مجله هیدرولیک
سال نوزدهم شماره 2 (تابستان 1403)

The shallow water equations are a set of hyperbolic balance laws that describe the behavior of water flow in shallow regions such as rivers, lakes, and oceans. Solving hyperbolic balance laws poses significant challenges due to the presence of non-conservative terms, shocks and discontinuities. Analytical solutions are limited to simplified cases, so numerical methods are often employed to solve these equations. Numerical schemes addressing these balance laws must ensure the well-balanced property (Bermudez and Vázquez 1994), ensuring that discretized numerical fluxes must exactly balance by the approximated source terms. These types of numerical schemes utilize upwind/flux splitting techniques to handle wave propagation and discontinuities. Such well-balanced approaches work well for supercritical or subcritical regions but are known to struggle when Riemann problem includes both (LeFloch and Thanh 2011)- particularly in trans-critical flows and hydraulic jumps. To address this, various treatments, such as entropy fixes, shock fitting techniques, have been developed. Notably, Akbari and Pirzadeh (2022) (Akbari and Pirzadeh, 2022)introduced a set of shockwave fixes to cure the numerical slowly moving shock anomaly. Their approach is advantageous in accurately capturing the hydraulic jump. However, such scheme is only first-order accurate, as higher-order schemes progress, it becomes necessary to extend such technique to greater accuracy in high-resolution schemes. A second order well balanced numerical scheme has been designed for the shallow water equations using a semi-discrete MUSCL reconstruction. The first step in the semi-discrete finite volume method is to discretize the governing equations in space. For the one-dimensional shallow water equations, this involves dividing the computational domain into a set of control volumes and approximating the integral form of the conservation equations over each control volume. By considering the fluxes at the control volume interfaces and accounting for the source terms, a system of ordinary differential equations (ODEs) can be obtained. To ensure accurate and stable solutions, a second-order finite volume approach is employed for spatial discretization. The proposed approach aims to exactly preserve all steady states of shallow water equations while maintaining the second order of accuracy. To achieve this, we extend a recently developed fully well-balanced scheme, called HLL-MSF, to higher-order of accuracy. To upgrade the first-order HLL-MSF scheme to second order while maintaining the same well-balanced property of the first order one, a MUSCL reconstruction approach with a suitable weighted technique is proposed. The weighted approach allows the numerical scheme to revert to the first order scheme with shockwave fixes at hydraulic jumps or at trans-critical points. Appropriate flux limiters are also introduced to ensure the well-balanced property of the numerical scheme in smooth steady state cases. The method's accuracy and stability are attributed to these carefully chosen flux limiters and weighted coefficients. The final step in the semi-discrete finite volume method involves time integration to advance the solution in time. In this paper, the third order explicit Runge-Kutta method is chosen as the time integration scheme. By combining the second-order finite volume spatial discretization and the third-order explicit Runge-Kutta time integration scheme, the proposed finite volume method ensures higher-order accuracy in both space and time. To verify the well-balanced property and the second order of accuracy of the proposed numerical scheme several numerical examples and benchmarks found in the literature including both steady and unsteady cases are presented. For numerical experiments that have analytical or reference solutions, numerical errors are calculated using L1 and L∞ norms. The first test case is devoted to the simulation of steady state at rest or the lake at rest situation. Numerical errors demonstrate that the proposed scheme is exactly well-balanced in this case. The second test case addresses a smooth steady state of trans-critical flow over a bump. The proposed second order scheme is confirmed to capture the smooth steady state precisely (Table 1). We also perform experiments on trans-critical flow with hydraulic jump to see how the proposed scheme behaves when the solution contains a shock discontinuity. Unlike the traditional higher-order schemes which often use the pre-balanced shallow water formulation to achieve the exact conservation property on steady state cases at rest, the proposed second order scheme can capture both smooth and non-smooth (Hydraulic jump) parts exactly with no smears and oscillations (Table 1). An additional test case is conducted to confirm the second order accuracy of the numerical scheme. Table 2 Illustrates that the intended accuracy is clearly achieved. Finally, three numerical experiments are conducted in quasi-steady and unsteady conditions including slowly moving shocks over flat or discontinuous topography. The higher-order approximate solvers are known to achieve better accuracy for such flows than the first order counterparts. In this paper, second-order well-balanced numerical schemes are developed for the solution of one-dimensional shallow water equations. The approach accurately models different regimes of the flow accurately. The advantage of the proposed scheme over existing higher-order schemes is the fully well-balanced and entropy satisfying properties, where all steady states solutions are exactly preserved.

For a bridge pier in the flow path, a three-dimensional and complex flow pattern is formed around the pier leading to the formation of a scour hole around it. The development of the scour hole will cause the instability of the bridge pier and ultimately the destruction of the pier and the bridge. This problem becomes very important during the floods, when the flow in the river increases rapidly and has the highest potential of destruction. Most studies have investigated scouring in steady flow conditions. The maximum scour depth that occurs under a flood hydrograph can be much smaller than the equilibrium depth resulting from steady flow under peak discharge conditions (Kothyari et al., 1992; Lai et al., 2009). Therefore, the use of flood peak discharge for design can greatly overestimate the maximum scour depth compared to the actual flood conditions (Chang et al., 2004). Considering the importance of scouring investigation in the conditions of unsteady flow and the limited available studies in this regard, more research in this field is necessary. The purpose of this research is to investigate the effect of the time of the rising and falling limbs of the hydrograph, as well as the duration of the hydrograph on the temporal variations of scour depth and its maximum value around the cylindrical pier.

The experiments were carried out in a rectangular flume with glass walls and a straight length of 10 m, a width of 0.74 m and a depth of 0.6 m in the Physical and Hydraulic Modeling Laboratory of Shahid Chamran University, Ahvaz. The test section in the flume was covered with uniform sand with an average size of d50=0.7 mm and geometric standard deviation σ=1.3. In order to achieve the goals of this study, a total of 13 experiments were examined. In order to investigate the effect of the time of the rising and falling limbs of the hydrograph on the temporal variation and the maximum scour depth, a number of 6 hydrographs with a constant duration of 100 minutes and the ratio of the time to reach the peak (Tp) to the duration time (Td) of the hydrograph (skewness) equal to 0.1, 0.2, 0.4, 0.6, 0.8 and 0.9 were designed. Also, 7 hydrographs with Gaussian distribution and duration times (Td) of 10, 20, 45, 80, 100, 120 and 160 were simulated to investigate the effect of flood duration on scouring (Fig. 3 and Table 1). A hydrograph generation system was used to create unsteady flow in the flume. This system included a programmed inverter that was used to adjust the variable flow rate of the hydrograph. The inverter was connected to the pump on one side and to the electromagnetic flow meter on the other side and was run by a computer through a software.

The results of the temporal variations of scouring showed that scouring starts from the sides of the pier and reaches the nose of the pier over time, and finally, the maximum depth of scouring occurs in the nose of the pier. The results showed that the maximum scour depth in the hydrograph with a Gaussian distribution occurs after the peak time (about 10% of duration time) (Fig. 5). Investigating of the effect of the rising limb of the hydrograph showed that in hydrographs with similar duration, the time to reach the peak of the hydrograph has no effect on the maximum scour depth, but it has a significant effect on the temporal changes of shear stress and scour depth. By reducing the time of the rising limb of the hydrograph from 90 minutes to 10 minutes, the shear stress change rate increases 9 times and the scouring rate increases about 6 times. Investigating the effect of the falling limb of the hydrograph on the maximum scour depth showed that the effect of the falling limb on the maximum scour depth increases with the decrease of the time of the rising limb of the hydrograph. The results also showed that the maximum scour depth in hydrographs with skewness of 0.1, 0.2, 0.4, 0.6 and 0.8 is 51.35, 21.28, 12, 5.56 and 1.79 percent more than the scour depth at peak discharge, respectively (Fig. 6). It was also observed that in hydrographs with the same peak time but different duration time, the time of the falling limb is effective on the value of the maximum scour depth. With the increase of 9, 4, and 1.5 times the time of the falling limb of the hydrograph, the maximum scour depth increases by 30.23, 14, and 1.82 percent, respectively. Investigating the duration time of the hydrograph showed that the increase in the duration time increases the depth and dimensions of the scour hole around the pier. It was observed that the maximum scour depth for hydrographs with a duration of 20, 45, 80, 100, 120 and 160 minutes were 19.44, 38.89, 52.78, 58.33, 66.67 and 69.44% more than a hydrograph with a duration of 10 minutes, respectively (Fig. 7). In addition, the slope of the time variations of the scour depth decreases with the increase of the duration time due to the lengthening of the flow rate changes interval along the rising limb of the hydrograph (Fig. 8).

In this study, scouring around a cylindrical pier was investigated under unsteady flow conditions. The results showed that for hydrographs with similar peak discharge and duration time, the time of the rising limb of the hydrograph has a significant effect on the temporal variation of shear stress and scour depth, but it has almost no effect on the maximum scour depth. In addition, it was found that by reducing the time of the rising limb, the influence of the falling limb of the hydrograph on the maximum scour depth increases. Investigating the results of the effect of hydrograph duration time on local scour showed that with the increase of hydrograph duration time, the maximum scour depth increases and the slope of temporal variations of scour depth decreases.

Floods can cause significant damage to goods and people, particularly in densely populated urban areas with high asset values. Flood risk is typically assessed using flow depth, flow velocity, and water level parameters (de Moel et al., 2009). Meja-Morales et al., (2021) investigated the impact of flow exchanges between a porous urban block and surrounding streets and found that porosity significantly affects urban flood flow characteristics. In another study, Meja-Morales et al., (2023) examined the effect of flow instability and open areas in urban blocks on key flood characteristics and reported that the instability level of incoming hydrographs greatly affects the volume of flood water stored in urban blocks. This research aims to evaluate the distribution of flow depth, velocity, and flow patterns in non-porous urban block streets by considering changes in stable inflow. The study seeks to understand multidirectional flow paths caused by the street network and develop a flood risk map for humans using Flow3D software. The validation results of the numerical model showed that the turbulence model had the highest correlation with the laboratory model, with a relative error of 3% and 6.8% for the velocity profile near the water surface and averaged velocity at depth, respectively. In all models, the right and upstream streets had the highest and lowest depth, speed, and human stability number, while the downstream street had the largest range of flood parameters, with 2 to 3 times the average speed and 3 to 4 danger zones for pedestrians. Increasing the flow rate at Inlet 1 for a constant flow rate at inlet 2 increased the flooding characteristics of the right and downstream streets while decreasing the speed in the left street. Conversely, increasing the flow rate at Inlet 2 for a constant flow rate at Inlet 1 increased the flooding characteristics of the left street, decreased the speed in the right and downstream streets, and had minimal effect on the flood characteristics of the upstream street.

As we know, the calculation of hydraulic gradient is highly important in the analysis of steady flow inside rockfill materials. Binomial and exponential relationships are used to calculate the hydraulic gradient based on the non-Darcy flow velocity, and the binomial relationship is more accurate and efficient than the exponential relationship.Since it is necessary to use an exponential relationship in the two-dimensional analysis of non-Darcy flow in coarse porous media, in the past, researchers have provided relationships to calculate the coefficients m and n of the exponential relationship based on the coefficients a and b of the binomial relationship. In some previous studies, Vmax= 1 has been considered, even though the maximum flow velocity depends on the physical characteristics of the pebbles and the characteristics of the flow and is not necessarily equal to one. For this reason, in this research, by designing and equipping the laboratory and recording the laboratory data, the maximum velocity based on the values of a, b and Re of the analytical model of Ahmed and Sunada(1969) is proposed.As mentioned above, various researchers tried to calculate the coefficients of the exponential relationship using the values of a and b in the binomial relationship. One of the most important relationships is presented by George and Hansen (1992) as follows.n=(5a+6bV_max)/(5a+3bV_max ) (1)m=(5a+4bV_max )(4a+3bV_max )/(4(5a+3bV_max ) (V_max )^(n-1) ) (2)Further, by stating that in the coarse-grained porous medium, the slope of the energy line (Sf) is equal to the hydraulic gradient (i), it can be stated that one of the most important parameters in the investigation of the flow in the gravel medium in free flow and under pressure is the calculation of it is a hydraulic gradient. In this research, using the coefficients of the binomial relationship, we presented a solution to calculate the values of m and n in the exponential relationship with better accuracy. Therefore, considering that the exponential relationship is used in the two-dimensional analysis of the non-Darcy flow in porous gravel media, this can play a significant role in reduction of the error of hydraulic gradient calculation.

In the current research, the laboratory data recorded in the hydraulic laboratory of the Faculty of Civil Engineering of Zanjan University were used. For this purpose, an attempt was made to design and set up a test device and perform tests on different gravel materials. Experiments were carried out in a laboratory flume with the ability to tilt, with dimensions of 1m×1m and a length of 15m, and the length of 2.2m of the mentioned flume is filled with rockfill. The walls of the flume are made of plexiglass, and to measure the piezometric height along the porous media, 23 piezometers are used on the bottom of the channel, which are arranged at certain distances from each other and along them. The water flow in the channel is created by a pump with a maximum flow capacity of 90 liters per second. In order to create a porous media, three types of rockfill materials with small, medium and large diameters have been used in the experiments. During the tests, to ensure a stable flow, the pump was working for about 10 minutes with the desired flow and after the stability of the flow, the desired parameters were measured. These parameters include the piezometric height at the location of 23 piezometers as well as the water depth at the location of each piezometer. Piezometric values are read using a calibrated table. The water depth was also measured and recorded directly by a ruler.

Since the exponential relationship is only accurate for a certain range of Reynolds numbers and the user area recommended for this relationship by its providers is only non-quiet flow conditions, therefore, if the exponential relationship is used in the two-dimensional solution of the equations, there will be a large error will enter the calculations. To avoid this problem, various researchers have tried to convert the binomial relationship into an exponential relationship. If the minimum flow velocity Vmin and the maximum flow velocity Vmax in the conversion area of the binomial relationship is in the form of an exponential relationship. In order to convert the two mentioned relations, relations (1) and (2) can be used. According to the conducted tests, in most cases, Vmin is considered zero and Vmax value is assumed to be equal to one, while the maximum flow velocity depends on the physical characteristics of the pebbles and the characteristics of the flow and is not necessarily equal to one. Therefore, Vmax can be calculated from the following relationship according to Ahmed and Sunada's(1969) analytical model and the definition of the Reynolds number as Re=ρVd/μ.V_max=Re_max a/b (3)By using relations (1), (2) and (3), it is possible to take advantage of the accuracy of the binomial relation and the practical property of the exponential relation in the two-dimensional analysis in porous media.If relations (1) and (2) are used in the calculation of the coefficients of the exponential relationship of steady flow in gravel materials, the average relative error between the calculated and recorded hydraulic gradients in the laboratory assuming Vmax=1 (according to previous research) in fine gravel materials, medium and coarse are calculated to be 21.95%, 22.98% and 21.97%, respectively. While if relation (3) is used (the solution presented in the current research), the average relative error values of the hydraulic gradient are equal to 11.39, 14.69 and 19.72%, respectively.

In general terms, by using relations (1) and (2) and using the relation proposed in the present study instead of Vmax=1, the average values of the relative error of the hydraulic gradient in fine, medium and coarse rockfill materials have decreased to 10.56, 8.29 and 2.25%, respectively, which indicates the high accuracy and efficiency of the proposed solution.

Rivers are complex systems in which different chemical, biological, and physical processes occur in it. When the flow moves along the river, there is an exchange between the surface flow and the subsurface flow. The hyporheic zone is a saturation zone below the riverbed, which plays an important role in many biological and chemical processes. Residence time is the most important characteristic of the hyporheic zone. Because the chemical and biological reactions that occur inside the sediments depend on the time at which flow paths remain in the bed for a while and then return back to the surface flow. Hyporheic exchange is the mixing of surface and subsurface flow just beneath the river bed. Such exchanges can be caused by the presence of different bedforms in the river. Dunes are one type of river bed that can be observed in straight, meander, and braided rivers. The pressure gradient between the upstream and downstream of a dune leads to hyporheic exchanges. The wavelength, amplitude, and slope of the lee side and stoss side can affect the rate of exchanges. In the present study, the effect of the dune lee side slope at angles of 10, 20, and 30 degrees on the characteristic of the hyporheic zone (i.e., residence time, exchange flow, and hyporheic depth) has been investigated numerically.

The FLOW3D software is used for the numerical simulation of surface flow. The simulation domain consists of a flume with 2.7m length, 0.1m width, and 0.3m height. The model running time was 120 seconds for surface flow simulation, which, with the passing of this time, the flow in the channel becomes stable. The pressures along dunes are introduced as a Dirichlet boundary condition on top of the groundwater model, i.e., MODFLOW. Then, the effect of the dune lee side slope at angles of 10, 20, and 30 degrees on the characteristic of the hyporheic zone (i.e., residence time, exchange flow, and hyporheic depth) has been investigated.

The results show that the maximum and minimum pressure occurred on the stoss side and the crest of the dune, respectively. By increasing the dune lee side slope, the distance between the maximum and minimum pressure is reduced, the depth of hyporheic exchange decreases, and the exchange rate and residence time increase. Also, for all three angles, with a constant ratio of the subsurface to surface flow, the depth of hyporheic exchange increases with the increase of the hydraulic conductivity to the dune length ratio (K/A). Increasing the velocity of the subsurface flow causes the subsurface flow to dominate the surface flow and the flow in the subsurface flow moves towards the surface flow. As a result, by increasing the ratio of subsurface flow velocity to surface flow velocity, the exchange flow increases, and the depth of hyporheic exchange decreases.

The results show that as the lee side slope increases, the residence time, and exchange flow increase, and hyporheic depth decreases. Also, by increasing the hydraulic conductivity, the hyporheic exchange depth increases, but by increasing the subsurface flow velocity and the porous media thickness, the hyporheic exchange depth decreases.

One of the important challenges in investigating the hydraulic, morphological and ecological behavior of rivers is to accurately determine the geometrical dimensions of river bed particles. The information extracted from the grain size curve of river bed particles has many applications in the field of river engineering, such as modeling of sediment transport, changes in sediment deposition or river bed erosion, and changes in river morphology (Hasannejad Sharifi et al., 2015).Nowadays, despite the fact that determining the granularity of the bed particles and the boundaries of rivers is important, the removal of sediments from the natural environment leads to disturbing the sedimentary bed and causes changes in the river system (Sadeghi and Qara Mahmoudoli, 2013). Therefore, it is very important to use a method that can calculate the results of granulation with less cost and quickly. Image processing methods are known as new method have been taken into consideration in order to increase rapidity and accuracy, along with the method of field measurement of granularity. In this research, by examining the morphological and sedimentary conditions of the rivers of the Zanjan province, as well as the ease of access, Zanjanrud river was selected as the study area. Then three cross-sections for sampling with frame dimensions of 20×20 cm were determined from the bed of Zanjanrud river and after photographing the surface of the samples, painting by spray paint in order to identify the surface grains, sampling was done from the determined sections. The samples were taken to the laboratory and the sieve test was performed. Then, in the laboratory environment, each of the samples was arranged in a metal frame with dimensions of 20×20 cm and was installed in a channel with a length of 8 meters and a width of 20 cm. By applying the conditions of stagnant water at three depths of 4, 8 and 12 cm and water flow with different depths and velocities, the samples were re-photographed and the grain size curve were obtained by Image J and Hydraulic Toolbox softwares. Based on the results of Sieve, in sample A, almost 85% of the particles have a diameter equal to and less than 31.7 mm. In sample B, almost 85% of the particles passed through the sieve of 1.38 mm. In sample C, more than 97% of the particles passed through the 38.1 mm sieve. According to the classification system of the United States, the sediments of all three sections of Zanjanrud are gravel. Also, according to the results, it can be seen that the standard deviation of the samples is less than 2, the uniformity coefficient is less than 4, and the sorting coefficient is less than 2, which shows that the examined samples are almost uniform. In sample A, atdepth of 8 cm, the average relative absolute error values were obtained similar to those at depth of 4 cm. In sample B, the average error value of the fine grains is the same \coarse-grain. The reason for this can be due to the low value of the standard deviation of this sample compared to the other two samples. The average error value at depth of 8 cm was found to be 14% for fine grains and 12% for coarse grains. In sample C, similar to sample A, the average error value in the fine grains is less compared to the coarse grains.The average relative error at depth of 8 cm was found to be 9% for fine grains and 14% for coarse grains. Results showed that Image J software has better performance than Hydraulic Toolbox software in determining the particles granulation curve of the Zanjanrud river.The effect of flow velocity and depth on the K values of the index diameter of 50% showed that at depth of 4 cm, the average values of K in samples A, B, and C are 1.01, 1.19, and 1.08, respectively. Also, at depth of 8 cm, the values of K for samples A, B, C are equal to 1.05, 1.12, and 1.11, respectively, and at depth of 12 cm,there are equal to 1.08, 1.18, and 1.13, respectively. According to these results, it can be seen that the K values of sample A are lower than the other two samples in all three depths of 4, 8, and 12 cm. At depth of 8 cm, the K values of sample B at different velocities coincide approximately with those of sample. Also, the K values of sample C at two depths of 4 and 12 cm are lower than the K values of sample B. The average K values of the 50% index diameter at three depths of 4, 8, and 12 of sample A are 1.01, 1.05, and 1.08, respectively (with an average value of 1.046); the mean K values of sample B are 1.19, 1.12 and 1.18 respectively (with an average value of 1.16) and also the mean K values of sample C are 1.08, 1.11 and 1.13 respectively (with an average value of 1.1) (with a total average of 1.1 for all three samples). In most of the hydraulic flow conditions, the granulation curve obtained by Image J software is close to the sieve granulation curve. In contrast, the Hydraulic Toolbox granulation curve is significantly different from the sieve curve and the curve of Image J software. The flow velocity and flow depth does not have a noticeable effect on the K values of the index diameter of 50%.

The use of modern methods of recycling effluent with high treatment efficiency in compare to the traditional methods (the use of sedimentation ponds) is of great importance. One of the modern methods of treating these effluents is the use of filtering presses. A newer method is using hydro cyclones as a result of centrifugal hydrodynamic forces for separating solid particles from the effluent, e.g., stone powder in the current research, (Naderi et al.2019). In this regards, many researches have been done in the field of separation of two different phases of fluid using hydro cyclone in different hydraulic conditions and geometric parameters.

Materials used in research
The materials used in this research were solid particles in the wastewater of the stone cutting factory of Mahmoudabad industrial town of Isfahan. The resulting slurry was collected in the factory and after drying and separation was used for the experiments. In this study, Granite and Marble stone powder with a density of 2750 and 3020 kgm-3 and a particle diameter distribution less than 600µm was used. Also, the effect of hydraulic parameters such as flow division ratio (under flow discharge to inlet flow discharge), wastewater weight concentration and inlet pressure on hydro cyclone performance with a cyclone diameter of 140 mm in the separation of Granite and Marble powder stone from wastewater was investigated.Design of experiments Experiments were designed by Design Expert.10 software with response level method (RSM). In conducting experiments, first, a solution of Granite and Marble powder stone with the desired concentrations (0.64 to 7.36% by weight) in a volume of 20 liters was prepared in a tank, and then this solution was circulated to the hydro cyclone by pressure with a centrifugal pump.In each experiment, inlet perssure and overflow pressure were recorded using pressure guages and the volumetric flow rate from overflow and underflow was measured. Samples of overflow and underflow were taken with a specified volume and then stored and dried in an oven at 105ºC for 24 hours. The residual dry mattter mass was measured from the overflow and underflow samples, and the solid particle separation efficiency was computed from Eq. (1):E=M_u/M×100 (1) In this Equation, Mu is mass per unit volume of Granite and Marble stone powder inside the underflow (grl-1); M is total mass per unit volume of examined Granite and Marble stone powder in hydro cyclone inlet (total overflow and underflow; grl-1) and E is the efficiency of separation of solid particles from the feed.

Influence of operating parameters of weight concentration of the feed (Cw), inlet pressure of the hydro cyclone (Pi), and the Ratio of outlet flow) under flow) from the discharge to inlet flow to the hydro cyclone (Rf) on the efficiency of the examined hydro cyclone is given in the form of Eq. (2) for Granite powder stone and Eq. (3) for Marble powder stone.(2) Efficiency=93.16-1.01R_f-4.36C_w+10.01P_i+0.04R_f C_w-0.04R_f P_i+0.96 C_w P_i+8.12E-003*〖R^2〗_f -0.08*〖C^2〗_w-2.41* P_i ²(3) Efficiency=116.26-2.11R_f-9.13C_w+6.22 P_i+0.11R_f C_w+0.17R_f P_i+1.22 C_w P_i+5.54E-003*〖R^2〗_f+0.12*〖C^2〗_w-2.55* P_i ²In order to validate the obtained model, two normal probability diagram comparing the predicted data of the model with the real values were used. The results showed that with decreasing flow distribution ratio, particle separation efficiency increases, also separation efficiency decreased with increasing inlet pressure, and with increasing effluent concentration.

In this study, the effect of hydraulic parameters including flow separation ratio, inlet pressure and feed concentration was investigated on the performance of a semi industrial hydro cyclone. The analysis of the results using Response Surface Methodology (RSM) showed the effectiveness of the statistical method for obtain the optimum operational condition of the examined hydro cyclone. Based on the RSM method, the highest separation efficiency was obtained 86 and 88%, for Granite and Marble powder stone respectively.

In natural rivers and certain urban channels, when flood events occur, the flow diverts from the main channel and inundates the surrounding floodplains. These particular configurations are termed compound channels. Floodplains can exhibit both symmetrical and asymmetrical patterns. Depending on geometric factors, lateral slope, and differences in elevation, multiple floodplains can manifest on each side of the main channel. These intricate structures are known as multi-stage compound channels. Multi-stage compound channels not only possess enhanced flow conveyance capabilities compared to simpler classic compound channels but also their second or third floodplains may offer prospects for recreational utilization or landscape enhancement within urban settings. Historically, the examination of flow parameters and the computation of conveyance capacities for compound sections were carried out using conventional methodologies, such as divided channel methods and traditional flow resistance equations like Manning, Chezy, Darcy-Weisbach, and others. However, these approaches often disregarded the momentum exchange arising from interactions between the main channel and floodplains, as well as the impact of secondary flows. Consequently, the estimated flow rates for compound sections tended to be higher than actual values. Sellin (1964) played a pioneering role in acknowledging the interaction between the main channel and floodplains, laying the groundwork for subsequent investigations into the evolution of conventional techniques. In contrast to the extensive research on classic compound channels, multi-stage compound channels have received limited attention and exploration in the scientific literature.

In this research, a numerical solution of the Navier-Stokes equations using the finite volume method and The RNG k-ε turbulence model has been employed to simulate various hydraulic characteristics of multi-stage compound channels. These characteristics encompass the three-dimensional flow pattern, distribution of transverse velocity, secondary flows, turbulence energy, and the stage-discharge relationship. The RNG k-ε turbulence model is adept at reproducing rotational flows and large vortices, addressing the limitations of the standard k-ε model in representing non-circular channels at corner locations and rotational flows.To verify the validity of this mathematical model, laboratory data obtained from a channel with a three-stage asymmetric rectangular compound section (Singh, 2021) were utilized. The experimentation carried out in a channel featuring a main section width of 0.445 meters. On one side of the main section, two floodplains of widths 10 and 20 centimeters were established. The bed of the main channel was constructed using glass, while the bed of the first and second floodplains were covered with a uniform layer of syntetic grass to introduce channel roughness.The channel itself is 20 meters in length, with a longitudinal slope of 0.003. The total height and width of the channel are 0.5 and 0.745 meters, respectively. The bankful height is set at 0.0425 meters. The flow rate within this channel varies between 20 and 60 liters per second.

Overall, the comparison of the three-dimensional model's computational results with experimental data in terms of the positions and values of maximum and minimum velocities indicates the satisfactory accuracy of the proposed mathematical model in this study.The turbulence intensity and momentum exchange at the interface between the first and second floodplains are lower compared to the interface between the main channel and the first floodplain. This discrepancy is attributed to the greater velocity difference at the interface plane of the main channel and the first floodplain. The influence of secondary currents at the main channel and first floodplain interface diminishes as the water level rises. However, significant secondary currents persist at the boundary between the first and second floodplains across all investigated relative depths. This underscores the significance of flow dispersion in contrast to convection in the second floodplain, particularly in cases of shallow relative depths.The highest flow velocity is observed at the midpoint of the main channel, inclined toward its right wall, situated far from the channel bed and close to the water surface. The computed transverse profiles of stream-wise velocity are satisfactorily accurate in both the main channel and floodplains (especially the second floodplain). Nevertheless, the modeling error is relatively notable at the interface of the main channel and the first floodplain. Predicting flow discharge for this channel using the mathematical model yields an average error of approximately 3.9% and a maximum error of 6.2%.

Due to the lack of experimental data on height and width variations of the second floodplain and their impact on flow characteristics, expressing the effects of these conditions is challenging. Further research involving precise laboratory measurements is required to comprehensively understand the influence of these changes.Considering that one of the applications of multi-stage compound channel is in urban areas and the first floodplain has a smaller width and is designed with the aim of increasing the channel conveyance capacity, and the second floodplain is intended to beautify the urban landscape and use it as a recreational and tourism environment. Therefore, it usually has tree vegetation. In addition to creating high shear stresses in the bed of the second floodplain, this causes high energy loss and a significant decrease in the conveyance capacity of the multi-stage compound channel. Therefore, it is recommended to design rivers or manmade flood control channels in the urban areas in the form of multi-stage compound channels so that the first floodplain is to be significantly increase the conveyance capacity of the channel and the second floodplain is for urban and public landscapes. This provides nearby people to escape from the danger zone during severe urban floods.