Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications
Introduction
Currently, most of flow problems can be simulated by finite difference, finite volume and finite element methods. When a complex geometry is immersed in the flow domain, these approaches often involve the tedious grid generation and complicated solution process arising from strong coupling between the discretization of governing equations and implementation of boundary conditions. To simplify the solution process, it is desired to develop an approach which can decouple the solution of governing equations and the implementation of boundary conditions. The immersed boundary method (IBM) is such an approach. It was initially proposed by Peskin [1] in 1970s for simulation of blood flows in the heart. This method uses a fixed Eulerian mesh for the flow field, and a set of Lagrangian points to represent the boundary immersed in the fluid. The basic idea of IBM is to treat the physical boundary as deformable with high stiffness. A small distortion of the boundary will yield a force which tends to restore the boundary into its original shape. The balances of such forces are distributed into the Eulerian nodes and the Navier–Stokes (N–S) equations with a body force are solved on the whole domain including exterior and interior of the body.
Based on the work of Peskin [1], various improvements have been made recently [2], [3], [4], [5]. Among the remarkable works, Lai and Peskin [2] proposed a so-called second-order accurate immersed boundary method which was used to simulate flows over a circular cylinder. The interaction between the fluid and the immersed boundary is modeled using a well-chosen Dirac delta function. As compared to their first-order model [2], numerical viscosity is reduced in this model. However, due to the use of the first-order Dirac delta function interpolation, the model does not truly have the second-order accuracy. By employing the feedback forcing to represent the solid body, Goldstein et al. [3] developed a model called the virtual boundary method which was used to simulate laminar and turbulent flows. Different from the way in Peskin [1] where Hook’s law is used to calculate the restoring force at the boundary, feedback forcing method directly computes the restoring force by using fluid and boundary velocities. However, there are two user-specified parameters in this model, which may lead to inconvenience for its application. Ye et al. [4] proposed a method named the Cartesian grid method which combines the cell-merging technique with the finite volume method to simulate two-dimensional unsteady incompressible viscous flows. Due to irregular shapes of cells cut by the boundary, complex interpolation is required to calculate the fluxes, which may affect the computational efficiency. Lima E Silva et al. [5] proposed a version named physical virtual model to simulate an internal channel flow and the flow around a circular cylinder. This model is very similar to the work of Peskin [1] except that the restoring force is calculated by applying the momentum equations at the boundary points. The process involves tedious derivative approximation and interpolation of velocity and pressure.
In the above IBM versions, the solution of flow field is obtained by solving incompressible Navier–Stokes (N–S) equations. As an alternative computational technique to the N–S solvers, the lattice Boltzmann method (LBM) [6] has been proven to be an efficient approach for simulation of flow field. LBM is a particle-based numerical technique, which studies the dynamics of fictitious particles. The major advantage of LBM is its simplicity, easy implementation, algebraic operation and intrinsic parallel nature. Like the IBM, the standard LBM is usually applied on the Cartesian mesh. Due to this common feature, it is desirable to combine these two methods together. Many efforts have been made in this aspect. The first attempt was given by Feng and Michaelides [7], [8]. They successfully applied IB-LBM to simulate the rigid particle motion. Niu et al. [9] proposed the momentum exchange-based IB-LBM for simulation of several incompressible flows. Peng et al. [10] developed the multi-block IB-LBM for simulation of flows around a circular cylinder and an airfoil. Both works adopt the multi-relaxation LBM to get the flow field.
The key issue in the IBM is the computation of restoring force. The popular way is the penalty method [1], [7]. This method introduces a user-defined spring parameter which may have a significant effect on the computational efficiency and accuracy. Another way is the direct forcing method, which was first introduced by Fadlun et al. [11]. As direct forcing method needs to solve N–S equations to compute the force at the boundary point, it may spoil the merits of LBM when IBM is combined with it. Recently, a simple method for computing the restoring force was proposed by Niu et al [9], in which the momentum exchange at the boundary is used to compute the force. As compared to the work of Ladd [12], the force computation of Niu et al. [9] is simpler and more convenient since the force is computed at the boundary points and one does not need to care the details of the boundary position and mesh points.
In the conventional IBM, the non-slip boundary condition is not enforced in its solution process. This is different from the conventional body-fitted solvers, where the non-slip condition is imposed at the boundary. For example, in the LBM, the non-slip condition is implemented by the bounce-back (BB) rule. Ladd [12] well applied BB-LBM to simulate solid–fluid suspensions. Since then, many efforts have been made to develop high-order BB-LBM to simulate flows around complex geometries and moving objects. Bouzidi et al. [13] presented quadratic interpolation to implement bounce-back boundary condition on the boundary of moving objects. The implementation has the second-order accuracy, and the circular Couette flow and steady flow over a periodic array of circular cylinders are accurately simulated. Chun and Ladd [14] proposed an equilibrium interpolation LBM for simulation of flows in narrow gaps. Only the equilibrium part of distribution function is interpolated on the boundary to achieve the second-order accuracy. With the help of multi-relaxation time model, this method can accurately simulate highly viscous flows. As compared to the body-fitted solvers, the major drawback of IBM is that the non-slip boundary condition is only approximately satisfied at the converged state. As a consequence, some streamlines may penetrate the solid body. To remove this drawback, Kim et al. [15] introduced the mass source or sink into the computation. Using this way, the non-slip condition can be well kept, but the complexity is introduced into the computation. Recently, Shu et al. [16] found that unsatisfying of non-slip boundary condition in IBM is in fact due to pre-calculated restoring force. Using the fractional step technique, they concluded that, adding a body force in the momentum equations is equivalent to make a correction in the velocity field. To enforce the non-slip boundary condition, the velocity correction (or restoring force) should be considered as unknown, which is determined in such a way that the velocity at the boundary interpolated from the corrected velocity field satisfies the non-slip boundary condition. In the work of Shu et al. [16], the velocity correction is made at adjacent points to the boundary along the horizontal and vertical mesh lines. The approach is very simple. However, it only has the first-order accuracy and the computed forces at the boundary have some oscillations. The reason may be that the linear relationship is applied along the horizontal/vertical mesh lines and the smooth Dirac delta function is not used.
In this work, we will follow the idea of Shu et al. [16] to propose a variant of IB-LBM. To effectively consider the body force in the lattice Boltzmann equation (LBE), we adopt the model proposed by Guo et al. [17]. As shown in [17], the conventional LBE with body force such as the one used in [7], [8] cannot properly consider the discrete lattice effects to the density and momentum. To remove this drawback, Guo et al. [17] proposed a representation of forcing term in the LBE which is similar to the form in the work of He et al. [18]. In this model, the velocity is naturally contributed by two parts. One is from the density distribution function, while the other is from the body force. Similar to the work of Shu et al. [16], we can term the velocity from the density distribution function as the intermediate velocity, and the velocity from the body force as the velocity correction. In the conventional IB-LBM, the velocity correction is pre-computed and cannot be manipulated to satisfy the non-slip boundary condition. In this work, it is considered as unknown and is determined from enforcement of non-slip condition. Through the relationship between the force density and velocity correction, the drag and lift forces can be directly computed from the obtained velocity correction. The solution process is exactly the same as the conventional IB-LBM except that the body force is determined by enforcing the non-slip boundary condition in this work. The present method is validated by its application to simulate the steady and unsteady flows past a circular cylinder and airfoil. The obtained results basically agree well with available data in the literature. Since the boundary condition is accurately satisfied, the obtained numerical results do not show any streamline penetration to the solid body.
Section snippets
Conventional immersed boundary-lattice Boltzmann method
For the viscous incompressible flows in a two-dimensional domain Ω containing an immersed boundary in the form of closed curve Γ, as shown in Fig. 1, the governing equations of immersed boundary method can be written asHere x, u, p and f are the Eulerian coordinates, fluid velocity, fluid pressure and force density acting on the fluid phase, respectively. X and F stand for the
Implicit velocity correction-based immersed boundary-lattice Boltzmann method
We start with the LBE. As shown in [17], [20], the LBE with external force (Eq. (6)) cannot properly consider the discrete lattice effects to the density and momentum. In order to correctly recover the viscous and incompressible N–S equations involving the external force, the contribution of the force to both momentum ρu and momentum flux ρuu should be considered. The first attempt to propose a better representation of forcing term in LBE was given by He et al. [18]. Later, Guo et al. [17]
Numerical test of overall accuracy
LBM is adopted in the present work to obtain the basic flow field. As we know, LBM has the second-order accuracy in time and space. However, when LBM is combined with IBM, the Dirac delta function interpolation and the smoothing kernel are used to get the velocity correction at Eulerian points and the velocity at boundary points. The interpolation only has the first-order accuracy. Although it is only applied in the region nearby the boundary, it may have effect on the global accuracy of
Conclusions
This paper presents a variant of immersed boundary-lattice Boltzmann method (IB-LBM). To well consider the effect of external force to the momentum and momentum flux as well as the discrete lattice effect, the lattice Boltzmann equation with external forcing term proposed by Guo et al. [17] is adopted. In this lattice Boltzmann model, both the density distribution function and the external force contribute to the fluid velocity. As the non-slip boundary condition at solid surface is usually for
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