A weakly compressible smoothed particle hydrodynamics (WCSPH) method was developed to model openchannel flow over rough bed. An improved boundary treatment is proposed to quantitatively characterize bed roughness based on the ghost boundary particles (GBPs). In this model, the velocities of GBPs are explicitly calculated by using evolutionary polynomial regression with a multiobjective genetic algorithm. The simulation results show that the proposed boundary treatment can well reflect the influence of wall roughness on the vertical flow structure. A fully developed open channel is established, and its flume length, processing time, and turbulent model are discussed. The mixedlengthbased subparticle scale (SPS) turbulence model is adopted to simulate uniform flow in open channel, and this model is compared with the Smagorinskybased one. For the modified WCSPH model, the results show that the calculated vertical velocity and turbulent shear stress distribution are in good agreement with experimental data and fit better than the calculations obtained from the traditional Smagorinskybased model.
Computational fluid dynamics (CFD) is widely used in hydraulic engineering simulations with good computational efficiency and accuracy. As an important branch of CFD, meshless methods such as smoothed particle hydrodynamics (SPH) method, moving particle semiimplicit (MPS) method, and discrete element method (DEM) have received wide attention due to their ability to consider large deformations of free interface and multiple interfaces without mesh distortion. The SPH method is a widely used Lagrangian meshless method that was originally developed by Gingold and Monaghan [
Boundary conditions can directly affect the calculation accuracy and efficiency for the accurate simulation of openchannel flows. However, it is quite challenging to achieve strict boundary conditions in the SPH framework due to the interpolation procedures. Previous studies of numerical simulations of free surface flows mainly focused on boundary treatments of inflow and outflow. A variety of nonreflecting boundary schemes such as open boundary condition [
An additional complication is that openchannel beds in nature have different forms and different scales of roughness, such as fine sand bed, pebble river bed, or vegetation river bed. The bed roughness can exert an important influence on the flow field characteristics, such as average flow rate, timeaveraged pressure, Reynolds stress, and turbulent energy. The slip/noslip treatment on the bottom boundary directly affects the overall accuracy of the simulation results for the openchannel flow. Therefore, the bottom boundary should be carefully described. To the best of our knowledge, there are two major types of approaches to accurately account for the effects of bed roughness on the openchannel flow. One approach is to express the effect of the rough bed on the fluid by adding a source term in the fluidgoverning equation [
In addition to boundary conditions, the turbulence model is equally important in simulating openchannel flows. The turbulence phenomena are quite complex, preventing full resolution through fullproof theory. Existing turbulence models are primarily based on semiempirical assumptions. For example, the Reynolds stress caused by momentum exchange can be related to the average flow field by the eddy viscosity assumption [
The remainder of this paper is organized as follows: After the introduction, the methodology of the SPH model is presented in detail and an improved bottom boundary treatment is proposed for smallscale rough bed. Next, a numerical open channel is established with a revised turbulence model for openchannel flow in Section
SPH is a purely Lagrangian method, using a series of particles to discretize the continuum. For the fluid dynamics, the value of field functions such as mass, density, velocity, and pressure at each discrete particle can be approximated based on the physical properties of surrounding particles:
A commonly used smoothing kernel is a quintic function, as proposed by Wendland [
The Reynoldsaveraged Navier–Stokes (RANS) equations can be expressed as
In the SPH notation, the governing equations can be rewritten as [
To close the governing equations (
In the original SPS turbulence model, the eddy viscosity coefficient is taken as
Following Dalrymple and Rogers [
To limit spurious highfrequency noise in the density field, the dissipative term proposed by Molteni and Colagrossi [
The computing domain is divided into three parts of inﬂow domain, internal fluid domain, and outﬂow domain (Figure
Schematic diagram of inlet and outlet conditions.
The properties of the internal fluid particles are determined by the governing equations, and the properties of the inflow and outflow particles are controlled by the specified boundary conditions. The velocities of particles in the inflow and outflow domains are controlled by a specified flow velocity, which is analogous to the physical pumping process. The density and pressure conditions for the inflow and outflow particles are both determined by the hydrostatic pressure of the free surface flow. Because the velocity and pressure of particles in the buffer layer are forcibly allocated, particle instability and diverging velocity field near the interface between the internal fluid and the buffer layer could arise. Therefore, the flume length should be long enough to avoid undesirable boundary effects and to reduce any overconstrained issues in inflow and outflow regions.
Similar to the periodic boundary condition proposed by GomezGesteira et al. [
No special attention is paid to the particles at the free surface because the continuous density method (equation (
Openchannel water flow is always affected by fluid resistance and energy dissipation, especially for rough bed. The roughness of the bed surface significantly influences the formation and development of the water flow structure, i.e., the vertical flow velocity distribution. To quantitatively describe the bed surface roughness, equivalent roughness is generally introduced, also known as Nikuradse’s equivalent roughness [
The treatment of bed surface roughness in the SPH method has always been a complex issue. If the governing equations of equations (
Schematic diagram of the bottom boundary condition.
For the hydraulic smooth flow and the hydraulic transition flow, the wall function method can be adopted. For the hydraulic rough flow, the rough scale should be evaluated, and the smallscale rough bed can be adopted. The scale of roughness in the SPH method is a relative concept, where smallscale rough is if
The ghost particle velocity of the bottom boundary must be determined by trial calculation. When the numerical result of the crosssectional velocity distribution is closest to the theoretical flow velocity distribution, the ghost particle velocity corresponds to the equivalent roughness.
A twostage predictorcorrector algorithm is used for the numerical integration scheme in this study [
As suggested by Monaghan and Kos [
To validate the proposed model, a case with water depth
Schematic diagram of numerical water flume, where
In the test scenario, the numerical sensors were evenly arranged in the middle section to monitor the sectional water level process (Figure
Numerical water elevation for a specific current velocity of 0.312 m/s.
Gauge number, 
WG1  WG2  WG3  WG4 

Water elevation, 
0.05004  0.05002  0.04997  0.05003 
Figure
Instantaneous velocity (a) and pressure (b) fields for the model including the wall function.
The fluid boundary layer and openchannel turbulence can be fully developed under a sufficiently long computational domain in the streamline direction. However, it is necessary to minimize the length of the computing domain due to limited computing resources. The suitable flume length must be determined by trial calculation and is initially taken as up to 100 times the water depth. The length of the front, middle, and rear sections of the numerical water flume is taken as 50, 20, and 30 times the water depth, respectively (Figure
Simulated results of current velocity
The development of openchannel flow requires a certain amount of time, and the simulation time is directly related to the boundary and initial conditions. Therefore, for a specific model, it is crucial to determine the appropriate simulation time. Figure
Numerical current velocity in the streamline direction
Since fluid turbulence can cause spatiotemporal fluctuation in fluid parameters such as current velocity and water pressure, especially for fluid near the flume bottom, time and space averaging are required to reduce the oscillation of flow parameters. The particle approximation process (equation (
The average current velocities under four other flow conditions were analyzed in the same way, with case I (
Averaged current velocity
The original SPS model [
Comparison of current velocity and turbulent shear stress with water depth under different Smagorinsky constants. (a) Velocity profile for different
For cases with
The
Effect of particle spacing on current velocity (a) and shear stress (b) for the Smagorinskybased SPS turbulence model.
Mayrhofer et al. [
To obtain a reasonable shear force, some researchers have tried to modify the original SPS model based on other perspectives. For example, Gotoh et al. [
Simulated results of mixed length (a), shear stress (b), current velocity (c), and dimensionless current velocity (d) for the mixedlengthbased SPS turbulence model.
In Section
To investigate openchannel flow with wide bed roughness, a series of numerical cases were designed with reference to the actual size of river sediment particles [
River sediment classification (cosmid and boulder sizes were not considered in this simulation).
Type  Cosmid  Silt  Sand  Gravel  Cobble  Boulder 

Size (×10^{−3} m)  <0.004  0.004∼0.062  0.062∼2.0  2.0∼16.0  16.0∼250.0  >250.0 
The numerical cases were divided into hydraulic smooth flow, hydraulic rough flow, and hydraulic transition flow based on calculation of the friction Reynolds number. Here, a total of 159 sets of cases were performed. Among them, 51 sets made up the hydraulic smooth flow experimental group, 39 sets made up the hydraulic transition flow experimental group, and 69 sets made up the hydraulic rough flow experimental group (see Supplemental data (available
The distribution of current velocity in the entire vertical section should be consistent with that of the theoretical curve by specifying the appropriate ghost particle velocity, so the ghost particle velocity can be matched with flow condition and bed surface roughness. Figure
Simulated results of the dimensionless velocity distribution for the model with the wall function method. (a) H50s001ks1 and (b) H100s004ks005.
Figure
Effect of particle spacing on the wall function method.
We next performed an error analysis of the velocity distribution of all the test cases (Figure
Error analysis of the wall function model for hydraulic smooth flows a, c, d, e, and i and hydraulic rough flows b, f, g, h, j, k, and l.
According to the above findings, the bed slope and the equivalent roughness are positively correlated determinants of the numerical error. Obviously, for the same hydraulic condition, the result will be more accurate for smaller particle spacing, but more computing resources will be needed.
From the analysis presented above, a reasonable current velocity distribution can be obtained by adjusting the velocity of the ghost particles on the bed surface. However, this method would be cumbersome if it was necessary to determine the ghost particle velocity every time by trial and error. Therefore, it is necessary to establish relationships between ghost particle velocity and case condition such as bed slope, bed surface roughness, water depth, and particle resolution.
As a powerful datamining method, the evolutionary polynomial regression with multiobjective genetic algorithm (MOGAEPR) technique is widely used to construct nonlinear polynomial expression [
Based on the simulation results presented in Section
From Table
Statistical value of empirical formulae from EPRMOGA.
Formulae  Candidate variables  Complexity  Accuracy  










SSE  CoD  
Equation ( 
−  —  +  −  +  +  +  6  4  0.34  0.88 
Equation ( 
—  −  +  −  +  +  —  6  2  1.03  0.77 
Equation ( 
—  −  +  —  —  +  +  4  3  0.28  0.86 
In this study, an indepth study on the simulation of the openchannel flow over roughness bed based on WCSPH was performed. Through the numerical test, the influence of bed surface roughness on the flow structure was clarified. Numerical tests showed that the Smagorinskybased SPS turbulence model performs poorly near the rough bed surface in the openchannel flow, showing deviation of the vertical velocity and turbulent shear stress distribution from the theoretical values, but the improved mixedlengthbased SPS turbulence model achieved satisfactory fluid structure.
Based on the principle of the wall function model in the grid method, an improved meshless wall function was developed for smallscale rough boundary and the sensitivity of this method was determined and error analysis was performed. There was relatively small numerical error in the mainstream and nearsurface regions, but the error varied widely for the inner region. This error is positively correlated with the channel bed slope and the equivalent roughness. In addition, for the convenience of model application, empirical formulae are proposed to calculate the ghost particle velocity under different hydraulic conditions using data mining. The proposed rough bed model is expected to be useful for openchannel turbulence simulation.
The training data of the multiobjective genetic algorithm used to support the findings of this study are included within the supplementary information file.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The improved WCSPH code used in this paper was developed from the open source SPHysics code developed by Prof. M. GomezGesteira at University of Vigo, Prof. R.A. Dalrymple at Johns Hopkins University, and others. This research was supported by the National Key Research and Development Project (grant nos. 2017YFC0403600 and 2016YFE0201900), the State Key Laboratory of HydroScience and Engineering, Tsinghua University, Beijing, China (grant no. 2017ky04); and the China Postdoctoral Science Foundation (grant no. 2018M641372).
Supplemental test results including average currently in inflow and outflow regions, friction velocity, Reynolds number, Froude number, friction Reynolds number, and the velocity of ghost particle in the wall function model can be accessed from the online version of the paper. Totally, 159 sets of training data in terms of hydraulic smooth flow, hydraulic rough flow, and hydraulic transition flow are opensourced for a potential researcher.