Numerical Modeling of Wave-Current Flow around Cylinders Using an Enhanced Equilibrium Bhatnagar-Gross-Krook Scheme

1The Key Laboratory of Water and Sediment Sciences of Ministry of Education, School of Environment, Beijing Normal University, Beijing 100875, China 2State Key Laboratory of Hydraulics and Mountain River Protection, College of Water Resource and Hydropower, Sichuan University, Chengdu 610065, China 3State Key Laboratory of Simulation and Regulation ofWater Cycle in River Basin, China Institute ofWater Resources andHydropower Research, Beijing 100038, China


Introduction
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 non-streamline 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 [1] introduced a virtual boundary technique to simulate uniform flows around cylinders, and the oscillations caused by this method can be attenuated by high-order finite differences.Based on this, Lima E Silva et al. [2] proposed the physical virtual model in which this immersed boundary was represented with a finite number of Lagrangian points, distributed over the solid-fluid interface.Ofengeim and Drikakis [3] presented numerical research on the interaction of plane blast waves and a cylinder, revealing that the blast-wave duration significantly influenced the unsteady flow around the cylinder.Breuer [4] computed the turbulent flow around a cylinder (Re = 3900) via large eddy simulation.Meneghini et al. [5] used a fractional step method to simulate laminar flows between two cylinders.Hu et al. [6] built a fully nonlinear potential model based on a finite element method to investigate the wave motion around a moving cylinder, and it provided certain important features that were absent in the linear theory.Wu and Shu [7] proposed a local domain-free discretization method that is able to simulate flow around an oscillating cylinder easier due to its advantage of handling the boundary.Claus and Phillips [8] used spectral/hp element methods to study the flow around a confined cylinder.The nonconforming spectral element method and adaptive meshes method were tested by Hsu et al. [9], demonstrating its feasibility on curve surfaces of cylinder.
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 [10] introduced a new kind of boundary conditions, improving the accuracy close to the quasianalytical reference solution.Jiménez-Hornero et al. [11] used LBM to simulate the turbulent flow structure in an open channel with the influence of vegetation.Liu et al. [12] established a two-dimensional multiblock lattice Boltzmann model for solute transport in shallow water flows.Based on the Chapman-Enskog process, Liu and Zhou [13] proposed a lattice Boltzmann model to simulate the wetting-drying front in shallow flows.
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 [14] presented a numerical strategy to handle curved and moving boundaries for simulating viscous fluid around a rotating isothermal cylinder with heat transfer.Rabienataj Darzi et al. [15] used the LBM to analyze mixed convection flow and heat transfer between two hot cylinders.However, up to now, there is no LBM model for wave-current flow around cylinders.
In this study, considering wave-current interaction, a two-dimensional 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.

Methodology
where   represents the wind shear stress and   represents the bed shear stress.
Wave Radiation Stress (  ).Longuet-Higgins and Stewart [16] defined the difference between the time-average momentum value and the static water pressure on the water column per unit area, known as the wave radiation stress.In (3), the wave radiation stresses   ,   ,   , and   are determined via local wave parameters.The wave radiation stress along the direction of wave propagation is   = (2  / − 1/2), and the lateral one is   = (  / − 1/2), where  = (1/8) 2  ,  is wave velocity,   represents the group velocity, and   represents the wave height.The conversion is conducted in the Cartesian coordinate system [17]: where  represents the angle between the wave direction and the -axis.
Bed Shear Stress (  ).Bed shear stress (  ) is generated by the wave-current interaction in the  direction, calculated as follows [18]: in which   represents the bed friction coefficient, which may be either constant or calculated from   = / 2  , where   represents the Chezy coefficient given based on the Manning coefficient   , represents the wave bottom frictional velocity;   represents the wave-current influence factor, which is equal to 0.917 for the waves and currents are in the same direction, −0.1983 for perpendicular relation and 0.359 for other angles [19]; and   represents the wave friction factor, which is from 0.006 to 0.001 in practice [20].
Wind Shear Stress (  ).Wind shear stress (  ) is usually expressed as where   is the density of air;   is the resistance coefficient; and   is the component of the wind velocity in  direction.

Lattice Boltzmann Method. 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   is treated in local equilibrium function at collision step.
The discrete evolution process in the LBM with the enhanced force term [12,21] can be written as where the external force term can be written as where   represents the weight factor:   = 4/9 for  = 0;   = 1/9 for  = 1, 3, 5, 7;   = 1/36 for  = 2, 4, 6, 8.   represents the distribution function of particles;  eq  represents the local equilibrium distribution function; Δ represents the time step;  represents the single relaxation time; and   represents the velocity vector of a particle in the  link.
For the D2Q9 lattice shown in Figure 1, each particle moves one lattice at its direction.The velocity of each particle is defined by where  = Δ/Δ and Δ is the lattice size.An equilibrium distribution function  eq  can be expressed as Therefore, the equilibrium distribution function can be written as where there must be due to symmetry.Moreover, the local equilibrium distribution function must satisfy the following three conditions: Hence, the relations among  0 , , and Ã are

Mathematical Problems in Engineering
We can obtain Therefore, the enhanced equilibrium distribution function  eq  is From Chapman-Enskog expansion, we have Substitution of ( 17) and ( 18) into (7), one can obtain To order Δ, it is To order Δ 2 , it is Substitution of ( 20) into (21), we have Taking ∑[(20) + Δ × (22)] about  provides Taking ∑   [(20) + Δ × (22)] about  provides According to the law of conservation of mass, we know If the center-scheme for the force term is applied, evaluation of the other terms in the above equations using ( 13) and ( 25) simplifies ( 23) and ( 24) and obtains with Substitution of ( 27) into (26) leads to the following equations which were referred to as wave-current coupling equations

Wave-Current Flow around a Circular
Cylinder.This model is built based on the verified LBM hydrodynamic model [22].The layout diagram of the channel is shown in Figure 2. The length is 7 m, and the width is 2 m.The bottom is flat and a solid cylinder with a 0.12 m radius is located at 2 m, 1 m.The initial water depth is 1 m and the flows go from the left to the right.The computational domain is divided by 140 × 40 computational grids.The time step is 0.01 s.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 1 ( 0 is initial horizontal velocity and V 0 is initial vertical velocity).
Test 1 (driven by the current).It can be seen that the water depth and flow velocity obviously varied due to the presence of the middle column (see Figure 3).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.
Test 2 (driven by the wave).The initial water is still and a wave maker is set at the inlet, where the incident waves are parallel generated in the -axis.The water depth is intuitively depicted in Figure 4, where one can find regular wave propagation although there is a deformation caused by the cylinder.
In terms of the longitudinal velocity , it is not always positive, as the flow is only driven by the wave (see Figures 4 and 5).This phenomenon is further described in Figure 6.
Test 3 (driven by both wave and current).Under the interaction effects of waves and currents, the wave run-up is pushed higher than before (see Figures 7 and 8), and the deformation process is more apparent (see Figure 9).
To illustrate the effects of currents and waves, the comparisons of the velocity  and the water depth ℎ are plotted in Figures 10 and 11, respectively.It can be found that the wavecurrent interaction is not a simple superposition of waves and currents, and furthermore, wave-current interaction effects are greater than summation of these two effects separately.

Solitary Waves around
Cylinders.This case is a classic cylinder model that has been simulated by many researchers before [23,24].In this section, a solitary wave around a cylinder is simulated first.The whole water channel is 60 m long and 30 m wide, and there is a circular cylinder with  = 1.5875 m in the center of the channel.The initial solitary wave with amplitude of 0.4 m is incident from left.Lattice size is 0.4 m, and the time step is 0.01 s.
Figure 12 shows the plots of three-dimensional perspective view of water surface at  = 8.7 s and 16 s.The solitary wave climbs up and a sequence of significant disperse waves after initial wave encountering the cylinder can be observed.At  = 16 s, the solitary wave is about to propagate out of the area.At the same time, disperse waves are fully developed   to cover almost all the channel behind the frontal wave.The results are consistent with previous research [23].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 13).The whole domain is divided by 150 × 100 computational grids.The time step is 0.01 s.

Conclusion
This paper proposes a two-dimensional hydrodynamic model to investigate the wave-current 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.

Figure 2 :
Figure 2: The layout sketch of the channel.
The magnitude of V
is the three-dimensional water depth of the wave around four cylinders at different times.The climbing up of water on the first cylinder can be observed at  = 7 s.At  = 9 s, the solitary wave encounters two middle cylinders and then runs up the front sides.Furthermore, a circular back disperse wave begins to turn up and propagates along the channel.The height of the middle part of the solitary wave decreases significantly due to obstruction of the frontal cylinder.The solitary wave encounters the rear cylinder at  = 10.5 s.The results show that the back disperse waves induced by the frontal cylinder form a circular wave pattern propagating towards the left open boundary.At the same time, the circular disperse waves, emerging from the two

Figure 12 :Figure 13 :Figure 14 :
Figure 12: The plots of three-dimensional perspective view of water surface.

Table 1 :
The flow variables and wave parameters.