Transactions on Ecology and the Environment vol 48, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 Secondary flow in a sine-generated open channel M.F. Maghrebi Civil Eng. Dept., Faculty ofEng., Ferdowsi Univ. ofMashhad, Iran Abstract Experimental works in a sinuous open channel were performed to identify the vortical structures. Although the nature of such vortices are 3D and they mainly associated with low-speed and low-pressure zones in channel, the hypotheses of 2D models for the prediction of phase lag and bed topography seems to be satisfactory. In the theoretical part of the present paper by the use of the dynamic equation in the lateral direction for a curved open channel an equation for the velocity distribution in the lateral direction is extracted. Then, assuming a sine-generated channel whose curvature is a function of its longitudinal position, the phase lag between the secondary flow and the main flow is obtained. Next, by the use of different assumptions for the flow characteristics, different phase-lag formulas are extracted. Finally using the concept of incipient motion, variation of bed topography in a complete wavelength of a sine-generated open channel is presented. 1 Introduction Investigation of secondary flow in a curved channel is one of the ways to study the formation of the bed topography. In a curved open channel, due to centrifugal forces, secondary flow forms. Interaction between main flow and secondary flow causes that the point of maximum secondary current to be driven further to the apex of channel, which is called 'phase lag' and usually expressed in term of degree. Formulations of phase lag have presented by a number of researchers. Ikeda and Nishimura[S] presented a formula based on a depth averaged method of Saint-Venant equations in shallow flow. Johannesson and Parker[6] reported the phase lag as a function of flow depth and bed roughness. Zhou et a1.[13] focused on the growth and decay of secondary flow through meanders. They have reported that secondary flow is responsible for the redistribution of longitudinal velocities and bed scour in mobile boundaries. Different assumptions for the variation of secondary flow directly or indirectly leads to different expressions for phase lag which are derived herein from the equation of motion for secondary flow. Field observation show that the maximum erosion rate along the curved channel occurs at a section downstream of apex. Transactions on Ecology and the Environment vol 48, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 228 Water Resources .Uanagement 2 Experimental works For the determination of potentially erodible zones, in a uniform bed, velocity components in a fixed boundary sine-curve channel were measured by a magnetic current velocity meter with a diameter of 5rnm and a sampling frequency of 20Hz for a duration of 7min in two horizontal directions simultaneously. Geometric specifications and hydraulic parameters of the channel are summarized as follows (see Fig. l): .b = 40cm width of channel in y direction .Q = 8.67x discharge .l. = 120cm wave length of sinuous channel .d = Ikm depth of flow .So = 0.0006 bed slope In a Y4 wavelength of the sinuous channel 440 points in four different elevations were measured. In order to identify vortices, it was realized that the best way is to draw iso- contour lines of vorticity magnitude. Hence, vorticity was calculated based on the following formula: dv du 0 = dx dy' where u and v are depth-averaged velocity components in X and y directions, respectively and o is vorticity. The result is given in Fig.2. It can be seen that the vorticity magnitude near the concave bank is larger and vortical structures are expanded and formed a coalition with each other to form a low pressure zone. In the zone of high vorticity that is associated with low velocity and low pressure 3D vortical structures are present. In addition to lateral component of velocity v, vertical component of velocity W,is also Figure 1: Measured points and definition sketch for flow in a sine curved channel. v important. Although according to one circular core hypothesis the role of component of velocity in wide curved channel is more significant than W,the obtained results inspire that in low speed region the role of v and W components of velocity become important and they influence the bed topography. The presented result in Fig.2 may be referred to a uniform bed at a very initial stage of erosion. By passing the time, interaction of the main flow and secondary flow will affect the zone of potentially erodible of the channel. In the present work attention is particularly given to the secondary flow. Transactions on Ecology and the Environment vol 48, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 X cm Figure 2: Iso-contour lines of vorticity in a uniform bed fixed borders sinus channel. 3 Governing equations The presented model can be applied to a subcritical, turbulent flow in an alluvial channel curves with uniform bed sediment. The geometry of channel is in such ways that the bank effects on flow pattern are insignificant. According to Brice[l] classification of rivers, this model is only applicable to sinus canaliform rivers, which are characterized by a flat slope, a lack of braiding, a notably uniform width, a uniform depth and a moderate to high sinuosity. The constraints can be listed as follows: b=const., r~>>b, d/b<il, v<<u. The equation governing the streamwise development of flow in a wide curved open channel under steady conditions may be derived from the lateral momentum equation in cylindrical coordinates: In which u is the longitudinal velocity component, v is the transverse velocity component, s is the longitudinal coordinate along channel centerline, r is the radius of curvature, g is the gravity acceleration, S, is the transverse water slope surface and E is the momentum diffusion coefficient in a turbulent flow. The velocity distribution along the streamwise direction is assumed to follow the power-law: where u is the mean velocity in the streamwise direction, m is the power of the velocity function which can be expressed in the following form: where f is Darcy-Weisbach fixtion factor and U* is the shear velocity. On the water 2 = d surface, velocity can be obtained by replacing in eq.3: Transactions on Ecology and the Environment vol 48, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 230 Water Resources Management in which subscript s represent the surface of water. By introducing q = %, eq.2 can be written in the following form: in which subscript c represents the center of the channel. Since vcs is only a function of S, partial differential equation in the form of eq.6 can be changed into an ordinary dv, B differential equation: -+AV,, = - . (7) ds rc The solution of this equation is as follows: - in which C is a constant, A = fi(lc~,d,,m,~,~) and B = fi(m,uc,rc). Langbien and Leopold (see [3]) have proposed that the path of many river meanders can be expressed in the following form: (9) 8 = Q,, sin(24 L) ' where 0 is the angular deviation of centerline from the line connected two consequent inflection points and L is wavelength measured along the centerline. Definition for 0 is shown in Fig. l and O,, is 0 at the inflection point of channel centerline. The curvature of the sine-generated curve can be calculated as follows: where RC = f&emx is the minimum curvature radius at the apex By replacing eq. l0 in eq.8 and integrating part by part we have: (1 1) 2dC 1+p2 -1 where P = , $ =tan P and C is the integral constant. To find the location of dvcs maximum radial velocity on the water surface, the relation of - = 0 should be satisfied. ds Differentiating eq. l I, concludes the following equation: p = AL). (12) In eq.12, is the phase lag in sine channels. It can be seen that the phase lag depends on the roughness of the channel bed, rate of suspended sediment, flow depth and the radius curvature of the channel. In Fig.3 the concept of phase lag of secondary flow along the centerline of a curved channel is shown. curvature hlauimum intensity of secondary Figure 3. Phase lag of secondq tlotr along thc: csnterl~neot a s~nechannel Transactions on Ecology and the Environment vol 48, © 2001 WIT Press, www.witpress.com, ISSN 1743-3541 4 Lateral velocity distributions In order to study the dishbution of the secondary flow, considering Boussinesq equation that represents the relation between the shear stress and the ratio of velocity variation along the depth, eq.2 can be written in the following form: where p is water density and 7, is shear component in the spanwise direction. Boussinesq equation in this direction can be written in the following form: Since in natural rivers d/h << l,to determine the spanwise velocity distribution in lateral direction KiMtawa et a1.[8] assumed that the shear stress in this direction is small. Considering an element of fluid which is under the action of hydrostatic pressure on two sides in r direction as well as under the action of acceleration to the center of the curvature, it can be shown that the spanwise water slope along the centerline is as follows: -2 Uc Src = -. (15) m At any point in a flowing fluid, eddy viscosity depends on the elevation of the considered point to the bed. According to the Rouse parabolic approach, variation of eddy viscosity is defined as E = ku, (d- z)% where U* is shear velocity.