Longitudinal variations of hydraulic characteristics of overland flow with different roughness*

WANG Xie-kang (王協(xié)康)

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China

Yangtze River Scientific Research Institute, Wuhan 430010, China, E-mail: wangxiekang@scu.edu.cn

YAN Xu-feng (閆旭峰)

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China

ZHOU Su-fen (周蘇芬), HUANG Er (黃爾), LIU Xing-nian (劉興年)

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China

Longitudinal variations of hydraulic characteristics of overland flow with different roughness*

WANG Xie-kang (王協(xié)康)

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China

Yangtze River Scientific Research Institute, Wuhan 430010, China, E-mail: wangxiekang@scu.edu.cn

YAN Xu-feng (閆旭峰)

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China

ZHOU Su-fen (周蘇芬), HUANG Er (黃爾), LIU Xing-nian (劉興年)

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China

(Received October 8, 2012, Revised December 4, 2012)

The evolution of the overland flow velocity along the distance downslope on smooth and granular beds in different cases is investigated by means of the electrolyte tracer via flume experiments. The results demonstrate that a non-uniform flow regime and a uniform flow regime exist in the development process of the overland flow. Owing to the different attributes of beds’ roughness, the position of those zones with different flow regimes varies correspondingly: (1) the overland flow on granular beds enters into the uniform regime much sooner, additionally, the roll waves tend to appear because of the presence of the proper flow resistance imparted by the roughness (coarse sands), and large slopes (20oand 25o) which makes the flow velocities and depths to undulate spatially. Furthermore, the flow resistance of the overland flows with different roughness elements, that is the non-sands, the fine sands and the coarse sands, is calculated. A quadratic interpolation method of the third order accuracy is employed in the calculation of the longitudinal flow resistance. The results show that it is rational to use the bed slope to approximate the hydraulic energy slope over a relative small roughness (the present roughness), however on the other hand, if the mean flow velocities and depths rather than the local parameters are used to calculate the flow resistance, a considerable error will be induced within the non-uniform regime of the overland flows, including the acceleration zone and the roll-wave zone.

overland flow, longitudinal hydraulic characteristics, flow resistance, quadratic interpolation

Introduction

The overland flow dominating the soil erosion, the scouring and the transport capacity[1-4]is one type of important natural flow patterns. Similar to the open-channel flow, the overland flow can be depicted by some hydraulic parameters. However, the overland flow is comparatively more complex than the openchannel flow due to the shallow sheet flow, the large slope and the changing roughness element. In many aspects, the overland flow can be characterized in terms of the flow patterns, the flow regimes and the flow resistance. On one hand, under the condition of a very low flow depth, the flow regime is more readily maintained as quasi-laminar, on the other hand, for relatively large and steep slopes, the high flow speed makes the overland flow absolutely turbulent. For the observation of the overland flow, the methods of observing traditional hydraulics and sediment transport mechanics are usually employed. For instance, the mean velocity, the mean flow depth, the discharge, theslopes and so on are measured to analyze the overland-flow characteristics. Nevertheless, due to the limitation of the flow depth, the investigation of the flow field and structure is relatively difficult, which hinders the overland flow studies.

For a realistic overland flow, due to the complexity of the boundary configuration, the characteristics of the overland flow can be very complicated. The roughness of the slope surface such as the vegetation on hillslopes and the large-scale stones on semiarid spots increases the total resistances, to reduce the mean flow velocity, the mechanical energy, and to increase the erosion. Those resistances in turn raise the flow depth. Thus the above aspects make it quite difficult to predict the overland-flow characteristics.

The mean velocity of the overland flow is initially associated with discharges and slopes in the form of a power function[5]. With a further study, the mean flow velocity is found to be less dependent on the varying slopes than the discharges in absence of largescale roughness elements; thereby the discharge is understood to be the main influencing factor of the mean flow velocity without large-scale roughness.

Similarly, the flow depth as one of most basic variables determines the overland-flow characteristics. In comparison with the mean flow velocity, the feature of the overland flow depth itself, however, makes it more difficult to carry out practical measurements. One factor responsible is that the flow is so shallow that either the fluctuation or the turbulence can produce a significant deviation in the flow depth, in particular, in cases of complicated boundaries like the uneven bottoms with vegetation and large-scale stones, and the characteristics of the fluctuation and the turbulence tend to be more intensive. Furthermore, the hydraulic phenomenon with roll waves in streams with relatively large-slope beds will make the flow velocities and the flow depths vary periodically in space, and the situations are also suitable for the overland flow[6,7]. To overcome above shortcomings, some particular measurements like the statistical approaches characterized by possibility functions[8,9]are applied to evaluate the mean flow depth. But, these approaches are usually rather complicated. A more conservative but effective method to dealing with the overland flow depth, however, is employed to calculate the flow depthhdescribed by the following equation[9,10]

whereqdenotes the unit discharge, andvdenotes the mean flow velocity. The results of the calculated flow depth are subsequently used to estimate other parameters such as the Froude number, the Reynolds number and the resistance coefficient.

The resistance as the essential parameter of the overland flow was widely in studied[4,9,12]. According to the available measured parameters, the resistance of the overland flow is well characterized by the Darcy-Weisbach coefficientf

wheregis the gravitational acceleration, andJsignifies the energy slope.

In accordance with literature[4,12], the Darcy-Weisbach resistance is empirically understood as the total resistance, which generally contains the surface (grain) resistance, the wave resistance, and the form resistance. Additionally, as with the rainfall and the mobility of the bed being taken into account, the rain resistance and the bed-mobility resistance should be included. Gary’s work[7]experimentally verified that the total resistance is not a simple linear superposition of those individual resistance components but in a more complicated non-linear relation. In the present paper, only the surface (grain) resistance is investigated since the fixed beds are designed to be smooth and granular fixed beds.

Instead of the mean hydraulic characteristics, the local ones along the distance downslope in the streamwise direction were rarely reported in literature[11]. In the conventional way, the overland flow for simplicity is technically treated as a uniform flow, which indicates that the developed overland flow does not vary spatially. And the mean flow velocities and depths are frequently evaluated to calculate the Darcy-Weisbach coefficient[12,13], with the assumption of replacing approximately the hydraulic energy slope with the bed slope. Nevertheless in actuality, the evolution of the overland flow with the roll waves generated due to the flow resistance is likely to make the longitudinal hydraulic features significantly fluctuate in space. More specifically, the treatment with the space-mean parameters in place of the local parameters seems inaccurate and may induce unavoidable error. In addition to roll waves, another factor affecting the result is the limitation of the length of the flume beds, the overland flow evidently includes the acceleration zone and the constant velocity zone on beds in laboratory experiments[5]. Furthermore, with significant achievements of the numerical simulations of open channel flows, the numerical models were more often used to analyze the overland flow[11,14]. Therefore, a new experimental approach is presented in this paper with the longitudinal hydraulic parameters being measured based on the above arguments, to investigate the spatially distributed variations of the hydraulic characteristics and the calculation of the flow resistance over different-roughness-element flumes.

Table 1 The range of parameters in different experimental cases

Fig.1 The Experimental system of overland flow

1. Apparatus and methods

Experiments were performed in the State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, China. The investigation includes three sets of experiments in a flume of 6.0 m in length, 0.5 m in width and 0.3 m in deepth with 0.003 m-thickness Plexiglas walls, and the bottom of flume made of Plexiglas. The end of the flume can be adjusted to change the bed slope. Figure 1 shows the schematic diagram of the experimental system. The first sets of experiments are on the smooth fixed bed, while the other two are on the rough fixed bed. The uniform sand is selected as the roughness adhered on the bed with its diameter varying between 0.00075 m and 0.0012 m. Correspondingly, the slope beds in experiments are referred to as the smooth-slope bed, the fine-sand slope bed and the coarse-sand slope bed, respectively. Three different slopes, 15o, 20oand 25oare adopted and 6 tentative discharges are applied; thus statistically a total of 162 experiments are needed as shown in Table 1.

As illustrated in Fig.1, the water supported by one pump enters the flume over a plate weir, which can make the input flow stable,θis the angle of slope. The tap of the pipe connected to the pump can be turned off and on to control the input discharge. The pebbles are deposited in front of the upstream head to allow the water to maintain stable before entering the flume. Main variables investigated include the flow discharges and the flow velocities. The flow discharges are evaluated by the means of measuring the water volume with a container. Based on the method of electrolyte tracer[15,16], an instrument produced by Institute of Water Conservancy of Chinese Academy of Sciences is applied to evaluate the flow velocity. The method is based on the mathematical-physical model of the solute transport within the steady water flow beneath a pulse boundary input, and in terms of the solute transport within the shallow water flow, the result is derived by the analytical solution of differential equations. The accuracy of the method is associated with the accuracy of the pulse function and the manual solute injection, but in general, it can provide more accurate measured velocities in comparison withother methods in evaluating velocities in terms of the shallow sheet flow. Its advantage is that it can evaluate velocities simultaneously at all targeted cross sections. The investigated flume spans from its upstream end to the location 5.5 m downstream, and ten velocity gauges are installed on the cross section every 0.5 m very close to the bed. The water depth is derived from Eq.(1) so as to avoid the trouble in measuring the flow depth.

Fig.2 The distribution of longitudinal overland flow velocities along slope in different cases

2. Results and discussions

2.1Variations of flow longitudinal velocity downslopeIn the overland flow study, a commonly-used method for investigating the mean flow velocities is to calculate the flow path in a measured time using the dye-speed measurement[17,18]. In order not to lose the accuracy of the observation, the tested zones of the flumes are supposed to ensure that the overland flow is developed to be comparatively uniform. However, the selection of the tested zones mostly qualitatively depends on a manual decision. The evolution of the overland flow on the slopes is somewhat associated with the geometric configuration among which the slope length is an essential factor. Hence, one purpose of the present research is to identify the spatial variations of the longitudinal flow velocity.

The actual nature is that the overland flow theoretically starts without a longitudinal velocity, such as the runoff-formed overland flow, and it subsequently accelerates on the slope beds for a distance, eventually reaching the stable state due to the combination of friction and gravity effects. The spent time and space of the whole process are determined by the surface configurations including the local morphological conditions, the patterns of the roughness elements and the length of the hillslope.

The tested data of the flow velocities are plotted against the slope length of the smooth and granular (fine-sand and coarse-sand) fixed beds under various slopes illustrated in Fig.2,vis the velocity of over-land flow,Lis the slope length. The flow velocities generally grow as the slope length increases with the acceleration of the overland flow being positive. On granular beds, the flow velocities remain stable as constants after the non-uniform zone while the flow velocities on smooth beds remain in the non-uniform state. The diagrams also indicate that the slope is a key factor to determine the position of the non-uniform zone, contributing to augment the flow acceleration as particularly apparent in terms of the flow velocities on granular fixed beds. Under different slopes (15o, 20oand 25o), the boundary positions between the non-uniform zone and the uniform zone are roughly at 2.0 m, 2.5 m and 3.0 m on fine-sand beds, while at 2.5 m, 3.5 m and 3.5 m on coarse-sand beds. Interestingly but not surprisingly, the tested smooth bed in particular under larger slopes is not long enough to develop the flow to be uniform, resulting in a flow accelerating on the whole bed. The coarse-sand bed providing a more significant resistance to retard the water can quickly stabilize the flow with constant velocities; however compared with the fine-sand bed, the stability positions of the flow velocities are postponed spatially. The rooted reason responsible for this phenomenon is that on coarse-sand beds the roll waves are generated, as frequently observed in man-made conduits like aqueducts and spillways. The generation mechanism of the roll waves is discussed and analyzed, and it is ascribed to the proper flow resistance that is neither little nor too much[9]. Clearly, since coarse sands are more frictional to protect the stability of roll waves effectively, coupled with the effect of the longitudinal gravitational component, which is increasingly more important with slopes rising, and the roll waves are generated.

Fig.3 Variations of overland flow depth in space along distance downslope (Discharge unit: 10–3m2/s)

As the roll waves appear, the overland-flow surface becomes dramatically wavy. Its longitudinal profile can fluctuate periodically, equivalent to undulations of the flow depths as illustrated in Fig.3,his the water depth of overland flow. With an appropriate flow resistance caused by coarse sands attached on the bed, the roll waves are formed with periodical variations of the flow depth, and the amplitudes are associated with the values of the slope. After the roll-wave zones, the flow goes back to the uniform state with flow velocities and depths approximately retaining constant. Thus the non-uniform zones should be identified and then avoided in investigating hydrodynamics of the overland flow.

2.2Calculation of flow resistance along the distance

The flow resistance as a rather essential hydraulic parameter for the overland flow is well characterized by the Darcy-Weisbach resistance coefficient. In comparison with other methods of evaluating the flow resistance such as the Manning coefficient and the Chezy factor, the Darcy-Weisbach resistance coefficient has a more clear physical meaning and can satisfy the principle of the dimensional consistency that cannot be achieved in the two other parameters. Thus the Darcy-Weisbach resistance coefficient is often selected as an indicator for the soil erosion, the scouring and the transport capacity on hillslopes. In the present research, the Darcy-Weisbach resistance coefficient is studied in a conventional way, that is, the variations of the flow resistance distribution along the distance downslope are derived from the longitudinal information like the flow velocity and the depth.

Rather than taking the average values, in the measurements of the longitudinal hydraulic parameters, the resistance along the distance of the overland flow can be calculated point to point using two approaches with different accuracies. The traditional approach is based on the presumption that the bed slope approximately in place of the energy slope is utilized to account for the feature of the flow resistance along the distance. Herein we assume that the total energy of the flow can be described by a function ()H xdepending on the slope lengthx, thus the hydraulic energy slope be obtained by the derivative of ()H xagainstxas,

In actuality, the local hydraulic energy slope is mathematically represented by the local curve slope of the function ()H x.

Fig.4 Calculated nodes for overland flow resistance along the distance

As shown in Fig.4, the complete hydraulic information at each measured across section is concentrated on a node system referred to as the main systemI,LΔ is the distance between two measured across scetions. Likewise, a sub-node system called the node systemiis employed to linearly express the information of hydraulic variables by spatially averaging between two neighboring cross sections, which are as follows,

2.3Approach A

To calculate the curve slope of the functionH(x), a quadratic interpolation is applied within the finite volume method with the third order accuracy, and the mathematical expressions are as follows.

Assume thatxaxis lies on the bed surface,Haxis is located normal to the bed surface, and the nodeIis located at the origin of the coordinates.

Hence we have,

wheremis a factor, satisfying the distribution in Table 2.

Table 2 The information of the quadratic interpolation

LetH(x) be a quadratic function, which can be determined byHi-1,HIandHI+1, orHI-1,HIandHi,

wherea,bandcare coefficients and can be calculated through substituting values of the ()H xwithin the function itself.

Then,

Hence, the hydraulic energy slope can be expressed as,

Substitute Eqs.(5), (6), (8) into Eq.(9),

Finally, the flow resistance along the distance downslope is derived as,

2.4Approach B

With approach B, the calculation of the hydraulic energy slopeJ, is approximated by the bed slopeS. Still, the longitudinal local flow velocities and depths are utilized in the calculation.

Hence, the flow resistance along the distance downslope is,

whereSis equivalent to sinθ.

2.5Approach C

Since the limitation of measurement instruments and the complexity of the shallow sheet flow, the local flow velocities can not readily be measured. Previousstudies frequently turn to the mean flow velocities, however, the present research considers the local flow velocities and depths. To point out the difference among approaches of the flow resistance calculation, the mean velocities and depths will be derived by averaging those in space,

In the mean time, the bed slopeSis also used to replace the hydraulic energy slopeJ, thus the flow resistance along the distance downslope is,

Fig.5 The comparison of calculation results of flow resistance among different approaches on the smooth fixed bed in different cases

Fig.6 The comparison of calculation results of flow resistance among different approaches on the fine-sand fixed bed in different cases

3. Calculation results

The comparisons of calculation results of the flow resistance with different approaches on fixed beds in different cases are presented in Figs.5, 6 and 7,f＇ andfare flow resistance coefficient calculated by approach B or C and approach A. Theoretically, approach A that employs a quadratic interpolation is more accurate in calculating the hydraulic energy slope in comparison with approaches B and C where the bed slope is used to approximate the hydraulic energy slope. Hence, the results of approach A are regarded as the standard for the flow resistance.

As shown in these illustrations, the largest magnitude of the flow resistance is on the coarse-sand bed, and this can be clearly explained by the roughness size. The correlation of the results of approaches A and B isstrong, on the other hand, there appears a dramatic deviation between the results of approaches A and C. The deviation degree of the results for the smooth bed is the smallest, followed by those for the fine-sand bed and then for the coarse-sand bed. It is quite obvious that the deviation of the results for the smooth bed is basically less than 10%, which indicates that in the 90 percent confidence interval the results of approach C are valid. Similarly, the majority data of the calculated flow resistance on the fine-sand bed are within the confidence interval, excluding some that are obviously biased. However, for the coarse-sand bed, a considerable proportion of the results are distributed outside the confidence interval. Besides, with the increase of the bed slope, the number of biased results also increases. According to the calculation, the biased scatters are those results investigated outside of the stable zone.

The above phenomena can be accounted for by applying the averaged parameters, including the average flow velocity and depth, in the equation for the Darcy-Weisbach resistance coefficient. In the nonuniform zone, particularly, in the zones where the roll waves are generated, the overland flow is in the state with significant variations of flow velocity and depth. Those averaged parameters sharply change with a large deviation from the realistic values, thereby the local flow velocities and depths should be utilized in the calculation. Unquestionably, the bed slope is one external drive force for the deviation, however, the approximation of using the bed slope in place of the hydraulic energy slope, surprisingly, does not produce a significant deviation in the Darcy-Weisbach resistance coefficient, which indicates that the loss of the total energy is not as important under a moderate roughness (smooth and granular beds). Therefore, it is rational to replace the hydraulic energy slope with the bed slope over a relatively modest roughness. But in presence of large roughness like cylinders and vegetation in which the total energy has a significant loss, this approximation is unadvisable.

Fig.7 The comparison of calculation results of flow resistance among different approaches on the coarse-sand fixed bed in different cases

4. Conclusions

For natural and laboratory-experimental overland flows, the local hydraulic parameters including the flow velocity and the depth are investigated, especially, the variations of their temporal and spatial distributions. In the development of an overland flow, it is clearly noted that there exist a non-uniform regime zone and a uniform regime zone as shown in experiments. Due to the flow resistance, the overland flow will eventually stabilize and become uniform, and the distance of the position where the flow is stabilized from the upstream end is proportional to the flow resistance. In addition to that, the bed slope plays a negative role in forming the uniform flow zone. However, in the case of a large slope and a proper flow resistance provided by the roughness (coarse beds), roll waves will appear, leading to a dramatic oscillation of the flow velocity and the depth and postponing the stable zone spatially. Owing to those characteristics distribution of the overland flows along the distance downslope, the effects on the soil erosion, the hillslope-surface scouring and the transport capacity are not simple and homogeneous but spatially variable. Hence, the traditional method of study on the overland flow has its limitation.

In order to calculate the longitudinal flow resistance, a new approach using a quadratic interpolation is employed. The approach retains mathematically the third order accuracy and is used to derive the local hydraulic energy slope with measured local overlandflow parameters. To make a comparison, the traditional approach is also used with the hydraulic energy slope approximated by the bed slope and the flowvelocity as well as the depth spatially averaged to calculate the flow resistance. Results show that with the selected roughness, the bed slope can roughly represent the hydraulic energy slope, while employing the mean flow velocity and depth to calculate the flow resistance can result in a significant deviation in the non-uniform flow zone, in particular, in the zone where roll waves are generated. Therefore, for simplicity, the mean flow velocity and flow depth are suggested to calculate the flow resistance within the stable zone, while the bed slope and the local flow velocity as well as the flow depth are applied within the unstable zone of the overland flow.

Acknowledgements

This work was supported by the Yangtze River Scientific Research Institute Open Research Program (Program SN. CKWV2012313/KY), the Open Foundation of State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University (Grant No. 1008).

[1] HU S., ABRAHAMS A. D. Resistance to overland flow due to bed-load transport on plane mobile beds[J]. Earth Surface Processes and Landforms, 2004, 29(13): 1691-1701.

[2] PAN C., SHANGGUAN Z. Runoff hydraulic characteristics and sediment generation in sloped grassplots under simulated rainfall conditions[J]. Journal of Hydrology, 2006, 331(1-2): 178–185.

[3] SHIH H.-M., YANG C. T. Estimating overland flow erosion capacity using unit stream power[J]. International Journal of Sediment Research, 2009, 24(1): 46-62.

[4] ZHANG G.-H., SHEN R.-C. and LUO R.-T. et al. Effects of sediment load on hydraulics of overland flow on steep slopes[J]. Earth Surface Processes and Landforms, 2010, 35(15): 1811-1819.

[5] ZHANG Guang-hui. Study on hydraulic properties of shallow flow[J]. Advances in Water Science, 2002, 13(2): 159-165(in Chinese).

[6] LIU Q. Q., CHEN L. and LI J. C. et al. Roll waves in overland flow[J]. Journal of Hydrologic Engineering, 2005, 10(2): 110-117.

[7] GARY L. Preliminary study of the interference of surface objects and rainfall in overland flow resistance[J]. Catena, 2009, 78(2): 154-158.

[8] PARSONS A. J., WAINWRIGHT J. Depth distribution of interrill overland flow and the formation of rills[J]. Hydrological Processes, 2006, 20(7): 1511-1523.

[9] SMITH M. W., COX N. J. and BRACKEN L. J. Modeling depth distributions of overland flows[J]. Geomorphology, 2011, 125(3): 402-413.

[10] HU S., ABRAHAMS A. D. Partitioning resistance to overland flow on rough mobile beds[J]. Earth Surfaces Processes and Landforms, 2006, 31(10): 1280-1291.

[11] YI Zi-jing, YAN Xu-feng and HUANG Er et al. The calculation of overland flow resistance and hydraulic characteristics along slope distance in a steep experimental flume[J]. Journal of Sichuan University (Engineering Science Edition), 2011, 43(Suppl.1): 43-47(in Chinese).

[12] BARROS A. P., COLELLO J. D. Surface roughness for shallow overland flow on crushed stone surfaces[J]. Journal of Hydraulic Engineering, ASCE, 2001, 127(1): 38-52.

[13] HOLDEN J., KIRKBY M. J. and LANE S. N. et al. Overland flow velocity and roughness properties in peatlands[J]. Water Resources Research, 2008, 44(6): W06415.

[14] RAZAFISON U., CORDIER S. and DELESTRE O. et al. A shallow water model for the numerical simulation of overland flow on surfaces with ridges and furrows[J]. European Journal of Mechanics B/Fluids, 2012, 31: 44-52.

[15] LEI T., XIA W. and ZHAO J. et al. Method for measuring velocity of shallow water flow for soil erosion with an electrolyte tracer[J]. Journal of Hydrology, 2005, 301(1-4): 139-145.

[16] LEI T., CHUO R. and ZHAO J. et al. An improved method for shallow water flow velocity measurement with practical electrolyte inputs[J]. Journal of Hydrology, 2010, 390(1-2): 45-56.

[17] DUNKERLEY D. Estimating the mean speed of laminar overland flow using dye injection-uncertainty on rough surfaces[J]. Earth Surface Processes and Landforms, 2001, 26(4): 363-374.

[18] ZHANG G.-H., LUO R.-T. and CAO Y. et al. Correction factor to dye-measured flow velocity under varying water and sediment discharges[J]. Journal of Hydrology, 2010, 389(1-2): 205–213.

10.1016/S1001-6058(14)60008-1

* Project supported by the National Natural Science Foundation of China (Grant No. 41171016).

Biography: WANG Xie-kang (1970-), Male, Ph. D., Professor