- Email: [email protected]
Numerical simulation of atomization rainfall and the generated flow on a slope
273
2012,24(2):273-279 DOI: 10.1016/S1001-6058(11)60244-8
NUMERICAL SIMULATION OF ATOMIZATION RAINFALL AND THE GENERATED FLOW ON A SLOPE* LIU Shi-he, TAI Wei, FAN Min State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: [email protected] LUO Qiu-shi Yellow River Engineering Consulting Co. Ltd, Zhengzhou 450003, China
(Received November 28, 2011, Revised March 6, 2012) Abstract: This article studies the atomization rainfall and the generated flow on a slope by numerical simulations. The atomization rainfall is simulated by a unified model for splash droplets and a suspended mist, and the distribution of the diameter of splash rain drops is analyzed. The slope runoff generated by the atomization rainfall is simulated by a depth-averaged 2-D model, and the localization of the rainfall intensity in space is specially considered. The simulation results show that: (1) the median rain size of the atomization rainfall increases in the longitudinal direction at first, then monotonously decreases, and the maximum value is taken at the longitudinal position not in consistent with the position where the maximum rain intensity is taken. In the lateral direction the median rain size monotonously decreases, (2) since the atomization rainfall is distributed in a strongly localized area, it takes a longer time for its runoff yield to reach a steady state than that in the natural rainfall, the variation ranges of the water depth and the velocity in the longitudinal and lateral directions are larger than those in the natural rainfall. Key words: atomized flow, unified model, slope runoff, numerical simulation
Introduction When the aerated jet shooting from a water release structure in high dams impacts against the water surface downstream, a large amount of water drops in the outer edge would be generated and then be thrown away, to form an atomized flow with unnatural and high-density rains as well as an unnatural mist. This atomized flow may threaten the safety of the normal operation of hydropower stations and the stability of the downstream river banks, and sometimes even affect the traffic safety, the production and the daily life near the dam site. Similar to the slope runoff generated by the natural rainfall, the slope runoff would also be generated when the water drops of the atomization rainfall accumulate on the slope, with the difference that the slope runoff generated by the
* Project supported by the Ministry of Water Resources special funds for Scientific Research on Public Causes (Grant No. 201101005). Biography: LIU Shi-he (1962-), Male, Ph. D., Professor
atomization rainfall is more strongly affected by the non-uniform distribution of the rainfall intensity, since the atomization rainfall is distributed in a strongly localized area with a great variation in space. Therefore, it would be very important to predict the atomization rainfall and to simulate the associated slope runoff in view of the downstream river bank stability. The prediction of the atomization rainfall was much studied through prototype observations, numerical simulations and model tests at different levels. For example, the characteristics of the jet flow was investigated in Ref.[1,2]. Sun et al.[3] and Liu et al.[4] studied the longitudinal variation of the rain intensity, and Liu et al.[5] proposed a unified model of the atomized flow to predict the rainfall intensity produced by the splash water drops and the suspended mist. So far the investigations of the atomized flow are mainly focused on how to determine the rainfall range and the rainfall intensity, with few results for the distribution of the radius of the atomized rain drops, but which is essential for a clear understanding of the slope runoff. There were many studies of the slope runoff generated by the natural rainfall. Among the experi-
274
mental studies, Zhang[6] studied the hydraulic characteristics of the slope runoff produced by the upstream inflow, Zhao et al.[7] studied the hydraulic characteristics of the slope runoff produced by heavy rains. In the theoretical analyses and numerical simulations, the slope runoff is often described by a kinematical wave model[8], a diffusion wave model or Saint Venant equations, and the coupling between the surface and subsurface flows is considered in the numerical simulation. For example, Zhang et al.[9] developed a coupled model by using a kinematical wave model to describe the surface flow, and the 2-D Richards equation to describe the subsurface flow, and Tung et al.[10] further developed the coupled model by using Saint Venant equations to describe the surface flow during the transient seepage analysis of a rainfall-infiltrated slope. There are a lot of experimental results available for the slope runoff, and a comprehensive analysis of the experimental data is important. Furthermore, the determination of the coefficients in the mathematical model, especially, the drag coefficient, is needed in an optimization consideration. Therefore, this article studies the distribution of the radius of atomized rain drops by a unified model of the atomized flow, and the generated slope runoff is simulated by a depthaveraged 2-D model, to get a clear understanding of the composition of the atomized rain drops and the generated flows, so as to provide a better technical guidance for taking engineering measures to avoid the harms of the atomized flow.
source is considered. Water is easy to deform, thence a series of deformations would happen during the impact of water drops against the downstream water surface, before splashed droplets are generated at last which jump from the downstream water surface with the initial velocity us , the angles T s (the angle between the initial velocity and the plane x1ox3 )and Is (the angle between the projection of the initial velocity in x1ox3 with the x1 axis). While a splashed water droplet leaves the downstream water surface, it moves under the action of the gravitational force, the buoyant force and the wind resistance. Let the velocity of the water droplet be u pi , then the governing equations for the splashed droplet with the diameter d p and the density U w , moving in the velocity field {u fi } =
{uw , 0, 0} of the jet driven wind with the density Ua are d u pi dt
§ U · 3 U a CD u p u f (u pi u fi ) ¨1 a ¸ gi (1) 4 Uw d p © Uw ¹
The governing equations for the position x pi of water droplets are d x pi dt
1. Simulation of atomized flow 1.1 Mathematical model and numerical method The water droplets in the atomized flow may move in two different states, those of a large geometrical size move in a splashed state, with the related source being called the splashed source, on the other hand, the water droplets with small geometrical size move in a suspended state, with the related source being called the suspended source. In Ref.[5] a unified model was developed for the atomized flow, in which the water droplets moving in the splashed state is described by a random simulation based on the Lagrangian description, and the water droplets moving in the suspended state is described by the theory of the air-water two-phase flow in the continuum mechanics based on the Eulerian description, and the demarcation diameter d m between the splashed and suspended states is determined similar to the classification of the bed load and the suspended sediment motion in the theory of river dynamics. Since in this article’s verification of the mathematical model, the jet velocity is about 6 m/s, the suspended source is very small and can be omitted[11], only the method to calculate the atomization rainfall generated by the splashed
=
(2)
= u pi
where CD is a coefficient depending on the relative Reynolds number § up u f d p · ¸ Red = ¨ ¨ ¸ Qa © ¹
and u p u f
is the relative velocity between the
water droplet and the jet driven wind. The initial velocities of the water droplet are
^u ` = ^u cosT cosI , u sin T , u cosT sin I ` (3) pi t =0
s
s
s
s
s
s
s
s
In the random model of this article, the probability distributions of d p , us , T s and Is are as follows[5,12]: d p obeys the * distribution, us and
T s obey the Weibull distribution, and Is obeys the normal distribution. The initial position of the water droplet is just at the outer edge of the aerated jet as it impacts with the downstream water surface. In this
275
article, the randomness of the initial position of water droplets is described by a uniform distribution at the outer edge of the aerated jet as they just impact with the downstream water surface. The solution steps are as follows: (1) Generate the water droplet with diameter d p . (2) Generate the initial velocity us and the splashing angles T s and Is of the water droplet. (3) Generate the initial position of the water droplet, and solve Eq.(1) by the fourth-order Runge-Kutta algorithm to get the velocity of the water droplet, the trajectories of the water droplet can be obtained by further numerical integration of the velocity, then the position where the water droplet falls into the downstream boundary (water surface or land) can be determined. Repeat the above process of random simulation until the simulating number of water droplets is N . The probability density function of the radius of the atomization rain drop, and the rainfall intensity P and other statistical quantities at the downstream boundary can be obtained by the statistical average of the random simulation results.
long and 1.3 m wide. While this jet impinges with the water surface of the pond, the atomized flow is generated. The rainfall intensity was measured by a graduated cylinder, supplemented with several transducers of electric balance. The diameters of the rain drops were measured by using the method of color speck[13]. The comparisons between the random simulation results and the experimental results of the rainfall intensity distribution in the longitudinal and lateral directions, the probability density function of the diameter of the atomization rain drops in the position of x1 = 0 m , x3 = 0.9 m are shown in Fig.1 and Fig.2, respectively. The two results are in close agreement with each other.
Fig.2 The probability density function of the diameter of atomization rain drops
Fig.1(a) The rainfall intensity distribution in longitudinal direction Fig.3 Variations of the median rain size in the longitudinal and lateral directions
Figure 3 also shows the variations of the median rain size d 50 in the longitudinal and lateral directions. It is seen that the random simulation results and the experimental results are also in agreement with each other, and the variation of d 50 has the following characteristics: (1) In the longitudinal direction, d 50 inFig.1(b) The rainfall intensity distribution in lateral direction
1.2 Verification The experimental results in Ref.[11] are used for verification. The experiment was conducted in a special equipment in Wuhan University. The high-speed aerated jet was generated by using a nozzle linked with a submersible pump installed on a pond of 2.3 m
creases with x1 at first, then monotonously decreases, and the longitudinal position for d50 to reach its maximum value is not in consistent with the position where the maximum rain intensity is reached. (2) In the lateral direction d 50 monotonously decreases with x3 .
276
2. Simulation of slope runoff generated by atomization rain drops 2.1 Mathematical model and numerical scheme The study of the slope runoff in hydrology is mostly based on the water balance to obtain the quantity of water in surface and subsurface flows, yet in the geotechnical engineering more attentions are paid to the change of the pore water pressure and the associated slope stability. In this article, the coupling between surface and subsurface water flows is not considered duo to the limited space of the article, and the special attentions are paid to the difference of the slope runoff due to the natural rainfall and the atomization rainfall. 2.1.1 Mathematical model The depth-averaged 2-D model is used to describe the slope runoff. The governing equations for the slope runoff on a slope with tilt angle T in the Cartesian coordinate system are wU wF w G + + =S wt w x w y
(4)
where uH § § H · ¨ g H 2 cos T ¨ ¸ U = ¨ u H ¸ , F = ¨ u2 H + ¨ 2 ¨ vH ¸ ¨ © ¹ uvH ©
· ¸ ¸, ¸ ¸ ¹
§ · ¨ ¸ vH ¨ ¸ G=¨ uvH ¸, ¨ ¸ 2 ¨¨ v 2 H + g H cos T ¸¸ 2 © ¹ R wT § · ¨ ¸ S = ¨ g H sin T 0.125O u 2 + v 2 u ¸ ¨¨ ¸¸ 0.125O u 2 + v 2 v © ¹
and H is the water depth, u and v are the depthaveraged velocities in x and y directions, respectively, O and g are the drag coefficient and the gravity acceleration, respectively, RwT is the atomization rainfall intensity. In Eq.(4), the static pressure is corrected as g H cos T for the steep slope runoff. The boundary conditions are as follows: the water depth is zero at the outer edge of the atomized flow on the slope, the flow is fully developed at the base of the slope.
2.1.2 Numerical scheme The control volume method is used for the discretization of Eq.(4). The discrete form can be obtained by choosing the rectangular mesh as the control volume and integrating Eq.(4) over the control volume similar to the procedures in Ref.[14,15], which is wU i 1 = wt Ai
N ed
¦E n
ij
' * ij + Si
(5)
j =1
where i and j denote the i th cell and the j th edge of the cell, respectively, U i is the average quantities stored at the center of the i th cell, nij is the edge, '* ij
unit outward normal vector at the j th
is the length of the j th edge, and E is the numerical flux through the edge. In the simulation of the slope runoff, the treatment of wet/dry boundaries is very important, therefore, the HLL Riemann solver is employed for the calculation of the normal flux E nij at the face of a control volume, which makes it possible to handle discontinuities. Using the first order accuracy scheme to calculate the time dependent terms, the discrete form of Eq.(5) becomes
U
m +1 i
§ 1 = U + 't ¨ © Ai
· E nij ' * ij + Si ¸ ¦ j =1 ¹ N ed
m i
m
(6)
2.2 Determination of key parameters In the mathematical model of Eq.(4), the drag coefficient is the key parameter, which is determined by a theoretical analysis and the experimental results. The slope runoff is one kind of thin sheet flows, if the theory of the steady uniform flow is used to describe it approximately in a short segment, the discharge per unit width becomes q = n 1 H 5 / 3 sin T , there-
fore H = n / sin T
3/ 5
q 3 / 5 , U = n / sin T
3 / 5
q2 / 5 ,
and the drag coefficient becomes
O=
8 g H sin T = 8 g n9 / 5 (sin T )1/10 q 1/ 5 U2
(7)
Zhang[6] carried out experiments for the variation of the drag coefficient against the discharge per unit width. In the experiments, the slope runoff was generated by the upstream inflow in a channel, and the experimental soil sample was adhered to the bottom of the channel. The water depth was measured by point gauges, and the velocity was measured by the dyeing method. Zhao et al.[7] also carried out experiments for the variation of the water depth, the velocity and the
277
discharge per unit width in a channel, unlike the experiments by Zhang[6], the slope runoff was generated by the rainfall, and the bottom of the channel is smooth and impermeable. In the experiments, the rainfall was generated by an automatically controlling system, and the flow velocity was measured by the method of electric pulses. Although the flow generated by the rainfall is a spatially varied flow, the characteristics of this flow in a local area are similar to those of the flow generated by an upstream inflow. Figure 4 shows the experimental results in Ref.[6,7] and Eq.(7) for the variation of the drag coefficient O c ( = O [8gn9 / 5
the natural rainfall to reach the steady state is about 400 s, while for the atomization rainfall, the duration time increases to 800 s since the distribution of the rainfall intensity is non-uniform, 2) in the initial stage of the runoff producing process, the discharge calculated by the kinematical wave model[8] is relatively larger than that by Eq.(4), yet the difference of the two results is very small, which seems that the kinematical wave model enjoys relatively high accuracy for the calculation of the runoff yield either by the natural rainfall or the atomization rainfall.
< (sin T )1/10 ]1 ) against the discharge per unit width. It can be seen from Fig.4 that the experimental results and the analytical results are in good agreement with each other. Therefore in this article, Eq.(7) is used for the determination of the drag coefficient.
Fig.5 Variation of the runoff yield with time at the base of the slope
Fig.4 Variation of the drag coefficient against discharge per unit width
2.3 Numerical simulation results 2.3.1 Computational domain The computational domain is 50 m long and 50 m wide, with the tilt angle of 30o. In the computation, the grid sizes in the longitudinal and lateral directions are all 1 m. The runoff producing processes for the natural rainfall and the atomization rainfall are simulated separately, and the averaged rainfall intensity in the computational domain is 0.02 m/h. The rainfall intensity for the natural rain is uniformly distributed in the computational domain, yet for the atomization rain, the rainfall intensity distribution of Fig.1 is used instead. 2.3.2 Analyses of the simulation results (1) Runoff producing process The slope runoff would gradually reach a steady state of a spatially varied flow while the rainfall duration is longer than the concentration time. Figure 5 shows the results for the variation of the discharge at the base of the slope against time in the natural and atomization rainfall conditions . It can be seen from this figure that: 1) the discharge at the base of the slope gradually increases from zero to a time independent value as the rain goes on, the duration time for
Fig.6(a) Variation of water depth in the longitudinal direction
Fig.6(b) Variation of water depth in the lateral direction at the base of the slope
(2) Variation of water depth Figure 6 shows the variation of water depth in the longitudinal and lateral directions for the steady slope runoff. It can be seen from this figure that: 1) the water depth increases in the longitudinal direction
278
owing to the rainfall supplement, with the variation range for the atomization rainfall greater than that for the natural rainfall, 2) the distribution of water depth at the base of the slope in the lateral direction is uniform for the natural rainfall, while for the atomization rainfall, this distribution is non-uniform, and the water depth increases in the area with the increase of the rainfall intensity.
the position to reach the maximum rain intensity. In the lateral direction the median rain size would monotonously decrease. (2) Since the atomization rainfall is distributed in a strongly localized area, the duration taken for its runoff yield to reach a steady state is longer than that for the natural rain, the variation ranges of the water depth and the velocity in the longitudinal and lateral directions are larger than those for the natural rainfall. References [1]
[2]
[3] Fig.7(a) Variation of flow velocity in the longitudinal direction
[4]
[5]
[6] Fig.7(b) Variation of flow velocity in the lateral direction at the base of the slope [7]
(3) Variation of velocity Figure 7 shows the variation of the velocity in the longitudinal and lateral directions for the steady slope runoff. It can be seen from this figure that: 1) the velocity increases in the longitudinal direction with the increase of the rainfall, and the variation range of the velocity for the atomization rainfall is larger than that for the natural rainfall, 2) the distribution of the velocity at the base of the slope in the lateral direction is uniform for the natural rainfall, while for the atomized flow rainfall, this distribution is non-uniform, and the velocity also increases in the area with the increase of the rainfall intensity, similar to the variation of the water depth.
3. Conclusions In this article, the unified model and the depthaveraged 2-D model are used to simulate the atomized flow and its generated flow on a slope. The conclusions are as follows: (1) The median rain size in the atomization rainfall increases in the longitudinal direction at first, then monotonously decreases, and the longitudinal position to reach its maximum value is not in consistent with
[8]
[9]
[10]
[11]
[12]
[13]
ZHOU Hua. Numerical analysis of the 3-D flow field of pressure atomizers with v-shaped cut at oriflce[J]. Journal of Hydrodynamics, 2011, 23(2): 187-192. XIAO Yang, TANG Hong-wu. Numerical study of hydrodynamics of multiple tandem jet in cross flow[J]. Journal of Hydrodynamics, 2011, 23(6): 806-813. SUN Shuang-ke, LIU Zhi-pin. Longitudinal range of atomized flow forming by discharge of spillways and oulet works in hydropower stations[J]. Journal of Hydraulic Engineering, 2003, (12): 53-57(in Chinese). LIU Hai-tao, SUN Shuang-ke and LIU Zhi-ping. Mathematical model for rain-fog transportation during flood discharge and its verification[J]. Advances in Science and Technology of Water Resources, 2011, 31(4): 4549(in Chinese). LIU Shi-he, SUN Xiao-fei and LUO Jing. Unified model for splash droplets and suspended mist of atomized flow[J]. Journal of Hydrodynamics, 2008, 20(1): 125-130. ZHANG Guang-hui. Study on hydraulic properties of shallow flow[J]. Advances in Water Science, 2002, 13(2): 159-165(in Chinese). ZHAO Xiao-e, WEI Lin and CAO Shu-you et al. Study on characteristics of overland flow with higher rain intensity[J]. Journal of Soil and Water Conservation, 2009, 23(6): 45-47(in Chinese). LI Zhan-bin, LU Ke-xin. Approximate analytical solution of rainfall process on permeable slope[J]. Journal of Hydraulic Engineering, 2003, (6): 8-13(in Chinese). ZHANG Pei-wen, LIU De-fu and ZHENG Hong et al. Coupling numerical simulation of slope runoff and infiltration under rainfall conditions[J]. Rock and Soil Mechanics, 2004, 25(1): 109-113(in Chinese). TUNG Yeou-koung, KWOK Yih-feng and NG Wang Wai Charles et al. A preliminary study of rainfall infiltration on slope using a new coupled surface and subsurface flow model[J]. Rock and Soil Mechanics, 2004, 25(9): 1347-1352(in Chinese). FAN Min, LIU Shi-he and ZHANG Kang-le. Experimental study and numerical simulation of the distribution of splashed raindrop size in atomized flow[J]. Proceedings of the 5th National Symposium on Hydraulics and Hydraulic Information. Xi’an, 322-327(in Chinese). LIU Shi-he, LIU Jiang and LUO Qiu-shi et al. Engineering turbulence[M]. Beijing: Science Press, 2010(in Chinese). LIAO Wei, WEI Miao-miao and HUANG Yu-yu. Research on raindrop diameter based on filter paper splash procedure[J]. Journal of Wuhan University of Technology (Transportation Science and Engineering), 2008, 32(6): 1165-1168(in Chinese).
279
[14]
MORITA M., YEN B. C. Modeling of conjunctive twodimensional surface–three-dimensional subsurface flows[J]. Journal of Hydraulic Engineering, 2002, 128(2): 184-201.
[15]
YOON Tae Hoon, ASCE F. and KANG Seok-Koo. Finite volume model for two-dimensional shallow water flows on unstructured grid[J]. Journal of Hydraulic Engineering, 2004, 130(7): 678-688.
2012,24(2):273-279 DOI: 10.1016/S1001-6058(11)60244-8
NUMERICAL SIMULATION OF ATOMIZATION RAINFALL AND THE GENERATED FLOW ON A SLOPE* LIU Shi-he, TAI Wei, FAN Min State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: [email protected] LUO Qiu-shi Yellow River Engineering Consulting Co. Ltd, Zhengzhou 450003, China
(Received November 28, 2011, Revised March 6, 2012) Abstract: This article studies the atomization rainfall and the generated flow on a slope by numerical simulations. The atomization rainfall is simulated by a unified model for splash droplets and a suspended mist, and the distribution of the diameter of splash rain drops is analyzed. The slope runoff generated by the atomization rainfall is simulated by a depth-averaged 2-D model, and the localization of the rainfall intensity in space is specially considered. The simulation results show that: (1) the median rain size of the atomization rainfall increases in the longitudinal direction at first, then monotonously decreases, and the maximum value is taken at the longitudinal position not in consistent with the position where the maximum rain intensity is taken. In the lateral direction the median rain size monotonously decreases, (2) since the atomization rainfall is distributed in a strongly localized area, it takes a longer time for its runoff yield to reach a steady state than that in the natural rainfall, the variation ranges of the water depth and the velocity in the longitudinal and lateral directions are larger than those in the natural rainfall. Key words: atomized flow, unified model, slope runoff, numerical simulation
Introduction When the aerated jet shooting from a water release structure in high dams impacts against the water surface downstream, a large amount of water drops in the outer edge would be generated and then be thrown away, to form an atomized flow with unnatural and high-density rains as well as an unnatural mist. This atomized flow may threaten the safety of the normal operation of hydropower stations and the stability of the downstream river banks, and sometimes even affect the traffic safety, the production and the daily life near the dam site. Similar to the slope runoff generated by the natural rainfall, the slope runoff would also be generated when the water drops of the atomization rainfall accumulate on the slope, with the difference that the slope runoff generated by the
* Project supported by the Ministry of Water Resources special funds for Scientific Research on Public Causes (Grant No. 201101005). Biography: LIU Shi-he (1962-), Male, Ph. D., Professor
atomization rainfall is more strongly affected by the non-uniform distribution of the rainfall intensity, since the atomization rainfall is distributed in a strongly localized area with a great variation in space. Therefore, it would be very important to predict the atomization rainfall and to simulate the associated slope runoff in view of the downstream river bank stability. The prediction of the atomization rainfall was much studied through prototype observations, numerical simulations and model tests at different levels. For example, the characteristics of the jet flow was investigated in Ref.[1,2]. Sun et al.[3] and Liu et al.[4] studied the longitudinal variation of the rain intensity, and Liu et al.[5] proposed a unified model of the atomized flow to predict the rainfall intensity produced by the splash water drops and the suspended mist. So far the investigations of the atomized flow are mainly focused on how to determine the rainfall range and the rainfall intensity, with few results for the distribution of the radius of the atomized rain drops, but which is essential for a clear understanding of the slope runoff. There were many studies of the slope runoff generated by the natural rainfall. Among the experi-
274
mental studies, Zhang[6] studied the hydraulic characteristics of the slope runoff produced by the upstream inflow, Zhao et al.[7] studied the hydraulic characteristics of the slope runoff produced by heavy rains. In the theoretical analyses and numerical simulations, the slope runoff is often described by a kinematical wave model[8], a diffusion wave model or Saint Venant equations, and the coupling between the surface and subsurface flows is considered in the numerical simulation. For example, Zhang et al.[9] developed a coupled model by using a kinematical wave model to describe the surface flow, and the 2-D Richards equation to describe the subsurface flow, and Tung et al.[10] further developed the coupled model by using Saint Venant equations to describe the surface flow during the transient seepage analysis of a rainfall-infiltrated slope. There are a lot of experimental results available for the slope runoff, and a comprehensive analysis of the experimental data is important. Furthermore, the determination of the coefficients in the mathematical model, especially, the drag coefficient, is needed in an optimization consideration. Therefore, this article studies the distribution of the radius of atomized rain drops by a unified model of the atomized flow, and the generated slope runoff is simulated by a depthaveraged 2-D model, to get a clear understanding of the composition of the atomized rain drops and the generated flows, so as to provide a better technical guidance for taking engineering measures to avoid the harms of the atomized flow.
source is considered. Water is easy to deform, thence a series of deformations would happen during the impact of water drops against the downstream water surface, before splashed droplets are generated at last which jump from the downstream water surface with the initial velocity us , the angles T s (the angle between the initial velocity and the plane x1ox3 )and Is (the angle between the projection of the initial velocity in x1ox3 with the x1 axis). While a splashed water droplet leaves the downstream water surface, it moves under the action of the gravitational force, the buoyant force and the wind resistance. Let the velocity of the water droplet be u pi , then the governing equations for the splashed droplet with the diameter d p and the density U w , moving in the velocity field {u fi } =
{uw , 0, 0} of the jet driven wind with the density Ua are d u pi dt
§ U · 3 U a CD u p u f (u pi u fi ) ¨1 a ¸ gi (1) 4 Uw d p © Uw ¹
The governing equations for the position x pi of water droplets are d x pi dt
1. Simulation of atomized flow 1.1 Mathematical model and numerical method The water droplets in the atomized flow may move in two different states, those of a large geometrical size move in a splashed state, with the related source being called the splashed source, on the other hand, the water droplets with small geometrical size move in a suspended state, with the related source being called the suspended source. In Ref.[5] a unified model was developed for the atomized flow, in which the water droplets moving in the splashed state is described by a random simulation based on the Lagrangian description, and the water droplets moving in the suspended state is described by the theory of the air-water two-phase flow in the continuum mechanics based on the Eulerian description, and the demarcation diameter d m between the splashed and suspended states is determined similar to the classification of the bed load and the suspended sediment motion in the theory of river dynamics. Since in this article’s verification of the mathematical model, the jet velocity is about 6 m/s, the suspended source is very small and can be omitted[11], only the method to calculate the atomization rainfall generated by the splashed
=
(2)
= u pi
where CD is a coefficient depending on the relative Reynolds number § up u f d p · ¸ Red = ¨ ¨ ¸ Qa © ¹
and u p u f
is the relative velocity between the
water droplet and the jet driven wind. The initial velocities of the water droplet are
^u ` = ^u cosT cosI , u sin T , u cosT sin I ` (3) pi t =0
s
s
s
s
s
s
s
s
In the random model of this article, the probability distributions of d p , us , T s and Is are as follows[5,12]: d p obeys the * distribution, us and
T s obey the Weibull distribution, and Is obeys the normal distribution. The initial position of the water droplet is just at the outer edge of the aerated jet as it impacts with the downstream water surface. In this
275
article, the randomness of the initial position of water droplets is described by a uniform distribution at the outer edge of the aerated jet as they just impact with the downstream water surface. The solution steps are as follows: (1) Generate the water droplet with diameter d p . (2) Generate the initial velocity us and the splashing angles T s and Is of the water droplet. (3) Generate the initial position of the water droplet, and solve Eq.(1) by the fourth-order Runge-Kutta algorithm to get the velocity of the water droplet, the trajectories of the water droplet can be obtained by further numerical integration of the velocity, then the position where the water droplet falls into the downstream boundary (water surface or land) can be determined. Repeat the above process of random simulation until the simulating number of water droplets is N . The probability density function of the radius of the atomization rain drop, and the rainfall intensity P and other statistical quantities at the downstream boundary can be obtained by the statistical average of the random simulation results.
long and 1.3 m wide. While this jet impinges with the water surface of the pond, the atomized flow is generated. The rainfall intensity was measured by a graduated cylinder, supplemented with several transducers of electric balance. The diameters of the rain drops were measured by using the method of color speck[13]. The comparisons between the random simulation results and the experimental results of the rainfall intensity distribution in the longitudinal and lateral directions, the probability density function of the diameter of the atomization rain drops in the position of x1 = 0 m , x3 = 0.9 m are shown in Fig.1 and Fig.2, respectively. The two results are in close agreement with each other.
Fig.2 The probability density function of the diameter of atomization rain drops
Fig.1(a) The rainfall intensity distribution in longitudinal direction Fig.3 Variations of the median rain size in the longitudinal and lateral directions
Figure 3 also shows the variations of the median rain size d 50 in the longitudinal and lateral directions. It is seen that the random simulation results and the experimental results are also in agreement with each other, and the variation of d 50 has the following characteristics: (1) In the longitudinal direction, d 50 inFig.1(b) The rainfall intensity distribution in lateral direction
1.2 Verification The experimental results in Ref.[11] are used for verification. The experiment was conducted in a special equipment in Wuhan University. The high-speed aerated jet was generated by using a nozzle linked with a submersible pump installed on a pond of 2.3 m
creases with x1 at first, then monotonously decreases, and the longitudinal position for d50 to reach its maximum value is not in consistent with the position where the maximum rain intensity is reached. (2) In the lateral direction d 50 monotonously decreases with x3 .
276
2. Simulation of slope runoff generated by atomization rain drops 2.1 Mathematical model and numerical scheme The study of the slope runoff in hydrology is mostly based on the water balance to obtain the quantity of water in surface and subsurface flows, yet in the geotechnical engineering more attentions are paid to the change of the pore water pressure and the associated slope stability. In this article, the coupling between surface and subsurface water flows is not considered duo to the limited space of the article, and the special attentions are paid to the difference of the slope runoff due to the natural rainfall and the atomization rainfall. 2.1.1 Mathematical model The depth-averaged 2-D model is used to describe the slope runoff. The governing equations for the slope runoff on a slope with tilt angle T in the Cartesian coordinate system are wU wF w G + + =S wt w x w y
(4)
where uH § § H · ¨ g H 2 cos T ¨ ¸ U = ¨ u H ¸ , F = ¨ u2 H + ¨ 2 ¨ vH ¸ ¨ © ¹ uvH ©
· ¸ ¸, ¸ ¸ ¹
§ · ¨ ¸ vH ¨ ¸ G=¨ uvH ¸, ¨ ¸ 2 ¨¨ v 2 H + g H cos T ¸¸ 2 © ¹ R wT § · ¨ ¸ S = ¨ g H sin T 0.125O u 2 + v 2 u ¸ ¨¨ ¸¸ 0.125O u 2 + v 2 v © ¹
and H is the water depth, u and v are the depthaveraged velocities in x and y directions, respectively, O and g are the drag coefficient and the gravity acceleration, respectively, RwT is the atomization rainfall intensity. In Eq.(4), the static pressure is corrected as g H cos T for the steep slope runoff. The boundary conditions are as follows: the water depth is zero at the outer edge of the atomized flow on the slope, the flow is fully developed at the base of the slope.
2.1.2 Numerical scheme The control volume method is used for the discretization of Eq.(4). The discrete form can be obtained by choosing the rectangular mesh as the control volume and integrating Eq.(4) over the control volume similar to the procedures in Ref.[14,15], which is wU i 1 = wt Ai
N ed
¦E n
ij
' * ij + Si
(5)
j =1
where i and j denote the i th cell and the j th edge of the cell, respectively, U i is the average quantities stored at the center of the i th cell, nij is the edge, '* ij
unit outward normal vector at the j th
is the length of the j th edge, and E is the numerical flux through the edge. In the simulation of the slope runoff, the treatment of wet/dry boundaries is very important, therefore, the HLL Riemann solver is employed for the calculation of the normal flux E nij at the face of a control volume, which makes it possible to handle discontinuities. Using the first order accuracy scheme to calculate the time dependent terms, the discrete form of Eq.(5) becomes
U
m +1 i
§ 1 = U + 't ¨ © Ai
· E nij ' * ij + Si ¸ ¦ j =1 ¹ N ed
m i
m
(6)
2.2 Determination of key parameters In the mathematical model of Eq.(4), the drag coefficient is the key parameter, which is determined by a theoretical analysis and the experimental results. The slope runoff is one kind of thin sheet flows, if the theory of the steady uniform flow is used to describe it approximately in a short segment, the discharge per unit width becomes q = n 1 H 5 / 3 sin T , there-
fore H = n / sin T
3/ 5
q 3 / 5 , U = n / sin T
3 / 5
q2 / 5 ,
and the drag coefficient becomes
O=
8 g H sin T = 8 g n9 / 5 (sin T )1/10 q 1/ 5 U2
(7)
Zhang[6] carried out experiments for the variation of the drag coefficient against the discharge per unit width. In the experiments, the slope runoff was generated by the upstream inflow in a channel, and the experimental soil sample was adhered to the bottom of the channel. The water depth was measured by point gauges, and the velocity was measured by the dyeing method. Zhao et al.[7] also carried out experiments for the variation of the water depth, the velocity and the
277
discharge per unit width in a channel, unlike the experiments by Zhang[6], the slope runoff was generated by the rainfall, and the bottom of the channel is smooth and impermeable. In the experiments, the rainfall was generated by an automatically controlling system, and the flow velocity was measured by the method of electric pulses. Although the flow generated by the rainfall is a spatially varied flow, the characteristics of this flow in a local area are similar to those of the flow generated by an upstream inflow. Figure 4 shows the experimental results in Ref.[6,7] and Eq.(7) for the variation of the drag coefficient O c ( = O [8gn9 / 5
the natural rainfall to reach the steady state is about 400 s, while for the atomization rainfall, the duration time increases to 800 s since the distribution of the rainfall intensity is non-uniform, 2) in the initial stage of the runoff producing process, the discharge calculated by the kinematical wave model[8] is relatively larger than that by Eq.(4), yet the difference of the two results is very small, which seems that the kinematical wave model enjoys relatively high accuracy for the calculation of the runoff yield either by the natural rainfall or the atomization rainfall.
< (sin T )1/10 ]1 ) against the discharge per unit width. It can be seen from Fig.4 that the experimental results and the analytical results are in good agreement with each other. Therefore in this article, Eq.(7) is used for the determination of the drag coefficient.
Fig.5 Variation of the runoff yield with time at the base of the slope
Fig.4 Variation of the drag coefficient against discharge per unit width
2.3 Numerical simulation results 2.3.1 Computational domain The computational domain is 50 m long and 50 m wide, with the tilt angle of 30o. In the computation, the grid sizes in the longitudinal and lateral directions are all 1 m. The runoff producing processes for the natural rainfall and the atomization rainfall are simulated separately, and the averaged rainfall intensity in the computational domain is 0.02 m/h. The rainfall intensity for the natural rain is uniformly distributed in the computational domain, yet for the atomization rain, the rainfall intensity distribution of Fig.1 is used instead. 2.3.2 Analyses of the simulation results (1) Runoff producing process The slope runoff would gradually reach a steady state of a spatially varied flow while the rainfall duration is longer than the concentration time. Figure 5 shows the results for the variation of the discharge at the base of the slope against time in the natural and atomization rainfall conditions . It can be seen from this figure that: 1) the discharge at the base of the slope gradually increases from zero to a time independent value as the rain goes on, the duration time for
Fig.6(a) Variation of water depth in the longitudinal direction
Fig.6(b) Variation of water depth in the lateral direction at the base of the slope
(2) Variation of water depth Figure 6 shows the variation of water depth in the longitudinal and lateral directions for the steady slope runoff. It can be seen from this figure that: 1) the water depth increases in the longitudinal direction
278
owing to the rainfall supplement, with the variation range for the atomization rainfall greater than that for the natural rainfall, 2) the distribution of water depth at the base of the slope in the lateral direction is uniform for the natural rainfall, while for the atomization rainfall, this distribution is non-uniform, and the water depth increases in the area with the increase of the rainfall intensity.
the position to reach the maximum rain intensity. In the lateral direction the median rain size would monotonously decrease. (2) Since the atomization rainfall is distributed in a strongly localized area, the duration taken for its runoff yield to reach a steady state is longer than that for the natural rain, the variation ranges of the water depth and the velocity in the longitudinal and lateral directions are larger than those for the natural rainfall. References [1]
[2]
[3] Fig.7(a) Variation of flow velocity in the longitudinal direction
[4]
[5]
[6] Fig.7(b) Variation of flow velocity in the lateral direction at the base of the slope [7]
(3) Variation of velocity Figure 7 shows the variation of the velocity in the longitudinal and lateral directions for the steady slope runoff. It can be seen from this figure that: 1) the velocity increases in the longitudinal direction with the increase of the rainfall, and the variation range of the velocity for the atomization rainfall is larger than that for the natural rainfall, 2) the distribution of the velocity at the base of the slope in the lateral direction is uniform for the natural rainfall, while for the atomized flow rainfall, this distribution is non-uniform, and the velocity also increases in the area with the increase of the rainfall intensity, similar to the variation of the water depth.
3. Conclusions In this article, the unified model and the depthaveraged 2-D model are used to simulate the atomized flow and its generated flow on a slope. The conclusions are as follows: (1) The median rain size in the atomization rainfall increases in the longitudinal direction at first, then monotonously decreases, and the longitudinal position to reach its maximum value is not in consistent with
[8]
[9]
[10]
[11]
[12]
[13]
ZHOU Hua. Numerical analysis of the 3-D flow field of pressure atomizers with v-shaped cut at oriflce[J]. Journal of Hydrodynamics, 2011, 23(2): 187-192. XIAO Yang, TANG Hong-wu. Numerical study of hydrodynamics of multiple tandem jet in cross flow[J]. Journal of Hydrodynamics, 2011, 23(6): 806-813. SUN Shuang-ke, LIU Zhi-pin. Longitudinal range of atomized flow forming by discharge of spillways and oulet works in hydropower stations[J]. Journal of Hydraulic Engineering, 2003, (12): 53-57(in Chinese). LIU Hai-tao, SUN Shuang-ke and LIU Zhi-ping. Mathematical model for rain-fog transportation during flood discharge and its verification[J]. Advances in Science and Technology of Water Resources, 2011, 31(4): 4549(in Chinese). LIU Shi-he, SUN Xiao-fei and LUO Jing. Unified model for splash droplets and suspended mist of atomized flow[J]. Journal of Hydrodynamics, 2008, 20(1): 125-130. ZHANG Guang-hui. Study on hydraulic properties of shallow flow[J]. Advances in Water Science, 2002, 13(2): 159-165(in Chinese). ZHAO Xiao-e, WEI Lin and CAO Shu-you et al. Study on characteristics of overland flow with higher rain intensity[J]. Journal of Soil and Water Conservation, 2009, 23(6): 45-47(in Chinese). LI Zhan-bin, LU Ke-xin. Approximate analytical solution of rainfall process on permeable slope[J]. Journal of Hydraulic Engineering, 2003, (6): 8-13(in Chinese). ZHANG Pei-wen, LIU De-fu and ZHENG Hong et al. Coupling numerical simulation of slope runoff and infiltration under rainfall conditions[J]. Rock and Soil Mechanics, 2004, 25(1): 109-113(in Chinese). TUNG Yeou-koung, KWOK Yih-feng and NG Wang Wai Charles et al. A preliminary study of rainfall infiltration on slope using a new coupled surface and subsurface flow model[J]. Rock and Soil Mechanics, 2004, 25(9): 1347-1352(in Chinese). FAN Min, LIU Shi-he and ZHANG Kang-le. Experimental study and numerical simulation of the distribution of splashed raindrop size in atomized flow[J]. Proceedings of the 5th National Symposium on Hydraulics and Hydraulic Information. Xi’an, 322-327(in Chinese). LIU Shi-he, LIU Jiang and LUO Qiu-shi et al. Engineering turbulence[M]. Beijing: Science Press, 2010(in Chinese). LIAO Wei, WEI Miao-miao and HUANG Yu-yu. Research on raindrop diameter based on filter paper splash procedure[J]. Journal of Wuhan University of Technology (Transportation Science and Engineering), 2008, 32(6): 1165-1168(in Chinese).
279
[14]
MORITA M., YEN B. C. Modeling of conjunctive twodimensional surface–three-dimensional subsurface flows[J]. Journal of Hydraulic Engineering, 2002, 128(2): 184-201.
[15]
YOON Tae Hoon, ASCE F. and KANG Seok-Koo. Finite volume model for two-dimensional shallow water flows on unstructured grid[J]. Journal of Hydraulic Engineering, 2004, 130(7): 678-688.