共查询到20条相似文献，搜索用时 15 毫秒
1.
V. P. SINGH 《水文研究》1996,10(7):955-969
Error equations for the kinematic-wave and diffusion-wave approximations were derived under simplified conditions for space-independent flows occurring on infiltrating planes or channels. These equations specify error as a function of time in the flow hydrograph. The kinematic-wave, diffusion wave and dynamic-wave solutions were parameterized through a dimensionless parameter γ which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow and channel roughness when the initial condition is non-vanishing; and it reflects the effect of bed slope, channel roughness, lateral inflow and infiltration when the initial condition is vanishing. The error equations were found to be the Riccati equation. 相似文献
2.
《Advances in water resources》2003,26(11):1189-1198
A two-dimensional finite element based overland flow model was developed and used to study the accuracy and stability of three numerical schemes and watershed parameter aggregation error. The conventional consistent finite element scheme results in oscillations for certain time step ranges. The lumped and the upwind finite element schemes are tested as alternatives to the consistent scheme. The upwind scheme did not improve on the stability or the accuracy of the solution, while the lumped scheme provided stable and accurate solutions for time steps twice the size of time steps needed for the consistent scheme. A new accuracy based dynamic time step estimate for the two-dimensional overland flow kinematic wave solution is developed for the lumped scheme. The newly developed dynamic time step estimates are functions of the mesh size, and time of concentration of the watershed hydrograph. Due to lack of analytical solutions, the time step was developed by comparing numerical solutions of various levels of discretization to a reference solution using a very fine mesh and a very small time step. The time step criteria were tested on a different set of problems and proved to be adequate for accurate and stable solutions. A sensitivity analysis for the watershed slope, Manning’s roughness coefficient and excess rainfall rate was conducted in order to test the effect of parameter aggregation on the stability and accuracy of the solution. The results of this analysis show that aggregation of the slope data resulted in the highest error. The roughness coefficient had a smaller effect on the solution while the rainfall intensity did not show any significant effect on the flow rate solution for the range of rainfall intensity used. This work pioneers the challenge of providing guidelines for accurate and stable numerical solutions of the two-dimensional kinematic wave equations for overland flow. 相似文献
3.
The errors in surface runoff prediction by neglecting the relationship between infiltration rate and overland flow depth 总被引：9，自引：0，他引：9
Many simplifications are used in modeling surface runoff over a uniform slope. A very common simplification is to determine the infiltration rate independent of the overland flow depth and to combine it afterward with the kinematic-wave equation to determine the overland flow depth. Another simplication is to replace the spatially variable infiltration rates along the slope i(x, t) due to the water depth variations h(x,t) with an infiltration rate that is determined at a certain location along the slope. The aim of this study is to evaluate the errors induced by these simplications on predicted infiltration rates, overland flow depths, and total runoff volume. The error analysis is accomplished by comparing a simplified model with a model where the interaction between the overland flow depth and infiltration rate is counted. In this model, the infiltration rate is assumed to vary along the slope with the overland flow depth, even for homogeneous soil profiles. The kinematic-wave equation with interactive infiltration rate, calculated along the slopy by Richard's equation, are then solved by a finite difference scheme for a 100-m-long uniform slope. In the first error analysis, we study the effect of combining an ‘exact’ and ‘approximate’ one-dimensional infiltration rate with the kinematic-wave equation for three different soil surface roughness coefficients. The terms ‘exact’ and ‘approximate’ stand for the solution of Richard's equation with and without using the overland flow depth in the boundary condition, respectively. The simulations showed that higher infiltration rates and lower overland flow depths are obtained during the rising stage of the hydrograph when overland flow depth is used in the upper boundary condition of the one-dimensional Richard's equation. During the recession period, the simplified model predicts lower infiltration rates and higher overland flow depths. The absolute relative errors between the ‘exact’ and ‘approximate’ solutions are positively correlated to the overland flow depths which increase with the soil surface roughness coefficient. For this error analysis, the relative errors in surface runoff volume per unit slope width throughout the storm are much smaller than the relative errors in momentary overland flow depths and discharges due to the alternate signs of the deviations along the rising and falling stages. In the second error analysis, when the spatially variable infiltration rate along the slope i(x, t) is replaced in the kinematic-wave equation by i(t), calculated at the slope outlet, the overland flow depth is underestimated during the rising stage of the hydrograph and overestimated during the falling stage. The deviations during the rising stage are much smaller than the deviations during the falling stage, but they are of a longer duration. This occurs because the solution with i(x, t) recognizes that part of the slope becomes dry after rainfall stops, while overland flow still exists with i(t) determined at the slope outlet. As obtained for the first error analysis, the relative errors in surface runoff volume per unit slope width are also much smaller than the relative errors in momentary overland flow depths and discharges. The relation between the errors in overland flow depth and discharge to different mathematical simplifications enables to evaluate whether certain simplifications are justified or more computational efforts should be used. 相似文献
4.
Predicting the behavior of overland flow with analytical solutions to the kinematic wave equation is appealing due to its relative ease of implementation. Such simple solutions, however, have largely been constrained to applications on simple planar hillslopes. This study presents analytical solutions to the kinematic wave equation for hillslopes with modest topographic curvature that causes divergence or convergence of runoff flowpaths. The solution averages flow depths along changing hillslope contours whose lengths vary according hillslope width function, and results in a one-dimensional approximation to the two-dimensional flow field. The solutions are tested against both two-dimensional numerical solutions to the kinematic wave equation (in ParFlow) and against experiments that use rainfall simulation on machined hillslopes with defined curvature properties. Excellent agreement between numerical, experimental and analytical solutions is found for hillslopes with mild to moderate curvature. The solutions show that curvature drives large changes in maximum flow rate qpeak and time of concentration tc , predictions frequently used in engineering hydrologic design and analysis. 相似文献
5.
Although rainfall is assumed spatially uniform in conventional hydrological modelling for rainfall–runoff simulations, moving storms have been shown to have substantial influence on flow hydrographs. In this study, criteria for attainment of the equilibrium discharge from watersheds subjected to moving storms were examined. Non-linear numerical kinematic-wave models were developed to simulate runoff from an overland plane and from a V-shaped catchment. Dimensional analysis was applied to obtain the independent variables to be used as control factors in performing a series of numerical tests. The results indicate that, for storms moving downstream, runoff can attain equilibrium discharge even though the storm length is shorter than the watershed length and the rainfall duration is less than the time to equilibrium of the watershed for stationary uniform storms. The phenomenon of attainment of equilibrium discharge from watersheds subjected to moving storms is contradictory to conventional hydrologic design, which assumes the storm duration must equal the time to equilibrium to attain the maximum discharge. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
6.
Vegetation and soil properties and their associated changes through time and space affect the various stages of soil erosion. The island of Ishigaki in Okinawa Prefecture, Japan is of particular concern because of the propensity of the red‐soil‐dominated watersheds in the area to contribute substantial sediment discharge to adjacent coastal areas. This paper discusses the application of remote sensing techniques in the retrieval of vegetation and soil parameters necessary for the distributed soil‐loss modelling in small agricultural catchments and analyses the variation in erosional patterns and sediment distribution during rainfall events using numerical solutions of overland flow simulations and sediment continuity equations. To account for the spatial as well as temporal variability of selected parameters of the soil‐loss equations, a method is proposed to account for the variability of associated vegetation cover based on their spectral characteristics as captured by remotely sensed data. To allow for complete spatial integration, modelling the movement of sediment is accomplished under a loose‐coupled GIS computational framework. This study lends a theoretical support and empirical evidence to the role of vegetation as a potential agent for soil erosion control. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
7.
Two primary concerns in performing watershed overland flow routing are the numerical instability and computational efficiency. The stability of executing an explicit scheme has to be maintained by observing the Courant–Friedrich–Lewy criterion, which is adopted to confirm that the numerical marching speed is larger than the wave celerity. Moreover, there is another criterion of time step devised in previous studies to avoid back‐and‐forth refluxing between adjacent grids. The situation of refluxing usually occurs on flat regions. In light of this, the selection of a small time increment to honor both restrictions simultaneously is believed to decrease the computational efficiency in performing overland flow routing. This study aims at creating a robust algorithm to relax both restrictions. The proposed algorithm was first implemented on a one‐dimensional overland plane to evaluate the accuracy of the numerical result by comparing it with an analytical solution. Then, the algorithm was further applied to a watershed for 2D runoff simulations. The results show that the proposed integrated algorithm can provide an accurate runoff simulation and achieve satisfactory performance in terms of computational speed. 相似文献
8.
Second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is obtained by using the cumulant expansion ensemble averaging method and by taking the time dependent sure part of the multiplicative operator into account. It is shown that the satisfaction of the commutativity and the reversibility requirements proposed earlier for linear stochastic differential equations without forcing are necessary for the linear stochastic differential equations with forcing when the cumulant expansion ensemble averaging method is used. It is shown that the applicability of the operator equality, which is used for the separation of operators in the literature, is also subjected to the satisfaction of the commutativity and the reversibility requirements. The van Kampen’s lemma, which is proposed for the analysis of nonlinear stochastic differential equations, is modified in order to make the probability density function obtained through the lemma depend on the forcing terms too. The second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is also obtained by using the modified van Kampen’s lemma in order to validate the correctness of the modified lemma. Second-order exact ensemble averaged equation for one dimensional convection diffusion equation with reaction and source is obtained by using the cumulant expansion ensemble averaging method. It is shown that the van Kampen’s lemma can yield the cumulant expansion ensemble averaging result for linear stochastic differential equations when the lemma is applied to the interaction representation of the governing differential equation. It is found that the ensemble averaged equations given for one the dimensional convection diffusion equation with reaction and source in the literature obtained by applying the lemma to the original differential equation are restricted with small sure part of multiplicative operator. Second-order exact differential equations for the evolution of the probability density function for the one dimensional convection diffusion equation with reaction and source and one dimensional nonlinear overland flow equation with source are obtained by using the modified van Kampen’s lemma. The equation for the evolution of the probability density function for one dimensional nonlinear overland flow equation with source given in the literature is found to be not second-order exact. It is found that the differential equations for the evolution of the probability density functions for various hydrological processes given in the literature are not second-order exact. The significance of the new terms found due to the second-order exact ensemble averaging performed on the one dimensional convection diffusion equation with reaction and source and during the application of the van Kampen’s lemma to the one dimensional nonlinear overland flow equation with source is investigated. 相似文献
9.
Beatriz Quintal Eva Caspari Klaus Holliger Holger Steeb 《Geophysical Prospecting》2019,67(8):2196-2212
In interconnected microcracks, or in microcracks connected to spherical pores, the deformation associated with the passage of mechanical waves can induce fluid flow parallel to the crack walls, which is known as squirt flow. This phenomenon can also occur at larger scales in hydraulically interconnected mesoscopic cracks or fractures. The associated viscous friction causes the waves to experience attenuation and velocity dispersion. We present a simple hydromechanical numerical scheme, based on the interface-coupled Lamé–Navier and Navier–Stokes equations, to simulate squirt flow in the frequency domain. The linearized, quasi-static Navier–Stokes equations describe the laminar flow of a compressible viscous fluid in conduits embedded in a linear elastic solid background described by the quasi-static Lamé–Navier equations. Assuming that the heterogeneous model behaves effectively like a homogeneous viscoelastic medium at a larger spatial scale, the resulting attenuation and stiffness modulus dispersion are computed from spatial averages of the complex-valued, frequency-dependent stress and strain fields. An energy-based approach is implemented to calculate the local contributions to attenuation that, when integrated over the entire model, yield results that are identical to those based on the viscoelastic assumption. In addition to thus validating this assumption, the energy-based approach allows for analyses of the spatial dissipation patterns in squirt flow models. We perform simulations for a series of numerical models to illustrate the viability and versatility of the proposed method. For a 3D model consisting of a spherical crack embedded in a solid background, the characteristic frequency of the resulting P-wave attenuation agrees with that of a corresponding analytical solution, indicating that the dissipative viscous flow problem is appropriately handled in our numerical solution of the linearized, quasi-static Navier–Stokes equations. For 2D models containing either interconnected cracks or cracks connected to a circular pore, the results are compared with those based on Biot's poroelastic equations of consolidation, which are solved through an equivalent approach. Overall, our numerical simulations and the associated analyses demonstrate the suitability of the coupled Lamé–Navier and Navier–Stokes equations and of Biot's equations for quantifying attenuation and dispersion for a range of squirt flow scenarios. These analyses also allow for delineating numerical and physical limitations associated with each set of equations. 相似文献
10.
Sub‐grid scale parameterization of hillslope runoff and erosion processes for catchment‐scale models of semi‐arid landscapes 
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下载免费PDF全文 The processes of hillslope runoff and erosion are typically represented at coarse spatial resolution in catchment‐scale models due to computational limitations. Such representation typically fails to incorporate the important effects of topographic heterogeneity on runoff generation, overland flow, and soil erosion. These limitations currently undermine the application of distributed catchment models to understand the importance of thresholds and connectivity on hillslope and catchment‐scale runoff and erosion, particularly in semi‐arid environments. This paper presents a method for incorporating high‐resolution topographic data to improve sub‐grid scale parameterization of hillslope overland flow and erosion models. Results derived from simulations conducted using a kinematic wave overland flow model at 0.5 m spatial resolution are used to parameterize the depth–discharge relationship in the overland flow model when applied at 16 m resolution. The high‐resolution simulations are also used to derive a more realistic parameterization of excess flow shear stress for use in the 16 m resolution erosion model. Incorporating the sub‐grid scale parameterization in the coarse‐resolution model (16 m) leads to improved predictions of overland flow and erosion when evaluated using results derived from high‐resolution (0.5 m) model simulations. The improvement in performance is observed for a range of event magnitudes and is most notable for erosion estimates due to the non‐linear dependency between the rates of erosion and overland flow. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
11.
如何有效压制数值频散是有限差分正演模拟研究中的关键问题之一.近年来,许多学者对二阶声波方程的差分算子开展了大量的优化工作,在压制频散方面取得不错的效果.一阶压强-速度方程广泛用于研究地震波在地下变密度模型中传播规律,目前针对一阶方程的优化工作大多只是在空间差分算子上展开.本文在前人研究的基础上,推导出一阶声波方程中压强场与偏振速度场之间的解析关系,据此在传统交错网格基础上给出一种高精度的显式时间递推格式,该递推格式将时间差分与空间差分算子结合在一起,并采用共轭梯度法得到精确时间递推匹配系数,实现时空差分算子的同时优化.在编程实现算法的基础上,通过频散分析与三个典型模型测试表明:本文方法能够较为有效地压制时间频散与空间频散,提高数值计算精度;同时对复杂模型也有很好适用性. 相似文献
12.
A transversely isotropic material in the sense of Green is considered. Using a series of potential functions proposed in [Eskandari-Ghadi M. A complete solution of the wave equations for transversely isotropic media. J Elasticity 2005; 81:1–19], the solutions of the transient wave equations within a half-space under surface load are obtained in the Laplace–Hankel domain for axisymmetric problems. The solutions are investigated in detail in the special case of a surface point force pulse varying with time as Heaviside function. Using Cagniard–De Hoop method, the inverse Laplace transform and inverse Hankel transform of the solutions are then obtained in the form of integrals with finite limits. For validity of the analytical results, the final formulations for surface waves are degenerated for an isotropic material and compared with the existing formulation obtained by Pekeris [The seismic surface pulse. Proc Natl Acad Sci USA 1955;41:469–80], to show that they are exactly the same. The numerical evaluations of the integrals for some transversely isotropic materials as well as an isotropic one are obtained. The present approach is then numerically verified by comparing a particular case of displacements for the surface of an isotropic half-space subjected to a point load of Heaviside function with the solutions obtained by Pekeris [The seismic surface pulse. Proc Natl Acad Sci USA 1955;41:469–80]. In addition, the wave equations for the mentioned medium are obtained on the vertical line directly under the applied surface load. The final formulations are degenerated for an isotropic material and compared with the existing formulation given in Graff [Wave motion in elastic solids. New York: Dover Publications Inc; 1975 [New Ed edition, November 1991]], to show that they are also exactly the same. Then equations are presented in graphical forms using an appropriate numerical evaluation. 相似文献
13.
将弹性波方程变换至Hamilton体系,构造适用于弹性波模拟的高效显式二阶辛Runge-Kutta-Nystrm(RKN)格式,运用根数理论得到此格式的阶条件方程组.通过给定系数的限定条件,得到方程的对称解.为了使时间离散误差达到极小,提出数值频率与真实频率比较,通过Taylor展开,得到关于辛系数的限定方程,求解方程组得到最小频散辛RKN格式.对比分析时间演进方程的稳定性,得到使库朗数达到极大值的限定方程,求解方程组得到最稳定辛RKN格式.发现此两种格式为同一格式.新得到的辛RKN格式不依赖于空间离散方法,为了对比的需要,选取有限差分法进行空间离散.在频散、稳定性分析中,与常见辛格式对比,从理论上分析了本文提出的格式在数值频散压制、稳定性提升等方面的优势,数值实验进一步证实了理论分析的正确性. 相似文献
14.
Fractals are famous for their self‐similar nature at different spatial scales. Similar to fractals, solutions of scale invariant processes are self‐similar at different space–time scales. This unique property of scale‐invariant processes can be utilized to translate the solution of the processes at a much larger or smaller space–time scale (domain) based on the solution calculated on the original space–time scale. This study investigates scale invariance conditions of kinematic wave overland flow process in one‐parameter Lie group of point transformations framework. Scaling (stretching) transformation is one of the one‐parameter Lie group of point transformations and it has a unique importance among the other transformations, as it leads to the scale invariance or scale dependence of a process. Scale invariance of a process yields a self‐similar solution at different space–time scales. However, the conditions for the process to be scale invariant usually dictate various relationships between the scaling coefficients of the dependent and independent variables of the process. Therefore, the scale invariance of a process does not assure a self‐similar solution at any arbitrary space and time scale. The kinematic wave overland flow process is modelled mathematically as initial‐boundary value problem. The conditions to be satisfied by the system of governing equations as well as the initial and boundary conditions of the kinematic wave overland flow process are established in order for the process to be scale invariant. Also, self‐similarity of the solution of the kinematic wave overland flow under the established invariance conditions is demonstrated by various numerical example problems. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
15.
V. P. Singh 《水文研究》1998,12(1):147-170
Using kinematic wave equations, analytical solutions are derived for flow owing to storms moving up and down a plane. By comparing the flow owing to a moving storm with that to an equivalent stationary storm, the influence of storm direction is investigated. The direction of storm movement exercises a significant influence on the peak flow and time to peak flow, as well as the shape of the overland flow hydrograph. © 1998 John Wiley & Sons, Ltd. 相似文献
16.
《中国科学:地球科学(英文版)》2021,(6)
The perfectly matched layer(PML) boundary condition has been proven to be effective for attenuating reflections from model boundaries during wavefield simulation. As such, it has been widely used in time-domain finite-difference wavefield simulations. The conventional PML has poor performance for near grazing incident waves and low-frequency reflections. To overcome these limitations, a more complex frequency-shifted stretch(CSF) function is introduced, which is known as the CFSPML boundary condition and can be implemented in the time domain by a recursive convolution technique(CPML). When implementing the PML technique to second-order wave equations, all the existing methods involve adding auxiliary terms and rewriting the wave equations into new second-order partial differential equations that can be simulated by the finite-difference scheme, which may affect the efficiency of numerical simulation. In this paper, we propose a relatively simple and efficient approach to implement CPML for the second-order equation system, which solves the original wave equations numerically in the stretched coordinate. The spatial derivatives in the stretched coordinate are computed by adding a correction term to the regular derivatives. Once the first-order spatial derivatives are computed, we computed the second-order spatial derivatives in a similar way; therefore, we refer to the method as two-step CPML(TS-CPML). We apply the method to the second-order acoustic wave equation and a coupled second-order pseudo-acoustic TTI wave equation. Our simulations indicate that amplitudes of reflected waves are only about half of those computed with the traditional CPML method, suggesting that the proposed approach has computational advantages and therefore can be widely used for forwarding modeling and seismic imaging. 相似文献
17.
This study develops a lattice Boltzmann method (LBM) with a two-relaxation-time collision operator (LTRT) to solve saltwater intrusion problems. A directional-speed-of-sound (DSS) technique is introduced to take into account the hydraulic conductivity heterogeneity and discontinuity, as well as the velocity-dependent dispersion coefficient. The forcing terms in the LTRT model are customized in order to recover the density-dependent groundwater flow and mass transport equations. Using the LTRT with the squared DSS achieves at least second-order accuracy. The LTRT results are verified with Henry’s analytical solution as well as compared with several numerical examples and modified Henry problems that consider heterogeneous hydraulic conductivity and velocity-dependent dispersion. The numerical results show good agreement with the Henry analytical solution and with the numerical solutions obtained by other numerical methods. 相似文献
18.
《Advances in water resources》2005,28(7):661-670
Use of characteristics method integrated with cubic-spline interpolation technique (CSMOC scheme) for computation of one-dimensional and two-dimensional kinematic overland flow has been examined in this study. The characteristic trajectory is allowed to fall on space line and time line for interpolating the corresponding values at the foot of trajectory in terms of neighboring grid points. The effects of different endpoint constraints on use of cubic-spline interpolation are investigated. As far as accuracy and simplicity are concerned, the not-a-knot constraint could be a better choice. Three hypothetical examples are used to examine the capabilities of CSMOC scheme through the comparison with the analytical solution and the well-known Preissmann scheme. Some degrees of numerical diffusion and numerical oscillation, attenuating and overestimating the peak discharge, are induced by the Preissmann scheme. In contrast, the CSMOC scheme gives convincing results for the kinematic overland flow computations. 相似文献
19.
Discussion about the validity of sharp‐interface models to deal with seawater intrusion in coastal aquifers 下载免费PDF全文
下载免费PDF全文 Analytical models have been exhaustively used to study simple seawater intrusion problems and the sustainable management of groundwater resources in coastal aquifers because of its simplicity, easy implementation, and low computational cost. Most of these models are based on the sharp‐interface approximation and the Ghyben–Herzberg relation, and their governing equations are expressed in terms of a single potential theory to calculate critical pumping rates in a coastal pumping scenario. The Ghyben–Herzberg approach neglects mixing of fresh water and seawater and implicitly assumes that salt water remains static. Therefore, the results of the analytical solutions may be inaccurate and unacceptable for some real‐complex case studies. This paper provides insight into the validity of sharp‐interface models to deal with seawater intrusion in coastal aquifers, i.e. when they can be applied to obtain accurate enough results. For that purpose, this work compares sharp‐interface solutions, based on the Ghyben–Herzberg approach, with numerical three‐dimensional variable‐density flow simulations for a set of heterogeneous groundwater flow and mass transport parameters, and different scenarios of spatially distributed recharge values and spatial wells placement. The numerical experiment has been carried out in a 3D unconfined synthetic aquifer using the finite difference numerical code SEAWAT for solving the coupled partial differential equations of flow and density‐dependent transport. This paper finds under which situations the sharp‐interface solution gives good predictions in terms of seawater penetration, transition zone width and critical pumping rates. Additionally, the simulation runs indicate to which parameters and scenarios the results are more sensitive. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
20.
A Lagrangian particle tracking scheme has been extended to simulate advective transport through coupled surface and subsurface flows. This extended scheme assumes a two-dimensional flow field for the overland domain and a three-dimensional flow field for the subsurface domain. Moreover it is assumed that the flow fields are simulated by a cell centered finite difference method. Pathlines in both the subsurface and the overland domain are simulated by classical particle tracking methods. Exchange of particles between the two domains is simulated by newly-developed algorithms presented in this study. Different algorithms are used depending on the direction of the exchange across the interface between the two domains. In the subsurface domain knowledge about a particle’s pathline is enough to detect a transfer to the surface domain and the solution is straightforward. However, in the two-dimensional overland domain pathlines are parallel to the land surface. Therefore the velocity field in the overland domain alone cannot be used to detect a transfer to the subsurface. We propose a relatively simple algorithm to estimate the probability of transfer to the subsurface domain. It is shown that this algorithm can also be used to handle the transfer from the overland domain to the atmosphere domain. The algorithm to estimate the transfer probabilities is based on the mass balance of water on a streamtube aligned with the particle’s pathline. This newly developed technique ensures that transit time distributions can be simulated accurately. These new relationships are implemented in an existing particle tracking code and are verified using analytical solutions for transit times. 相似文献