Flow around cylinders is a classic issue of fluid mechanics and it has great significance in engineering fields. In this study, a twodimensional hydrodynamic lattice Boltzmann numerical model is proposed, coupling wave radiation stress, bed shear stress, and wind shear stress, which is able to simulate wave propagation of flow around cylinders. It is based on shallow water equations and a weight factor is applied for the force term. An enhanced equilibrium BhatnagarGrossKrook (BGK) scheme is developed to treat the wave radiation stress term in collision step. This model is tested and verified by two cases: the first case is the flow around a single circular cylinder, where the flow is driven by current, wave, or both wave and current, respectively, and the second case is the solitary waves moving around cylinders. The results illustrate the correctness of this model, which could be used to analyze the detailed flow pattern around a cylinder.
The phenomena of flow around cylinders, which represent blunt bodies, widely exist in aviation, mechanical, and environmental engineering. In recent years, an increasing number of problems about complex flow around cylinders have been raised with the development of coastal engineering projects. Therefore, this topic attracts much attention among researchers.
Flow around cylinders is a classic and complicated problem. The cross section is contracted, the velocity increases, and the pressure decreases along the path when the flow encounters cylinders. The separation of the boundary layer is formed around cylinders due to the viscous force, which is called the flow around cylinders. Additionally, cylinders are nonstreamline objects, which influence the characteristics of flow around cylinders by many factors, such as the Reynolds number, the surface roughness, the turbulence intensity, and the cylinder size. All these lead to the complexity of flow around cylinders. The wave is one of the most common movement forms in water, and it is worth studying wave motion in shipping, coastal, and ocean engineering. Therefore, the research of wave propagation around cylinders is complicated, but significant.
With the development of the fluid mechanics theory and the continuous updating of computer equipment, computational fluid dynamics has been greatly developed and numerical simulation became an important tool in research. Saiki and Biringen [
The lattice Boltzmann method (LBM) is a promising numerical simulation method of recent decades. Compared to traditional methods, LBM has many advantages: the algorithm is simple; it can deal with complicated boundary conditions; and it is suitable for parallel processing. These superiorities lead to wide usage of LBM in many research fields. Ginzburg and D’Humieres [
At the same time, many scholars have investigated the flow around cylinders based on the LBM. However, most studies are related to the heat transfer around cylinders. Yan and Zu [
In this study, considering wavecurrent interaction, a twodimensional hydrodynamic numerical model is developed based on the LBM. The model couples three types of stresses, including wave radiation stress, wind shear stress, and bed shear stress. Meanwhile, an enhanced local equilibrium function is developed to treat the wave radiation stress. It is used to simulate the propagation of waves in the flow around cylinders, and then two classic examples are used for validation, which can provide characteristics of flow around cylinders.
The twodimensional shallow water equations including the continuity equation and momentum equation can be written in a tensor form as
In (
On account of the lattice Boltzmann method with a D2Q9 lattice, an enhanced equilibrium BGK Scheme is developed in this paper. The wave radiation stress
The discrete evolution process in the LBM with the enhanced force term [
For the D2Q9 lattice shown in Figure
D2Q9 lattice.
An equilibrium distribution function
Moreover, the local equilibrium distribution function must satisfy the following three conditions:
The recover deductions are following the ChapmanEnskog procedure.
Based on (
This model is built based on the verified LBM hydrodynamic model [
The layout sketch of the channel.
This case includes three different tests, which are driven by currents, by waves, and by both wave and current, respectively. The flow variables and wave parameters of three types situations are shown in Table
The flow variables and wave parameters.
Test 


Wave period (s)  Wave amplitude (m) 

1  1  0     
2  0  0  0.5  0.1 
3  1  0  0.5  0.1 
The magnitude of
The magnitude of
When the flow encounters the cylinder, it passes around and a weak area emerges just behind the cylinder, where the circulation and a drop of water surface can be found.
Threedimensional water depth diagram (
In terms of the longitudinal velocity
Velocity vector diagram (
The magnitude of
The magnitude of water surface (
Threedimensional water depth diagram (
The magnitude of
To illustrate the effects of currents and waves, the comparisons of the velocity
Comparison of
Comparison of
This case is a classic cylinder model that has been simulated by many researchers before [
Figure
The plots of threedimensional perspective view of water surface.
Furthermore, a solitary wave around four cylinders is simulated. The simulation is conducted in an area of constant water depth (1 m), being 60 m long and 40 m wide. The distance between the centers of any two adjacent cylinders is 7.17 m, and the radius of four cylinders is the same with 2 m (see Figure
The layout of the channel.
Figure
The plots of threedimensional perspective view of water surface.
This paper proposes a twodimensional hydrodynamic model to investigate the wavecurrent interaction around cylinders. The lattice Boltzmann method was used to discretize the mathematical model in numerical simulation. A BKG scheme with an enhanced equilibrium is used to treat the wave radiation stress. The numerical results of both cases are in good agreement with practicalities and previous studies, demonstrating that this new model is able to produce reliable results for studying cylinders problems.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was supported by the National Natural Science Foundation of China (51379001) and the Open Fund of State Key Laboratory of Hydraulics and Mountain River Engineering (SKHL1518).