Secondary Flow in a Sine-Generated Open Channel

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.

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.

important. Although according to one circular core hypothesis the role of v 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.

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

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

surface, velocity can be obtained by replacing 2 = d in eq.3:

Water Resources Management 230

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:

8 = Q,, sin(24 L) ' (9) 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 P and C is the integral constant. To find the location of where P = , $ =tan 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

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 direction as well as under the action of acceleration to the center of the r 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. For this profile depth averaged eddy viscosity is given as:

Doing simple mathematical operation and integrating, an expression for the rate of variation of secondary velocity in the z direction is obtained:

It should be noted that in determination of integral constant, boundary conditions at bed was zr = 0. The above equation can be expressed in term of dimensionless elevation

'1 = as follows: %

Integrating and applyng boundary conditions obtain the dimensionless form of lateral velocity:

--vc - (m + 2)(3m + 2) 2 3(m + I) k+2)- (5m +4) v,, 2(2m + 1) [37 -m'7 (3m + 2)(m + 2) l

Observing carefully the above equation, one can understand that the net mass flux across the central plane in the main flow direction is zero. The other lateral velocity distribution can be obtained by the assumption that the variation of the longitudinal shear stress is linear[3] i.e. T, = 'Ksb(1-q) . Doing almost the same procedure as done for the previous velocity distribution, will lead to the following results:

The other well known lateral velocity distribution is proposed by Odgaard[lO], which

shows a linear variation in respect to depth as:

The extracted velocity profiles as well as the one proposed by Odgaard are shown in Fig.4. Although the proposed velocity profile by Odgaard does not satisfy the non-slip condition on bed in the transverse direction, this feature of flow will not affect the calculation of phase lags. For the three presented velocity distributions in lateral direction which are called modeis 1 to 3, phase lags calculated by the use of eq. 12 can be listed as follows:

47r dc m2(m+1)

k2 ' L '(m + 2)(3m+ 2)

Falcon: v, = 2vcs(r7 - 112) Odgaard: v, = 2vcs (7 - 112) Kikkawa: s, = 0

Figure 4: Transverse velocity profiles.

Figure 5: A comparison between calculated phase lags (models 1 to 3) with experimental ones ex models 4 to 6) for two different f values.

The three proposed models for phase lags are compared with the models proposed by Kitanidis and Kennedy[7], Johnson and Parker[6], Ikeda and Nishimura[S] based on the experimental data which are named as models 4 to 6:

(26)

The results of six models in the range of 0 5 dc/ L 5 0.05 for two values of f = 0.01 and f = 0.05 are given in Fig.5 . From this figure it can be seen that with increasing the bed roughness, phase lag will be reduced. Also model 4 in prediction of phase lag shows a larger discrepancy. However, the results of calculated phase lags (1 to 3) show a good agreement with models 5 to 6.

5 Transverse bed slope

The distribution of flow depth throughout the meandering channels d = d(s,r), is determined on the basis of a simple relationship between vs and the transverse slope of the streambed, ST, obtained by the use of the incipient motion concept in a radial force balance for sediment particles on the bed surface. The obtained results by Durvin and Geldof, Ditrich and Smith, Odgaard and Kennedy and Tom et al. (see [g]) show that the transverse bed slope along the central axis of channel is almost constant and it can be shown as:

To obtain a relationship between the transverse bed slope along the entire of cross section it seems logical to make connection with the radius of curvature. Odgaard[9] has introduced a relationship as:

Since ST = $$ , by replacing it in eq.30, it yelds:

-=--dd IS~~lr~dr d dc r By integrating and rearranging: 4

rc in which 4 = ISTC~- and eq.32 can be expressed in the following form: dc

In order to examine the proposed relationship, it is required to compare it with the other relationships. To accomplish this, subscripts R,O and A are used for the proposed model,

234 Water Resources .Management

Odgaard model and Apman approach, respectively. Now, consider it is required to obtain bed topography of a sine-generated river with the information given below: ;,=0.6 %,Ds=o.~mm,4/h=1.3,m=3,k=O.4,~~=2.65,~~=10, &4=3,

.Proposed model: Dimensionless bed particle is calculated:

(G l) (2 65-1)9 8 L = D. = D, = ' 10.3~IOW' 7,589 [+] U [ 1 0-l2

By the use of Shield's diagram using given D*, critical parameter can be read: €lc= 0.038 Recalling path equation (eq.9), we have:

/ - cos(27tsl L) The relationship of - is used to obtain : Lc

Therefore, obviously we have:

.Odgaard model: The result of this model can be represented as:

.Apman model: In Apman model maximum flow depth at any cross section can be

calculated as:

To obtain the maximum flow depth in the proposed model and Odgaard model, we should replace n = 0.5 in the related equations.

In Fig.6 the ratio of for the three described models in the interval of () 0 5 d L 5 0.5 are plotted. It is clear that in the proposed model and the Odgarard model

Odgaard model

Proposed model

Apman model

Figure 6: Maximum flow depth in different cross sections based on three models. which the phase lag effects are considered, maximum flow depth occurred at a section downstream of the apex at the centerline (S/L = 0). While in the Apman model, which does not consider the effects of phase lag, maximum flow depth is predicted at the apex. From Fig.6, it can be seen that the proposed model and the Apman model show a better consistency with each other, and approximately the result of the proposed model is located between the two others. This implies the validity of the proposed model. In Fig.7 the contour lines of the dimensionless bed topography of the river which was described earlier are shown. These contour lines show the ratio of the flow depth to the depth of flow in the channel centerline. It can be observed that the variation of transverse bed profile around the apex is large. Going downstream, from the apex to crossover, it can be seen that the variation of the contour lines are reduced in such a way that around the crossover the transverse channel slope is reduced to form approximately uniform bed topography.

6 Conclusions

Secondary flow has an important role in configuration of meandering rivers. It affects the bed topography in different ways. The experimental results, which obtained in a non- erodible channel shows the position of potentially erodible zones. However, field observations reveal that the maximum erosion rate along meandering bend curves occurs at a cross section downstream of maximum channel curvature. This inspire that should be a kind of interaction between bed topography and secondary flow which migrate the channel boundary toward downstream. Based on power-law velocity profile for main flow combined with the assumptions for shear stress and velocity profile in transverse direction. three formulas for phase lag are extracted and they are compared with the ones based on experimental formulas. The calculated results show a good agreement with the observations. Then in calculation of bed topography considering phase lag, a formula is obtained. The results of maximum water depth based on the extracted formula are compared with models of Odgaard and Apman. From these two, only the Odgaard model

236 Water Resources Management considers the phase lag effects. Comparison of the proposed model with two others shows that the proposed model is reliable.

d/d,

Figure 7: Iso contour lines of bed topography based on proposed model.

References

[l] Brice, J.C. (1983). "Planform Properties of Meandering Rivers. ", River Meandering Proceeding Conference The Rivers '83, New Orleans, Louisiana, University of Chicago Press, Chicago, 1- 15. [2] Change, H.H. (1984), "Analysis of River Meanders. " J. Hydr. Eng., ASCE, 110(1), 37- 50.

", [3] Falcon, M. (1984). "Secondary Flow in Curved Open Channels. Ann. Rev. Fluid Mechanics, Vol. 16, 179-193.

[4] Gottlieb, L. (1976), "30Flow Pattern and Bed Topography in meandering Cl~annels." Series Paper 11, Institute of Hydrodynamics and Hydraulic Engineering, Technical University Denmark, Copenhagen, Denmark. [S] Ikeda, S. and Nishimura, T. (1 985). "Bed Topography in Bends of Sand-silt Rivers. " J. Hydr. Eng., ASCE, 111(11), 1397-1411. [G] Johannesson, H. and Parker, G. (1989). "Secondary Flow in Mildly Sinuous Channel." J. Hydr. Eng., ASCE, 115(3), 289-308. [7] Kitanidis, P.K. and Kennedy, J.F. (1984), "Secondary Current and River Meander Formation. ", J. Fluid Mech. 144, 217-229. [g] Kikkawa, H., Ikeda, S. and Kitagawa, A. (1 976). "Flow and Bed Topography in Curved Open Channels. " J. Hydr. ASCE, 1O2(9), 1327- 1442. [g] Odgaard, A.J. (1984). "Meander Flow Model: Development." J. Hydr. Eng. ASCE, 112(12), 1117-1136. [l01 Odgaard, A.J. (1986a). "Bed Characteristics in Alluvzal Channel Bends." J. Hydr. Eng. ASCE, 108(1 l), 1268-1281.

[l l] Rozovskii, I.L. (1961), "Flow of Water in Bends of Open Channel." Acad. of Sci. of

the Ukraine, available from Off. of Tech. Services, U.S. Dept. of Commerce, Washington, D.C. [l21 Yen, B.C. (1972). "Spiral Motion of Developed Flow in Wide Curved Open Channels. " In: H.W. Shen (Editor), POB 606, Fort Collins, Colorado, Chapter 22. [l31 Zhou, J., Chang, H.H. and Stow, D. (1993). '2 model for Phase Lug Secondav Flow in River Meanders. " J. Hydro, 146,7348. [l41 Zimmermann, C., Kennedy, J.F., (1978), "Transverse Bed Slopes in Curved Alluvial Streams. " J. Hydr. Eng. ASCE, 104(1), 33-48.