate of 5Hz. An I-type electromagnetic probe was used to measure the longitudinal and transverse velocity components and a L-type probe for vertical velocity components. The probe of each current meter is 5mm in diameter and 1.4cm in length.

3  NUMERICAL ANALYSIS

Three dimensional standard k-e turbulence model was applied to generate the flow in over-bank shallow water and deep water flow conditions of the phase shifted meandering compound channel used in experiments. The differential equations are expressed below in the generalized coordinate system in tensor notation to simplify expression.  In numerical study, the boundary fitted coordinate system was used.

Continuity equation:         

Momentum equations:      

Where; G = gravitational acceleration, P = pressure, uⁱ =contravariant components of fluctuating velocity, Uⁱ = Contravariant  components of mean velocity, U_i = Covariant  components of mean velocity, z = elevation from a datum, r = density of water, n = kinematic viscosity, semicolon = covaraiant differentiation, comma = differentiation.

Turbulent shear stresses  are replaced by eddy-viscosity concept,

where; k is the turbulent kinetic energy and dij are metric tensors. Turbulence model;

In the standard k-e  model, the model coefficients in the above equations are assumed to be constant such as C_m=0.09, s_k=1.0, s_e=1.3, C_e1=1.44, C_e2=1.92. The k-e model is derived under a high Reynolds number condition and is based on the assumption that the eddy viscosity is the same for all Reynolds stresses, and the isotropic eddy viscosity concept is employed. The standard k-e model was applied to simulate the flow as pressure driven secondary currents are dominant in meandering channel flow.

The finite volume method was applied to obtain the discretized equations over the non-uniform and staggered grid. The QUICK (Quadratic Upstream Interpolation for Convective Kinetics) was applied in the numerical simulation. Boundary conditions must be specified for all variables except for pressure at the inlet, outlet, solid walls and free surface. At solid walls, the wall function technique was adopted. At the free surface, the rigid lid assumption was applied.

4  RESULTS

Figure 2 and 3 illustrate the experimental and numerical results of streamwise and secondary velocity fields of shallow and deep water over bank flow conditions. In this phase shifted meandering compound channel, maximum streamwise velocity in the main channel as well as over the flood plain occurs at the inner levee for both flow conditions in contrary to the single meandering channel flow where maximum velocity appears usually close to outer levees. The short wave length and weak curvature might cause the above flow behaviour as the effects of inertia force dominate remarkably over the centrifugal force. In addition, comparatively wide flood plains over the main channel width and phase shift between levees are the other factors contributed in this regard. Unlike in compound channels with parallel levees, horizontal circulation of flow was visible over the flood plain even for the deep water condition. As it forms nearly a dead flow zone, flow in the remaining area of the same region accelerates and velocity increases for mass balance. Hence, it is required to determine the most suitable phase shift angle taking into account the channel configuration in order to shorten the high velocity region that may adversely effect on channel levees. Even though flow velocity in the horizontal circulation zone over the flood plain is quite low, precautions for the inner levee of the main channel are required as strong vortex formation due to flow separation occurs close to the section 4 and propagates along the levee up to downstream of section 5.

The results of standard k-e model that is widely used in flow simulations, show that it can reproduce basic flow feature of compound meandering channel flow especially streamwise velocity fields with sufficient accuracy. Figure 2 shows the streamwise velocity fields at 1cm and 5.5cm above the main channel bed of shallow water over bank flow whereas velocity fields at 3cm and 9cm above the main channel bed of deep water over bank flow are given in Fig.3 (flood plain height is 5cm). Numerical results of streamwise velocity fields in the main channel specially below the flood plain level agree well with experimental results though above the flood plain level, velocity at the horizontal circulation zone close to the flood plain levee is overestimated. One of the drawbacks in the model is its inability to capture the circulation zone. Transverse component of secondary currents is produced satisfactorily even though vertical component is underestimated in the numerical results. One of the reasons for the underestimation would be the simplified boundary condition (rigid lid assumption) applied at the water surface.

In addition to above experimental conditions, the k-e model was applied for several channel configurations by changing the phase shift angle between channels. According to the bed shear distributions, it revealed that channel configuration can be designed to minimize the flow impact on channel levees by proper selection of the phase shift angle and flood plain width. 

5  CONCLUSIONS

Phase shift angle between main channel and flood plain levees can be adopted as a tool in river designs where it is applicable, in order to plan a desirable river configuration that minimizes flow impacts on river levees.

Standard k-e model can be applied to reproduce the flow fields in meandering compound channels with an adequate accuracy for basic river designs. However, the model does not well capture the horizontal circulation zones close to the flood plain levees.

Fig. 1  Measuring sections

Fig. 2  Experimental and numerical results for shallow water over bank flow.

Fig. 3  Experimental and numerical results of deep water over bank flow.