Calculation and Analysis of Momentum Coefficients in the Changjiang River Estuary

Third Chinese-German Joint Symposium on Coastal and Ocean Engineering National Cheng Kung University, Tainan November 8-16, 2006 Calculation and Analysis of Momentum Coefficients in the Changjiang River Estuary Zhiyao Song* and Jun Kong State key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Ocean College and Eco-environmental Modeling Center, Hohai University, Nanjing *Zhiyaosong@sohu.com Weisheng Zhang Nanjing Hydraulic Research Institute, Nanjing Abstract Based on the field data of velocity profile observed at 23 measuring sites distributed over the Changjiang River Estuary in March and September, 1996, the momentum coefficients are calculated and their temporal and spatial distribution characteristics are obtained through statistical analysis. For different typical time interval and different places in the Changjiang River Estuary, the results of linear regression show that the momentum coefficients are larger for flood flow than for ebb flow, and for flood season than for dry season, that they increase from spring tide, mean tide to neap tide on temporal distribution, and that they increase from the outside the mouth to inside the mouth for the North Branch, and increase at first and then decrease from outside the mouth to inside the mouth for the South Branch together with the North Passage and the South Passage, the maximum value appearing near the mouth where fresh water meets salt water on spatial distribution. Through analysis, the feature of frictional resistance in the motion of tide is better understood, and an easy approach is provideed for choosing the Manning coefficient for numerical simulation of plane 2-D tidal flow. 1 Introduction Momentum coefficients firstly defined by Boussinesq in 1897 is also called Boussinesq coefficients (Chow 1959). In fluvial flow, Boussinesq defined momentum coefficient as ( U A) 2 2 = uda (1) A

Table 1 The momentum coefficients (Chow 1959) flow environment Minimum Maximum Average Canals 1.03 1.07 1.05 Rivers 1.05 1.17 1.10 glaciers 1.07 1.33 1.17 streamlets 1.17 1.33 1.25 Where u is a velocity component normal to a cross-section, U is the section-averaged velocity, and A is the sectional area of river. Then the value of directly reflects the non-uniform feature of velocity profile in river. The value of is generally more than 1, and it equals to 1 only referring to uniform flow under the ideal flow. In the past, many researches on momentum coefficients were mostly concentrated on the 1-D water flow such as canals, rivers, glaciers and streamlets, some results were shown in table 1. It can be found that the variation of momentum coefficients is very distinct in different flow environment. In the last 10 years, with the development of hydrodynamic research, the plane 2-D momentum coefficients has been considered as the modified coefficients of convective terms (inertial forces) in the depth-averaged 2-D momentum equations, the application range of which has been widely extended. For instance, the research on wind-induced flow(koutitas,1988), numerical simulation of plane 2-D tidal flow (Falconer,1990;Lin,1995; Liu, 2000), sediment production model (Tang, 2001) and local scouring problem (Zhou,1998). Certainly, under many conditions, the momentum coefficients may be approximated as 1.0 for it is very close to 1.0. But when simulating plane 2-D flow involving some engineering constructions, such as submerged dikes or jetties, the introduction of specified momentum coefficients is essential and necessary (Song, 2003). Tidal estuary has a complicated hydrodynamical system, because of the existence of many flows such as river flow, tidal flow, wind-induced flow, wave-induced flow and density flow induced by fresh water and salt water mixed together, and their mutual actions. The non-linearity of these actions is so strong that the velocity profile is very non-uniform. On the viewpoint of physics, this phenomenon is actually attributed to the non-linear frictional resistance characteristic while water body moves. So, based on the study of momentum coefficients and understand deeply the non-linear characteristic of water flow, this is important on hydrodynamical research and application. The Changjiang River Estuary is a famous tidal estuary of the world, whose hydrodynamics research has a typical significance. In order to realize the sustainable utilization of Changjiang River Delta resources and environment, the deepen waterway regulation engineering has been executed. Certainly, much precious hydrographic field data has been observed and the last researches are mostly emphasized on analyzing water body mixing, section circulation and null point variation, etc. For these reasones, based on the field data of velocity profile obsearved in the Changjiang River Estuary in March and September 1996, the momentum coefficients are caluculated and their temporal and spatial characteristics are analyzed in this paper. Besides these, the feature of frictional resistance in the tide flow has been discussed here.

2. Calculation of Momentum Coefficients As to the problem of plane 2-D, non-uniform flow, momentum coefficients are defined on unit-width section normal to the different coordinate direction. By combining the east and north component of velocity, three momentum coefficients are defined in every profile. They represent respectively the amendment of inertial forces in the east or north direction. These momentum coefficients are defined and discretized for calculation as the following D K K 2 2 2 xx () t = u dz ( U D) D u Δ Δ 0 i zi ui zi i= 1 i= 1 D K K K t = uvdz ( UVD) D uvδz uδz vδz () xy 0 i i i i i i i i= 1 i= 1 i= 1 D K K 2 2 2 yy () t = v dz ( V D) D v Δ Δ 0 i zi vi zi i= 1 i= 1 2 2 (2) where, t is time, D is total water depth, u i and v i are velocity components in the i-th level layer. U and V represent the corresponding depth-average velocity components. z i is the thickness of i-th level layer. Based on the representation on the distribution of observed points and their time history, 23 observed points are selected for this study, which include the flood season, dry season with spring, mean or neap tide in their time series. In space, these points cover the whole estuary from entrance passage to the sea outside of estuary (see Figure 1). According to expression Eq.2, it is shown that all momentum coefficients are functions of time. In order to study their temporal and spatial characteristics, the time-averaged momentum coefficients in typical time interval should be calculated (for example, in flood or ebb flow period, in spring, mean or neap tide period, etc). Applying the linear regression method, we analyze the relationship between numerator and denominator in Eq.2, then the corresponding expressions and the time-averaged momentum coefficients are given. As an exmple, for one site in some specified time interval, the numerator (y i ) and denominator (x i ) of xx are calculated at every time t i during this period, thus we get the corresponding expression as here a is y = ax+ b (3) xx for b is little. The time-averaged momentum coefficients of all 23 observed sites in their observed duration are shown in table 2. all the results shows that these momentum coefficients are mostly concentrated between 1.00 and 1.24, and their mean value is about 1.07, which consists with the research results by Falconer (1980).

Figure 1 The Changjiang River Estuary and the sites for measurements Table 2 The time-averaged momentum coefficients Measured sites Observed duration (1996) Tide Still water depth momentum coefficients xx yy xy A 3.8.14:00~3.11.9:00 Mean 5.88 1.043 1.068 1.081 B 3.11.11:00~3.14.8:00 Neap 6.17 1.010 1.059 1.029 C 3.5.16:00~3.13.16:00 Spring,mean and neap 19.20 1.046 1.029 1.026 D 3.5.16:00~3.7.16:00 Spring 20.38 1.047 1.036 1.0355 3.12.10:00~3.14.8:00 Neap 12.11 1.054 1.052 1.054 E 3.8.17:00~3.12.8:00 Mean 11.47 1.034 1.038 1.037 F 3.12.10:00~3.14.8:00 Neap 5.28 1.014 1.157 1.056 G 3.5.16:00~3.7.16:00 Spring 7.27 1.030 1.047 1.046 H 3.10.10:00~3.12.8:00 Mean 10.57 1.062 1.021 1.006 I 3.12.13:00~3.14.8:00 Neap 9.32 1.026 1.022 1.034 J 3.5.16:00~3.7.13:00 Spring 8.11 1.041 1.051 1.026 K 3.10.13:00~3.12.7:00 Mean 10.63 1.031 1.003 1.029 L 3.5.19:00~3.7.15:00 Spring 8.58 1.038 1.010 1.036 M 3.10.12:00~3.12.8:00 Mean 8.42 1.016 1.000 1.015

N 3.12.10:00~3.13.23:00 Neap 9.39 1.053 1.006 1.036 O 3.5.16:00~3.14.8:00 P 3.10.16:00~3.14.7:00 Q 3.10.15:00~3.14.7:00 R 3.10.14:00~3.14.6:00 Spring,mean and neap Mean and neap Mean and neap Mean and neap 7.90 1.088 1.052 1.064 14.02 1.055 1.059 1.059 30.27 1.039 1.040 1.046 20.05 1.049 1.059 1.091 9.13.9:01~9.14.10:31 Spring 13.47 1.051 1.021 1.040 S 9.16.22:01~9.17.24:00 Mean 13.43 1.047 1.034 1.023 9.22.15:00~9.23.18:00 Neap 13.36 1.040 1.054 1.048 9.13.11:00~9.14.9:00 Spring 9.28 1.041 1.237 1.040 T 9.16.20:00~9.17.18:00 Mean 9.41 1.046 1.321 1.060 9.22.11:00~9.23.16:30 Neap 8.52 1.107 1.142 1.143 9.13.8:00~9.14.11:00 Spring 11.35 1.044 1.046 1.050 U 9.16.22:00~9.18.00:00 Mean 11.39 1.045 1.054 1.048 9.22.14:30~9.23.17:00 Neap 11.33 1.071 1.055 1.067 9.13.6:02~9.14.9:01 Spring 8.38 1.057 1.032 1.052 V 9.16.21:01~9.17.22:57 Mean 8.42 1.053 1.039 1.061 9.22.13:00~9.23.17:00 Neap 8.47 1.069 1.081 1.083 9.13.5:00~9.14.8:00 Spring 10.82 1.093 1.026 1.049 W 9.16.9:00~9.17.22:00 Mean 10.81 1.183 1.011 1.094 9.22.11:00~9.23.16:00 Neap 11.02 1.136 1.112 1.159 3 Temporal Characteristics 3-1 The Basic Feature About all 23 observed sites, the calculated momentum coefficients have similar characteristics with time. For observed site C, as seen in Figure 2, the comparison is shown between the momentum coefficient xx and depth-averaged velocity U in 192 hours, 8 days duration from spring to neap tide. The basic features can be obtained from this figure: (1) In much time, the momentum coefficients don t vary distinctly, which is very close to 1.0. (2) At slack water time,because denominator U is very small, usually the momentum coefficients will have a larger pulse value.

Figure 2 Comparison between the momentum coefficient and depth-averaged velocity in the site C 3-2 The Time-averaged Feature In order to study the time-averaged characteristic of momentum coefficients, we classified the velocity profile according to different season, different tide and different flow state, then used the linear regression method to calculate them. The results are shown in table 3. It is clear that the momentum coefficients are larger for flood season than for dry season, and for flood flow than for ebb flow, they are in proper turn for increase from spring tide, mean tide to neap tide. This results further reflect the feature of frictional resistance in motion of tide water with the specified time interval. Table 3 The momentum coefficients with the specified duration Momentum Flow state Tide Season coefficients Flood Ebb Spring Mean Neap Flood Dry xx 1.068 1.046 1.049 1.061 1.070 1.072 1.035 yy 1.059 1.034 1.061 1.071 1.087 1.088 1.038 xy 1.088 1.062 1.038 1.048 1.085 1.068 1.033

Table 4 The time-averaged momentum coefficients xx in whole estuary Entrances North Branch South Branch North South Passage Passage Outside Sea sites B P Q C D F I N R xx 1.056 1.055 1.020 1.006 1.047 1.054 1.054 1.048 1.039 4 Spatial Characteristics To all observed sites (also see Figure 1), the time-averaged momentum coefficients are shown in table 2. It is found that the distribution feature of momentum coefficients in whole estuary is not obvious because of the difference of measured duration and the flow conditions. So we chose sites B, P and Q in the North Branch, sites C, D in the South Branch, site F in the North Passage, site I in the south passage, and sites R, N in the outside sea in the same measured period (from 13:00, 12 March, to 16:00, 13 March,), then calculated and compared the time-averaged momentum coefficients xx, the results are list in table 4. From this table, we got (1) In the North Branch, the river discharge is very weak and the tide is very strong, so the time-averaged momentum coefficients increase from entrance to upstream. (2) In the South Branch, together with the North Passage and South Passage, the time-averaged momentum coefficients increase from upstream to entrance because of the influence of river flow, but in the outside sea, time-averaged momentum coefficients decrease, their maximum appears near the river mouth where the fresh water meets salt water. 5 Discussion To ideal uniform flow, the value of momentum coefficient is 1.0, which is widely used as an efficient in simplifying the 3-D flow to plane 2-D flow. While the fact can not be denied that all nature flow will be influenced by the friction in some degree and the velocity distribution along vertical direction is not uniform. So in essence, momentum coefficients actually reveal the characteristic of non-linear frictional resistance of water flow. In modeling 2-D depth-average water flow, the friction is usually written as the following uur τ = ρ + ( )(, ) 2 2 2 b g U V C D U V (4)

here ρ is water density, C=D 1/6 /n is Chezy coefficient, and n is Manning coefficient. Of expression (4), the frictional resistance is embodied by C and n. Until now, though the plane 2-D flow numerical model has been perfected, however in many actual application, the Manning coefficient is usually selected empirically, which is between 0.01 to 0.04 s/m 1/3. In tidal estuary, many researchers chose different Manning coefficient depending on the different flow state and different tide for fitting with the field data. For example, Liu (2000) adopted n=0.016 in flood flow and n=0.013 in ebb flow in modeling plane 2-D water flow in the Changjiang River Estuary. Actually, this chosen is reasonable. Since 1981, some scholars (Soulsby, 1981; Anwar,1981; Tagg, 1989) have revealed this resistance characteristic through experiments in the lab, on the site and his analysis. When tide propagates from deepwater to offshore, the acceleration of tide flow enlarges the resistance, and the opposite result will be induced by the deceleration of tide flow. Based on such results, Song (2002) educed the modified formula of the resistance coefficient which can better reflect such characteristics. Tidal flow is caused by the tide. As it propagates from deepwater to shallow water, with the influence of many factors including the bed friction, the velocity profile becomes non-uniform. The nearer to seabed, the less the velocity is. Many scholars have advanced formulas to describe this profile, such as logarithm distribution, exponent distribution, and logarithm-linear distribution. Among these, the logarithm theory is more famous which is based on Pandtl assumption and von Karman similar theory. It has been proved correct in many field study (Wang,1989; Bergeron, 1992; Ke, 1994; Anwar, 1996; Collins,1998). Further, on this theory, the momentum coefficient can be expressed as 2 2 2 2 3 ( k C ) = 1 gn ( k 1 ) = 1+ g + D (5) It is clear that the momentum coefficient is influenced by n 2. It will be enlarged by the increase of the Manning coefficient. Then the follow formula is obtained in the Changjiang River Estuary n f n e = xx f xx e ( 1) ( 1) = 1.22 (6) where subscript f represents the flood flow and e represents the ebb flow respectively. This result is very close to n chosen in much numerical simulation in the Changjiang River Estuary. So on the the other hand, the study of momentum coefficient reflect some characteristic of the Manning coefficient and deepen the understanding on the bed friction. It gives a reference to choose a proper Manning coefficient.

6 Conclusions Through the calculation and analysis of momentum coefficients in the Chajiang River Estuary, some conclusions were obtained, that is (1) In temporal distribution, the momentum coefficients are larger for flood flow than for ebb flow, and for flood season than for dry season, they are in proper turn for increase from spring tide, mean tide to neap tide. (2) In spatial distribution, the momentum coefficients increase from outside to inside mouth in the North Branch, but they increase at first and then reduce from outside sea to inside mouth in the South Branch, and their maximum appears near the river mouth where fresh water meets salt water. (3) From analysis of momentum coefficients, we can deepen the understanding on the bed friction, and this help us to choose a proper Manning coefficient in the plane 2-D tidal flow modelling. 7 References Chow, V.T. Open-channel Hydraulics, McGRAW-HILL Book Company, 27p, 1959. Koutitas, C. Mathematics Model in Coastal Engineering, London, Pentech Press, 1988. Wang, H.Z., Y.Z. Song and H.C. Xue. A Qusi-3D Numerical Model of Wind-driven Current in Taihu Lake Considering the Variation of Vertical Coeffiecient of Eddy Viscosity, J. of Lake Science, Vol. 13, No. 3, pp. 233-239, 2001 (in Chinese). Falconer, R.A. Numerical Model of Tidal Circulation in Harbors. J. of the Waterway Port Coastal and Ocean Division, ASCE. Vol. 106, No. 11, pp. 31-48, 1980. Lin, B. and R.A. Falconer. Modelling Sediment Fluxes in Estuarine Water Using a Curvilinaer Coordinate Grid System, Estuarine Coastal and Shelf Science, Vol. 41, pp. 413-428, 1995. Liu, H., W. Hu, Y.S. He and J.Z. Yue. Studies on Numerical Modeling of Water Environment in the Yangtze Estuary-numerical Simulation of Hydrodynamic Flows, J. of Hydrodynamics (Series A), Vol. 15, No. 1, pp. 17-30, 2000 (in Chinese). Tang, L.Q. Problems Needed to be Solved in Sediment Yield Model Based on Physical Processes, J. of Sediment Research, Vol. 5, pp.22-28, 1990 (in Chinese). Zhou, M.D. Study on Similarity of Local Scouring of Cohesive Material in Hydraulic Model Test, J. of Hydraulic Engineering, Vol. 7, pp. 60-63, 1998 (in Chinese). Song, Z.Y. and R.H. Li. Depth-averaged Mathematical Model for Calculation of Flow Field over Submerged Dikes, J. of Hehai University (Natural Sciences), Vol. 30, No. 3, pp. 24-26, 2002 (in Chinese). Soulsby, R.L. and K.R. Dyer. The Form of the Near-bed Velocity Profile in a Tidally Accelerating Flow, J. of Geophysical Research, Vol. 86, Issue C9, pp. 8067-8074, 1981.

Anwar, H.O. A Study of the Turbulent Structure in a Tidal Flow, Estuarine, Coastal and Shelf Science, Vol. 13, pp. 373-387, 1981. Tagg, A.F. and P.G. Samuels. Modelling Flow Resistance in Tidal Rivers, In: Hydraulic and Environmental Modelling of Coastal Estuarine and River Waters, Edited by R. A. Falconer et al., pp. 441-452, 1989. Kuo, A.Y., J. Shen and J.M. Hamrick. Effect of Acceleration on Bottom Shear Stress in Tidal Estuaries, J. of Waterway, Port, Coastal and Ocean Engineering, Vol. 122, No. 2, pp. 75-83, 1996. Song, Z.Y. and Y.F. Liu. Corrected Formula of Bed Resistance Coefficient for Plane Numerical Simulation of Tidal Current, China Ocean Engineering, Vol. 14, No. 2, pp. 221-227, 2002. Wang, F.C., S.Y. Gao and Z.Q. Su. A Preliminary Sudy of the Vertical Distribution of the Tidal Current and the Roughness Parameter of the Offshore Area of the Yellow River Estuary, J. of Ocean University of Qingdao, Vol. 19, No. 2, pp. 37-41, 1989 (in Chinese). Collins, M.B., X. Ke and S. Gao. Tidally-induced Flow Structure Over-intertidal Flats, Estuarine, Coastal and Shelf Science, No. 46, pp. 233-250, 1998. Anwar, H.O. Velocity Profile in Shallow Coastal Waters. J. of Hydraulic Engineering, Vol. 122, No. 4, pp. 220-223, 1996. Bergeron, N.E. and A.D. Abrahams. Estimating Shear Velocity and Droughness Length from Velocity Profile, Water Resources Research, Vol. 28, No. 8, pp. 2155-2158, 1992. Ke, X.K., M.B. Collins and S.E. Poulos. Velocity Struction and Sea Bed Roughness Associated with Intertidal Flats and Saltmarshes of the Wash, U. K., J. of Coastal Research, Vol. 10, No. 3, pp. 702-715, 1994.