Abstract: Recently pervious rockfill detention dam is used as a flood mitigating structure. Analysis of hydraulics of turbulent flow through this kind of dam is mostly done using Darcy-Weisbach equation. So far, many attempts have been made to study the friction coefficient as a function of Reynolds Number in turbulent flow of clean water through pervious rockfill detention dams, while this subject has remained intact for sediment laden flow of water. In this paper relationship between friction coefficient and Reynolds Number for sediment laden flow through highly pervious rockfill dams was investigated. Required data for a regression analysis obtained by conducting a series of laboratory tests to calibrate and validate a proposed power law friction coefficient-Reynolds Number relationship. A changeable bed slop Plexiglas flume, an adjustable rate sediment feeder and a recirculating flow electro pump system were used in present study. The tests were carried out on four different rectangular laboratory rockfill dams and three different non-cohesive suspended sediments. A power law relationship was obtained with a correlation coefficient of 0.74 using two thirds of laboratory measured friction coefficient and Reynolds Number. The obtained relationship was validated employing the remaining unused data with a Mean Square Error of 0.29 which is an acceptable agreement. A new power law relationship was found between friction coefficient and Reynolds Number in sediment laden flow through pervious rockfill dams. This new relationship is the only one thus has been proposed for the sediment laden flow of water through pervious rockfill dams.
INTRODUCTION
Rock materials, which are being widely used in the construction of reservoir dams, are also used in the construction of detention dams without any impervious shell at upstream face or core inside them. The governing hydraulic law of flow through rockfill materials in reservoir dams is quite different with that in detention dams. In the reservoir dams the governing hydraulic law is the Darcy’s law, while in the pervious rockfill detention dams due to a much higher velocity of water, the Darcy’s law is not valid (Mousavi et al., 2011). For instance, the flow net in rockfill detention dams varies with Reynolds Number (Li et al., 1998), the seepage force is not negligible anymore and the pipe theory (Darcy-Weisbach equation) should be used to determine the seepage discharge. This theory simulates the pores inside the rockfill as a pipe with more complicated configuration. This analogy provides a convenient tool to analyze flow through rockfill media under turbulent flow circumstances. Regarding the pipe theory approach in the analysis of hydraulics of flow through pervious rockfill dam, friction coefficient and Reynolds Number play important roles therefore, development of a non-Darcy relationship between them is a main issue (Li et al., 1998). Cheng et al. (2008) described that the relation of “F” and “Re” varies generally with the relative roughness length in the pore spaces of rockfill.
Regarding the relationship between Reynolds Number and coefficient of friction, so far many researchers such as Stephenson (1979), Herrera and Felton (1991), Ghazimoradi and Masumi (1993), Korki (1997), Li et al. (1998) and Samani et al. (2003) have conducted experimental studies on sediment free flow of water through rockfill media and have introduced two following forms of equations:
| (1) |
| (2) |
where, a, b, a1, b1 are the constants of equations and Re and f are Reynolds Number and friction coefficient, respectively. Constants of equations are the functions of size and shape of rock particles and fluid characteristics and are determined by means of experimental tests (Korki, 1997). For instance, Ghazimoradi and Masumi (1993), Li et al. (1998) and Samani et al. (2003) are the researchers whose scale of experiments were almost the same as that of present study and proposed the following relationship between f and Re respectively:
| (3) |
| (4) |
| (5) |
Mousavi et al. (2011) discussed that large amount eroded sediment particles due to surface flow as well as fluvial erosion of river bed may be carried by seasonal floods. As mentioned before, all previous works have been exclusively for clean water and nothing has been done for sediment laden flow. On the other hand, Joy et al. (1993) described that any model of the sediment particle transport process in porous media is an approximation of the actual process. They stated the complexity of the hydrodynamics, size and shapes of the particles, media, the paths through which the particles must travel as the main reasons of approximation. Therefore, there is a doubt and ambiguity to use the above formulas for sediment-laden flow through rockfill dams.
The objective of this study was to investigate the power law form Eq. 1 of f-Re relationship for non cohesive suspended sediment laden flow through pervious rockfill dam.
MATERIALS AND METHODS
In this study all laboratory tests were conducted in the hydraulic laboratory of Gorgan Agricultural University, Gorgan, Iran, from January 2009 to February 2011. To obtain required data all tests were performed, using a Plexiglas adjustable bed slope flume of 60 cm width, 60 cm height and 1100 cm length. A rectangular basket made of a steel mesh with an opening of 2 cm with a length of 78 cm and a width of 60 cm was placed in the flume. The basket was filled with rock particles of desired grading size and was considered as rockfill dam model. Four different grading particles with a mean diameter size (dm) of 5 or 12 cm and a standard deviation (σ) of 1 or 2 were used. A tail water gate was installed at the end of flume to provide required downstream water depth. To evaluate the effect of bed slope, all materials were tested with three flume bed slopes of 0.1, 0.5 and 1%. The tests were carried out with a discharge of 7.1 to 30.7 liter per second supplied by a recirculating electro pump system. A triangular weir was installed at the downstream end of flume to measure the water discharge. An adjustable rate sediment feeder was installed on the top of Plexiglas flume, adjacent to the upstream face of rockfill basket. All tests were run with a sediment discharge almost more than the maximum sediment transport capacity of the rockfill baskets, i.e. 143 g s-1. For each test, sediment feeding lasted almost 140 sec which was the possible maximum time due to the equipment restrictions. Forty six tests were carried out using uniform medium sand and uniform coarse sand with a d50 of 0.425, 0.6 and 0.85 mm as suspended sediment. Figure 1a presents schematic of the set up used in this research.
The test procedure started with adjusting the bed slope, turning on the pump and flowing water into the flume, then the discharge was adjusted to a desired value through regulating the downstream water level by means of the adjustable gate. The water discharge was measured by means of the triangular weir and once the steady state flow was established through the system, sediment feeder started working with a constant rate (143 g s-1). Water surface profile inside the rockfill dam was precisely measured by drawing it on the Plexiglas flume outside wall. Usually this was down when the equilibrium state of sediment transport through the rockfill dam was achieved.
| Fig. 1(a, b): | Schematic plan of hydraulic laboratory instruments, Schematic profile of sediment feeding into the water at the upstream end of rockfill basket |
The profile was exploited to calculate total head loss (Δh) and the average water depth (have) through the rockfill basket as follow:
| (6) |
| (7) |
| (8) |
Figure 1b shows schematic of sediment feeding. As shown, in this figure the water depths (h) were measured at 14 points along the rockfill basket; i.e. p1 to p14. The distance of two successive points is 6 cm. The friction coefficient and Reynolds Number are calculated according to Herrera and Felton (1991).
Forty six tests were carried out among them thirty tests were selected randomly to establish a relationship between f and Re, while sixteen others were used for evaluating the derived relationship.
Statistical analysis: Regression analysis was used to calibrate and validate the proposed power law equation (Eq. 1). The Least Squares method's system of equations which includes the basic equations must be solved to determine the unknown coefficient and constant (a and b) of the proposed equation. These equations along with Eq. 1 are written as below:
f = aReb |
Natural logarithm of Eq. 1:
Ln (f) = Ln (a) + b Ln (Re) Y = B + AX |
where, Y, A, X and B are Ln (f), b, Ln (Re) and Ln (a), respectively. A linear regression analysis (two equations system) is used to determine A and B as follow:
| (9) |
N is the number of observed data (f and Re). To facilitate statistical analysis the software SPSS was used in this research. Mean square error of predicted friction coefficients with the proposed power law equation versus laboratory measured friction coefficients was used for validation of equations:
| (10) |
where, n is the number of tests.
RESULTS AND DISCUSSION
As shown in the Table 1 the bed slope varied from mild to steep which enabled the tests to include the bed slope effects in the results. This table also indicated a wide rage of laboratory measured Reynolds Number from 1992.5 to 20937 which are close to the real field condition. A simple regression was found between f and Re with a pretty good coefficient of correlation of R2 = 0.74 using randomly selected thirty test results from Table 1.
| Table 1: | Characteristics of the rock particles and flow hydraulics and summary of the laboratory tests results |
| n: Test number, s bed slope of flume, Q: water discharge in each test, dm: mean diameter of rock particles, σ: standard deviation of rock particles, υ: water viscosity, Δh: flow headless across the rockfill basket, have: average water depth across the rockfill basket, Re: Reynolds Number of flow across the rockfill basket, f: friction coefficient of flow across the rockfill basket | |
| Fig. 2(a, b): | Correlation between friction coefficient and Reynolds Number for a sediment laden flow through pervious rockfill dams, (b): Validation of proposed equation |
As shown in Fig. 2a the relationship is in the form of power and presented as following:
| (11) |
To validate the equation 11, calculated values of “f” for 16 remained test results were plotted versus corresponding measured values. As shown in Fig. 2b with a mean square error of 0.29 between measured and calculated f values, the equation 11 is able to predict acceptable values of f based on Re for a sediment laden flow through highly pervious rockfill dam.
A comparison between previous researches and present study has been done in Table 2. Experiments of Stephenson (1979), Herrera and Felton (1991), Korki (1997), as well as the first experiment of Li et al. (1998), show smaller Re values than those of present study. These restrictions cause incapability for the resulted equations of mentioned researchers to predict the friction coefficient for the present study' range of Re. This is because in the resulted equations the extrapolation of Re out of the tested ranges is not allowed. Ghazimoradi and Masumi (1993), Samani et al. (2003) experiments and Li et al. (1998), second experiment show close ranges of Re to that of present study. Therefore, their resulted equations are capable to predict comparable friction coefficient of sediment free flow for the Re values in the range of present study. In Fig. 3 values of f calculated by means of sediment free flow formulas, which have been presented in previous literature, were compared with the laboratory derived f values and the values calculated by equation (11). As shown, a good precision is revealed in the results of present study (Eq. 11) and relatively in (Eq. 3), while the results of (Eq. 4, 5) are much far from the laboratory derived values; i.e. they predict almost two times higher than real values.
| Table 2: | Flow type, Reynolds Number and mean rock particle size of various researches |
| f-Re tests are the tests to be conducted to find the friction coefficient and Reynolds Number relationship | |
| Fig. 3: | A Comparison between f values calculated with different equations and the measured values in the laboratory tests |
In Fig. 3 values of Mean Square Error with respect to the laboratory derived values (MSE) for the results of foresaid equations are presented. This parameter for (Eq. 11) is very little, 0.29, for (Eq. 3) is also little but 2.2 times greater than present study; however the values of MSE derived from (Eq. 4, 5) are incredibly big such that they rise to 113 and 197, respectively. This difference is attributed to four restrictions in the experiments of Eq. 4 and 5: using fixed bed slope flume, neglecting standard deviation of rock particle sizes, limited number of tests in Samani et al. (2003). Comparison of values of MSE indicates the enhanced capability of Eq. 11 in prediction of friction coefficient (f) in sediment laden flow through rockfill media.
It should be noted that the rock particles used by Ghazimoradi and Masumi (1993) (Eq. 3) had a broken shape and sharp edges; but in the studies of Eq. 4 and 5 and also in the present study (Eq. 11) alluvial rounded edges particles were used. One may conclude that the friction coefficient in sediment free flow is very close to that of sediment laden flow if the former media is constructed of broken-sharp edge rock particles and the latter is constructed of the same size of rounded-edge alluvial rock particles. In other words, the influence of suspended sediment on the head loss is somewhat equivalent to the tortuosity created by the sharpness of stone edges.
The derived equation in this study (Eq. 11) facilitates usage of pipe theory to analyze non cohesive suspended sediment laden flow through pervious rockfill detention dams in non Darcy’s flow with a good precision.
CONCLUSION
The proposed friction coefficient and Reynolds Number relationship in this study indicates an acceptable agreement between its predicted friction coefficients and the laboratory measured friction coefficients. Proposed f-Re relationship is capable to analyze non cohesive suspended sediment-laden flow through rockfill dams in turbulent state with a good precision. The equation previously proposed for sediment free flow through sharp edge and broken rock particles is also able to predict relatively acceptable values. This may be due to existence of a substantial difference of flow resistance between two and one phase flow, i.e. sediment laden and sediment free flow.