Stability Improvement of the Immersed Boundary–Lattice Boltzmann Coupling Scheme by Semi-Implicit Weighting of External Force

Abstract

The immersed boundary–lattice Boltzmann (IB-LB) coupling scheme is known as an efficient scheme for fluid–structure interactions (FSIs). However, the conventional IB-LB schemes suffer from instability because they involve a high-Reynolds-number flow or a larger stiffness structure. An averagely weighted iteration approach is presented to improve the stability restriction in this paper. This new approach, which improves the stability by mitigating the high-frequency fluctuations, is implemented by iteratively calculating the external force, and averagely weighting the force obtained at every iterative step. Five cases are simulated to verify the accuracy and effectiveness of the present approach. Under the premise of maintaining the accuracy of the conventional IB-LB method, the implementation of the present approach can significantly enhance the numerical stability. Compared with the conventional IB-LB method, the present approach can significantly expand the material parameter range for simulation; in particular, this approach qualitatively improves the upper limit of the bending rigidity coefficient by approximately 8000 times. To use the outstanding stability of the present approach, the IB inertia force can be directly incorporated into the simulation. In addition, under the low-viscosity condition, the present approach can effectively simulate the large-deformation FSI problem.

1. Introduction

The immersed boundary (IB) method was first proposed by Peskin in the 1970s to simulate the fully fluid–flexible structure interaction problem. This method uses a fixed Eulerian mesh to simulate the flow field and a set of Lagrangian points to represent the boundary immersed in the fluid. The interaction between the fluid and the immersed boundary is implemented by a discrete Dirac delta function, which spreads the force of the deformable elastic boundary to the nearby fluid grid nodes and interpolates the boundary velocity from the local flow velocity to update the boundary position. In the most early studies [1,2,3,4], the immersed boundary is considered as a tensile structure, and hardly has any bending resistance. However, most real flexible materials are nearly incompressible (cannot deform under stretching or compression) and have bending rigidity (can resist bending and torsion). Therefore, the bending rigidity cannot be ignored, particularly when the deformation curvature is relatively large. The bending rigidity is ignored in the early literature, mostly because of the stability restraint of the IB method, where a small time step is required to maintain stability (much smaller than on the imposed time step by explicitly differencing the advective or diffusive term) [5,6]. The restraint may be particularly severe for a larger bending rigidity or a low fluid viscosity.

Much effort has been expended to alleviate this severe restriction. For example, the fluid viscosity has been artificially increased by two orders of magnitude [7]. However, this solution can only be used in special problems. A three-step approach to update the IB boundary velocity was presented by Shu [8], but it could not obtain an obvious effect [9,10]. A subspace iteration method was presented by Yin et al. [11], which could remedy the conventional IB scheme to a limited extent but was difficult to expand to three-dimensional cases. A subgrid resolution IB method that improved the stability against the slender fibers issue, was proposed by Maxian [12]. Compared with the above methods, an unconditional stability IB scheme presented by Newren et al. [13] has more advantages in terms of effectiveness and applicability. However, due to the need to iteratively solve large matrices, the scheme may consume more computational resources and time, so it has limited practicability.

The lattice Boltzmann (LB) method is an alternative simulation technique for complex fluid systems. Its easy implementation, intrinsic parallelism, and good suitability for numerous fluid flow problems have been demonstrated by many works. The immersed boundary–lattice Boltzmann (IB-LB) integrates the advantages of the two methods, and has extensive prospects in research and application [2,4,7]. However, the IB-LB coupling scheme also suffers from the stability restriction when it simulates a flexible structure. Some scholars have attempted to alleviate this instability. Tian et al. [14] integrated the three-step approach [8] with the IB-LB scheme, but this approach did not give the expected effect. Hao and Zhu [15] introduced an unconditional stable scheme, where the time step could be increased up to 30 times compared with that of the explicit method. Wu et al. [16] presented an implicit IB-LB coupling method based on the unconditional stability IB scheme [13]. This method can alleviate problems such as instability, restrictive choices of time step and boundary rigidity, and easily incorporate the boundary mass. Wu et al. [17] proposed a robust IB-LB method employing the fractional step technique to simulate fluid–structure interaction problems. Unlike the methods mentioned above, which require iteratively solving large matrices, Cheng et al. [18] improved the stability of the IB-LB coupling scheme by using an iterative external forcing method, which is the basis of the present study, but no thorough investigation was reported in their paper.

In this paper, an averagely weighted iteration approach is presented to improve the stability restriction of the conventional IB-LB coupling scheme. The present approach belongs to the semi-implicit schemes and mitigates the numerical oscillations by averagely weighting the external force term in the iteration process. The present approach is easy to use, can significantly improve the stability, and does not significantly increase the computing cost.

The remainder of the paper is organized as follows. Section 2 briefly describes the conventional IB-LB coupling schemes. Section 3 presents the new averagely weighted iteration approach. In Section 4, the accuracy and effect on the stability of the proposed approach are verified. Section 5 concludes the paper.

2. Immersed Boundary–Lattice Boltzmann Coupling Scheme

2.1. Immersed Boundary Method

In the IB method, the flow field is described on a fixed Eulerian mesh, and the IB is represented by sets of Lagrangian points. The formulation is as follows [4]:

$$\begin{array}{c}\hfill \frac{\partial \rho }{\partial t}+\nabla ·\left(\rho \mathbf{u}\right)=0\end{array}$$

(1)

$$\begin{array}{c}\hfill \frac{\partial \left(\rho \mathbf{u}\right)}{\partial t}+\nabla ·\left(\rho \mathbf{uu}\right)=-\nabla p+\upsilon \nabla ·\left[\rho (\nabla \mathbf{u}+{\left(\nabla \mathbf{u}\right)}^T)\right]+\mathbf{f}\end{array}$$

(2)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\mathbf{X}}_l(s,t)}{\partial t}}=\mathbf{U}({\mathbf{X}}_l,t)={\int }_{{\Omega }_f}\mathbf{u}(\mathbf{x},t)\delta (\mathbf{x}-{\mathbf{X}}_l(s,t))\mathrm{d}\mathbf{x}\end{array}$$

(3)

$$\begin{array}{c}\hfill \mathbf{f}(\mathbf{x},t)={\int }_{{\Gamma }_b}\mathbf{F}(s,t)\delta (\mathbf{x}-{\mathbf{X}}_l(s,t))ds\end{array}$$

(4)

$$\begin{array}{c}\hfill \mathbf{F}(s,t)={\mathbf{F}}_e(s,t)-{\mathbf{F}}_d(s,t)\end{array}$$

(5)

where $\mathbf{x}$, p, $\mathbf{u}$, and $\mathbf{f}$ are the Eulerian spatial coordinate, flow pressure, fluid velocity, and external force density, respectively; ${\mathbf{X}}_l$, $\mathbf{F}$, and $\mathbf{U}$ are the Lagrangian IB position, IB force density, and IB moving speed, respectively; and $\delta \left(r\right)$ is the delta function, which can be written as:

$${\delta }_h\left(x,y\right)=h^{-2}\varphi \left({\displaystyle \frac{x}{h}}\right)\varphi \left({\displaystyle \frac{y}{h}}\right)$$

(6)

where h is the mesh spacing, and $\varphi \left(r\right)$ is normally selected as:

$$\varphi \left(r\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{4}}\left(1+\cos ({\displaystyle \frac{\pi r}{2}}\right)),\hfill & \mathrm{if}\mid r\mid \le 2\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(7)

Equations (1) and (2) are the N–S equations with external force $\mathbf{f}$ in Eulerian form for the fluid flow in domain ${\Omega }_f$, whereas Equations (3)–(5) are the dynamic equations in Lagrangian form for the boundary ${\Gamma }_b$. Equations (3) and (4) describe the interaction between the IB and the fluid flow. The IB imposes the flow velocity on the boundary to obtain IB velocity $\mathbf{U}$ and the fluid flow spreads the Lagrangian IB force to the fluid to obtain the Eulerian force.

$\mathbf{F}$ in Equation (5) is the reaction force of IB, which is contributed to by the elastic force ${\mathbf{F}}_e$ and the inertial force ${\mathbf{F}}_d$, and given by:

$$\begin{array}{cc}\hfill {\mathbf{F}}_e(s,t)& ={\mathbf{F}}_s(s,t)+{\mathbf{F}}_b(s,t)+{\mathbf{F}}_f(s,t)\hfill \\ & =\frac{\partial }{\partial s}\left[K_s\frac{\partial \mathbf{X}}{\partial s}\right]-K_b\frac{{\partial }^4\mathbf{X}}{\partial s^4}+K_f(\mathbf{Z}-\mathbf{X})\hfill \end{array}$$

(8)

$$\begin{array}{c}\hfill {\mathbf{F}}_d(s,t)=m_s\frac{\partial \mathbf{U}(s,t)}{\partial t}\end{array}$$

(9)

The right-hand side in Equation (8) consists of a tension force ${\mathbf{F}}_s$, a bending force ${\mathbf{F}}_b$, and a fastening force ${\mathbf{F}}_f$, which represent the stretching/compression effect, the bending/torsion effect, and proscribing speed for certain structure points, respectively. The discrete coefficients $K_s$, $K_b$, and $K_f$ are the tension stiffness, bending rigidity, and fastening stiffness (an artificial coefficient), respectively. According to the literature [19], Equation (8) is discretized as follows:

$${\left({\mathbf{F}}_s\right)}_l={\displaystyle \frac{K_s}{{(\Delta s)}^2}}\sum _{m=1}^{n_f-1}(\left|{\mathbf{X}}_{m+1}-{\mathbf{X}}_m\right|-\Delta s){\displaystyle \frac{{\mathbf{X}}_{m+1}-{\mathbf{X}}_m}{\left|{\mathbf{X}}_{m+1}-{\mathbf{X}}_m\right|}}({\delta }_{ml}-{\delta }_{m+1,l})$$

(10)

$${\left({\mathbf{F}}_b\right)}_l={\displaystyle \frac{K_b}{{(\Delta s)}^4}}\sum _{m=2}^{n_f-1}({\mathbf{X}}_{m+1}+{\mathbf{X}}_{m-1}-2{\mathbf{X}}_m)(2{\delta }_{ml}-{\delta }_{m+1,l}-{\delta }_{m-1,l})$$

(11)

$${\left({\mathbf{F}}_f\right)}_l=K_f({\mathbf{X}}_l-{\mathbf{Z}}_l)$$

(12)

Here, ${\left({\mathbf{F}}_s\right)}_l$, ${\left({\mathbf{F}}_b\right)}_l$, and ${\left({\mathbf{F}}_f\right)}_l$ are the three Lagrangian force densities associated with the node l ($l=1,2,···,n_f$ is the serial number of the Lagrange points), ${\mathbf{Z}}_l(s,t)$ denotes the position of the fixed point. ${\delta }_{ml}$ is the Kronecker symbol, which is defined as:

$${\delta }_{ml}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}m=l\hfill \\ 0,\hfill & \mathrm{if}m\ne l\hfill \end{array}\right.$$

(13)

Equation (9) may be expressed in the finite difference form:

$${\left({\mathbf{F}}_d\right)}_l=m_s{\displaystyle \frac{{\mathbf{X}}^{n+1}-2{\mathbf{X}}^n+{\mathbf{X}}^{n-1}}{\Delta t}}$$

(14)

Here, ${\left({\mathbf{F}}_d\right)}_l$ is the Lagrangian force density associated with the node l ($l=1,2,···,n_f$ is the serial number of the Lagrange points); $m_s$ is the density of the flexible structure; $K_s$, $K_b$, $K_f$, and $m_s$ are the material parameters of the flexible structure, and the key factors to determine the boundary deformation and affect the simulation stability. However, the conventional IB-LB method suffers from severe instability restriction, which can hardly handle problems with a larger material parameter (particularly $K_b$ or $m_s$). To improve the problem is the main purpose of this paper. Without loss of generality, the material parameters are non-dimensionalized as follows:

$${\overline{K}}_s={\displaystyle \frac{K_s}{{\rho }_f{U}^{2}L}},{\overline{K}}_b={\displaystyle \frac{K_b}{{\rho }_f{U}^2{L}^3}},{\overline{K}}_f={\displaystyle \frac{K_f}{{\rho }_f{U}^{2}L}}$$

(15)

where ${\rho }_f$ is the fluid density, and U and L are the characteristic velocity and length, respectively.

2.2. The LB Method for the Fluid Flow

2.2.1. Basic Formulation of LBM with Multi-Relaxation-Time Collision

In this work, the LB model with the multi-relaxation-time (MRT) collision operator is used for the fluid flow simulation. The most common two-dimensional LB model uses a square lattice with nine discrete velocity directions (denoted as D2Q9). The MRT-LB model with external force may be written as:

$$f_a(\mathbf{x}+{\mathbf{e}}_a\delta t,t+\delta t)-f_a(\mathbf{x},t)=-M^{-1}\mathbf{S}[m_a(\mathbf{r},t)-m_a^{eq}({\mathbf{r}}_i,t)]+\delta t{F}_a$$

(16)

where $\{f_{\alpha }(x,t):\alpha =0,1,\cdots ,8\}$ are the discrete distribution functions at position $\mathbf{x}$ and time t; $\{F_{\alpha }:\alpha =0,1,\cdots ,8\}$ is the external force term; and $\{m_{\alpha }(\mathbf{r},t):\alpha =0,1,\cdots ,8\}$ are the moments of the distribution functions and expressed as $m={(\rho ,e,\epsilon ,j_x,q_x,j_y,q_y,p_{xx},p_{xy})}^T.$ The relation between m and distribution function f is expressed as $m=Mf$, where M is the transformation matrix [20]; $m^{eq}$ is the equilibrium moment, which is expressed as:

$$m^{eq}=\rho {\left(1,-2+3u^2,1-3u^2,u_x,-u_x,u_y,-u_y,u_x^2-u_y^2,u_x{u}_y\right)}^{\mathbb{T}}$$

(17)

The diagonal collision matrix $\mathbf{S}$ is given by:

$$\mathbf{S}=\mathrm{diag}\left(s_{\rho },s_e,s_{\epsilon },s_{\chi },s_q,s_{\chi },s_q,s_{\nu },s_{\nu }\right).$$

(18)

The fluid density $\rho$ and flow velocity $\mathbf{u}$ are defined as:

$$\begin{array}{c}\hfill \rho =\sum _{\alpha }{f}_{\alpha },\rho \mathbf{u}=\sum _{\alpha }{\mathbf{e}}_{\alpha }{f}_{\alpha }\end{array}$$

(19)

2.2.2. Introducing the External Force into the LB Model

The external force term of the LB model plays a significant role in the accuracy and stability of simulations. The literature [18] reports that, when the IB material parameters are relatively large or the fluid viscosity is small, severe numerical fluctuations can be detected in the external force term between adjacent timesteps if an explicit method is applied to discretize the IB-LB coupling scheme. The numerical fluctuations may initialize numerical instability. Therefore, an averagely weighted iteration (AWI) procedure is proposed in this paper, which improves the stability of the IB-LB coupling scheme by modifying a special introducing approach of the external force term (proposed by Cheng and Li [21], and will be called Cheng’s approach hereafter). The characteristic of Cheng’s approach, where the force term is split into two parts (the present and the next time step force effects), is the basis of the AWI procedure. Cheng’s force term in Equation (16) is written as:

$${\mathbf{F}}_a={\displaystyle \frac{1}{2}}[g_a(x,t)+g_a(x+{\mathbf{e}}_a\delta t,t+\delta t)]$$

(20)

where $g_a$ is defined as:

$$g_a={\omega }_a\left\{A+3\mathbf{B}·\left[\left({\mathit{e}}_a-\mathit{u}\right)+3({\mathit{e}}_a·\mathit{u}){\mathit{e}}_a\right]\right\}$$

(21)

Here, A is the source term in the continuity equation, and B is the external force term in the momentum equations.

In Equation (21), $g_a(x+e_a\delta t,t+\delta t)$ is an unknown variable, which can generally be treated using two types of approach: the explicit method, where $g_a(x+e_a\delta t,t+\delta t)$ is replaced by $g_a(x+e_a\delta t,t)$ for simplification, and the implicit method (or iterative approach), where $g_a(x+e_a\delta t,t+\delta t)$ is iteratively solved. To distinguish from the averagely weighted iteration (AWI) approach proposed in this paper, the traditional iterative approach is named the directly iterative (DI) approach.

3. Presentation of the Averagely Weighted Iteration Approach

In this section, a preliminary investigation is conducted to analyze the characteristics of the DI approach, and we show that for the strong coupling problems, instead of improving the stability, the DI approach decreases the stability compared with the explicit method. However, further investigation shows that the computed external force fluctuates around the analytical value when the DI approach is used. Thus, the AWI approach is proposed. By averaging over the fluctuated external force in each iteration cycle, the calculated external force is expected to eventually settle at the analytical value, and most factors that contribute to the divergence of the simulation can be eliminated.

3.1. Effectiveness of the DI Approach in Improving the Stability

According to the common view, the DI approach helps to improve the numerical stability. To verify whether this view is true on IBM, a stretched pressurized membrane immersed in a viscous fluid is simulated [13,22]. The resting shape of the membrane before inflation is a circular elastic curve with radius $R=51.491$, and the inflated and stretched shape is defined as an ellipse whose major and minor radii are $R_a=75$ and $R_b=50$, respectively (Figure 1). The membrane is located at the center of a square (width: $L=200$), and composed of $N_b=1200$ Lagrange points. The fluid viscosity is set as $\nu =0.001$. Only the tension stiffness is considered in the simulation and non-dimensionalized as ${\overline{K}}_s$ (following Equation (15)) to scale the stability of the iteration approach. In this case, the fluid density is set as ${\rho }_f=1$, and the characteristic length is set as the resting radius. Additionally, to avoid the transient velocity and facilitate a quantitative comparison, the characteristic velocity is prescribed uniformly as $U=0.1$. All of the above parameters are lattice units.

When ${\overline{K}}_s$ is gradually tuned, the following results are obtained:

(1)

With the explicit method, the executable program begins to run stably only at ${\overline{K}}_s\le 13.6$, when the membrane relaxes to a circular curve.

(2)

With the DI approach (the iteration time per evolution step is set as 10), the critical value is ${\overline{K}}_s\le 5.82$. In addition, the stability rapidly decreases with increasing iteration time (in Figure 2 the IB knots in four diagonal regions).

To further study this issue, a test simulation with ${\overline{K}}_s=13.6$ and $N=20$ is conducted. The simulation breaks down because of numerical overflow, and the external forces at probe points N and S in time step 1 are shown in Figure 3. Since probe points N and S locate at the endpoints of the ellipsoid, and their displacements are mainly in the y direction, the external forces at these points are thus mainly along the y direction. Severe fluctuation occurs after $N=8$ when the DI approach is applied. When the iteration continues, the fluctuation is amplified, which causes the simulation to diverge; thus, instead of improving the stability, the DI approach accelerates the divergence process.

3.2. Averagely Weighted Iteration (AWI) Approach

To resolve the convergence issue, the averagely weighted iteration approach is introduced into the iterative procedure. Combined with the AWI approach, Cheng’s external force term can be recast as:

$$g_a^{\left(s\right)}={\displaystyle \frac{s-1}{s}}{g}_a^{(s-1)}+{\displaystyle \frac{1}{s}}{\omega }_a\{3f^{(s-1)}·\left[\left(e_a-{\mathit{u}}^{\left(0\right)}\right)+3\left(e_a·{\mathit{u}}^{\left(0\right)}\right)e_a\right]\}$$

(22)

where $s=1,2,3\cdots N$ is the iteration index, and N is its upper limit. It can be proven that Equation (22) takes the average value of each iteration result with its former ones.

Figure 4 is the flow chart of the AWI approach. Each iteration cycle (s = 1, 2, ⋯ N) consists of the following steps: (1) Calculate the Lagrangian force density ${\mathit{F}}_{l,n+1}^{\left(s\right)}$ using Formula (5). When $s=1$, the superscript $s-1$ indicates that the variable is directly obtained by the explicit method (not obtained through iteration). (2) Spread the IB force ${\mathit{F}}_{l,n+1}^{\left(s\right)}$ to the fluid external force ${\mathit{f}}_{ij,n+1}^{\left(s\right)}$ using Formula (4). (3) According to Formula (22), calculate the force distribution function $g_{\alpha ,n+1}^{\left(s\right)}$ from the external force term ${\mathit{f}}_{ij,n+1}^{\left(s\right)}$ and the last iteration step $g_{\alpha ,n+1}^{(s-1)}$. (4) Calculate the fluid flow velocity in the IB layer using Formulas (16) and (19). (5) Interpolate the IB speed ${\mathit{U}}_l^{\left(s\right)}$ from the local fluid velocity ${\mathit{u}}_{ij,n+1}^{\left(s\right)}$ using Formula (3). (6) Stop the iteration step and go to next evolution step when s reaches the prescribed iteration time N. (7) If $s<N$, repeat steps (1)–(7) until convergence.

As shown in Figure 3, by introducing the AWI approach, the iterative results can be successfully stabilized to a fixed value, which avoids the oscillation divergence caused by the explicit method. Figure 5 shows the varying history of the Lagrange force (at point N) obtained by the explicit method and AWI approach. When the AWI approach is used, the non-physical force oscillation at the immersed boundary point is obviously improved.

4. Verification and Validation

4.1. Verification of the Accuracy of the Averagely Weighted Iteration (AWI) Approach

In this section, the volume conservation and pressure preservation are used to verify the accuracy of the AWI approach. Using identical parameters to those in the literature [22], the relaxing membrane is simulated by the explicit method and AWI method. Figure 6 shows the oscillating histories of the enclosed area and inside pressure, where lines of different colors represent the results simulated by different methods. The high-frequency boundary elastic waves are shown in the area and pressure curves.

Table 1. Parameters of the nearly relaxed membrane compared with analytical solutions.

Parameters		Enclosed Area, A		Pressure Jump, Δp
		t = 30,000	t = 100,000	t = 30,000	t = 100,000
Simulated	(Explicit)	1.1532	1.1461	0.00607	0.00596
	(AWI)	1.1544	1.1469	0.00601	0.00581
Analytical		1.1576	1.1576	0.00589	0.00589
Relative error (%)	(Explicit)	0.38	0.99	3.1	1.2
	(AWI)	0.28	0.92	2	1.4

presents the data of enclosed area A and the pressure jump at the evolution steps, for comparison with the analytical equilibrium values. Table 1 shows that the volume leakage and pressure loss have identical magnitudes, and both are notably small. For the volume leakage, the maximum relative error obtained by the AWI method is 0.92% and is slightly better than the error obtained by the explicit method (0.99%). For the pressure jump, at t = 30,000 the maximum relative error obtained by the AWI method is 2.0% and is slightly better than the error obtained by the explicit method (3.1%), however, at t = 100,000 the maximum relative error is (1.4%) and is not better than the error obtained by the explicit method (1.2%). Overall, the results simulated by the two methods are notably close, and their relative errors compared with the analytical solution are notably small. Therefore, the AWI approach method does not reduce the accuracy of the conventional explicit IB method.

4.2. Validation of the Stability Improvement

As previously noted, the AWI can improve the stability. However, the effects of introducing the AWI on the thresholds of material parameters or the fluid viscosity, and the effects of the number of iterations on the value of these parameters, are less understood. We study these effects in five typical cases in this section.

4.2.1. Tension Stiffness

The tension stiffness represents a material’s ability to revert to the original state when it is subjected to tension and compression. In this section, an identical case of a relaxing membrane to that in Section 3.1 is simulated by the AWI approach. The Figure 7 shows the maximal tension stiffness simulated with different iteration steps. With the explicit method, the maximal tension stiffness is ${\overline{K}}_s=13.6$, whereas it can be increased with different slopes using the proposed method. According to the curve in Figure 7, when the numbes of iterations are chosen as 5, 7, 10, and 20, the slopes of the increase are 2.88, 4.84, 1.69, and 0.07, respectively. Therefore, the iteration is highly efficient when the iteration number $N\le 10$; if $N=10$, the maximal tension stiffness ${\overline{K}}_s=42.7$ is 3 times that obtained by the explicit method. However, when the iteration number is larger than 10, the maximal ${\overline{K}}_s$ almost stops increasing.

4.2.2. Fastening Stiffness

The fastening force must be added to fix or actuate a certain part of an elastic boundary. Assuming that there is a ghost boundary and all variables defined on it are specified, the actual boundary is connected to this ghost boundary through a set of still springs. In practice, the fastening coefficient should be set to be sufficiently large so that the relative displacement between the actual boundary and the ghost boundary is negligible. However, instability occurs if the fastening coefficient exceeds a threshold. In this section, the effect of the AWI approach on the fastening coefficient is studied by simulating a classic benchmark case: 2D fluid flow around a circular cylinder.

A circular cylinder with diameter D = 80 is placed in the middle of a computational domain with dimensions $L\times W=400\times 400$. The cylinder is discretized by 500 Lagrangian points. The Dirichlet boundary condition of velocity $u_{inlet}=0.1$ is specified at the inlet, and the fluid viscosity is set as $\nu =0.1$. All quantities are expressed in lattice units.

Without loss of generality, the cylinder diameter and inlet velocity are selected as the characteristic length and velocity, respectively. The fastening coefficients are non-dimensionalized according to Equation (15). Under the premise of program stability, Figure 8 shows the attainable maximal fastening coefficient using the AWI approach with different iteration times. The maximal ${\overline{K}}_f$ is 10 with the explicit method, whereas it linearly increases with the iterations when the proposed method is used. The gradient is approximately 6.5 and the maximal ${\overline{K}}_f$ can reach 126 when the iteration time is $N=20$, which is approximately 12 times larger than that of the explicit method.

Since ${\overline{K}}_f$ is user-defined, to investigate the accuracy when different values of ${\overline{K}}_f$ are used, an average error $E_{d,ij}$ is introduced to quantify the deviation of the actual boundary from the ghost boundary. A relatively small and uniform $E_{d,ij}$ indicates a smaller introduction error in the fastening force.

$$E_{d,ij}={\displaystyle \frac{{\sum }_{N_b}\left|{\mathbf{X}}_{ij}^a-{\mathbf{X}}_{ij}^g\right|}{N_b}}$$

(23)

where ${\mathbf{X}}_{ij}$ is the coordinate of the immersed boundary in either the X or Y direction. Superscript a and subscript g indicate the actual and ghost boundaries, respectively.

The histories of $E_{d,ij}$ for different ${\overline{K}}_f$ are shown in Figure 9, where the eight curves correspond to the deviation of the actual boundary from the ghost boundary for ${\overline{K}}_f$ = 1, 10, 20, and 50 in the X and Y directions. Starting from the zeroth time step, all curves increase; when the time step exceeds 5000, the curves cease to increase except for the one with ${\overline{K}}_f=1$. Moreover, when ${\overline{K}}_f\ge 10$, the deviation settles below 0.025, whereas it fluctuates between 0.12 and 0.14 when ${\overline{K}}_f=1$. Thus, for the problem where a part or all of the boundary is fixed, ${\overline{K}}_f$ should be larger than 10. It is apparent that if ${\overline{K}}_f=10$ and the explicit method is used, no other forces can be included because of the instability issue, whereas the AWI approach can maintain stability.

4.2.3. Bending Rigidity

The bending rigidity is used to present the material’s ability to resist bending, and is indispensable to most elastic materials. Because of instability, the traditional explicit method can only simulate cases with small bending rigidity. In this section, by simulating the benchmark case, we will verify that the AWI approach can improve the upper limit of the bending coefficient.

A filament is placed vertically in the middle of a rectangular domain with dimensions $L\times W=400\times 200$. The filament length is $L_{fi}=100$ and is discretized with $N_b=301$ Lagrangian points. As shown in Figure 10, the filament middle point is fixed to (100, 100). The fluid flows from left to right, the inlet velocity is set to $U=0.01$, and the fluid viscosity is $\nu =0.01$. These quantities are expressed in lattice units.

By selecting ${\rho }_f$, U, and $L_{fi}$ as the characteristic density, velocity and length, the bending coefficient is non-dimensionalized according to Equation (15).

As shown in Figure 11, with the explicit method, the maximal bending coefficient is $2\times {10}^{-5}$, whereas it can increase to $1.75\times {10}^{-1}$ (up to 8000 times) with the proposed method. Figure 12 shows the streamlines after the flow field settles at ${\overline{K}}_b=1.75\times {10}^{-1}$. When the bending coefficient is relatively large, only a small curvature is generated along the filament, and the flow field resembles the one passing a bluff body, where opposite spinning vortices are generated in the wake. Figure 13 shows the vorticity field at ${\overline{K}}_b=1.75\times {10}^{-1}$. The streamlines and vorticity contours are notably smooth and symmetric, which indicates that the results of the AWI approach are notably stable when ${\overline{K}}_b=1.75\times {10}^{-1}$.

4.2.4. Density of the Flexible Structure

The inertial force is also called the d’Alembert force and significantly affects the amplitude of the vortex-induced vibration. This force is commonly discretized using a finite difference method, such as Equation (14). However, the stability can be maintained only for low-mass cases, and the critical mass is much smaller than the required mass to induce the vortex vibration. Thus, a penalty method is proposed to mitigate this problem, where a user-defined coefficient is required [14]. The proper value for this coefficient is much more difficult to determine than the fastening force. Thus, a filament in a uniform flow is simulated to study the effect of the weighted iterative method on the direct difference discretization of the inertial force.

A filament is placed horizontally in a rectangular domain with dimensions $L\times W=800\times 400$. The filament is 100 units long and is discretized with $N_b=301$ points. The filament upstream point is fixed to (100, 200). The fluid flows from left to right, the inlet velocity is 0.01, and the fluid viscosity is set to $\nu =0.0056$. All quantities are expressed in lattice units.

To make the filament almost inextensible and maintain a sufficiently small displacement of the upstream end, we select ${\overline{K}}_s=2.0\times {10}^3$ and ${\overline{K}}_f=2.0\times {10}^3$, by which the filament vibration is no longer sensitive to the bending rigidity [9]. The dimensionless filament mass is defined as $S=m_s/\rho L$, where $m_s$, $\rho$, and L are the filament linear density, fluid density, and filament length, respectively. As was well documented in [4,9], in a uniform flow field, a low-mass filament tends to be more stable than a high-mass filament, and a massless filament is always stable regardless of the magnitude of the initial perturbation. The computed critical mass using the proposed method is $S=0.28$, which is consistent with that reported in [23]. The vorticity field obtained with $S=0.28$ is shown in Figure 14. When other parameters are fixed and both explicit and weighted iterative methods are used, the computed results show the following: the explicit method will cause instability even without the filament mass, whereas the proposed method can obtain the maximal filament mass up to $S=0.8$. Compared to the original IB-LB coupling scheme using the penalty approach, the AWI approach avoids introducing the artificially fastening coefficient to impose the inertial force, which is relatively more accurate and efficient.

4.2.5. Fluid Viscosity

When the viscosity decreases, the stability issue of any IB method will become even more stringent, and the IB-LB coupling scheme is not an exception. In this section, we study the low-viscosity stability of the AWI method by simulating an impulsively started filament.

A filament is placed vertically in a square domain with dimensions $L\times W=500\times 500$. A uniform grid of 120 Lagrangian points is used to discretize the filament. The length of the filament is 120, of which the middle point is tethered to (250, 250). ${\overline{K}}_s$ and ${\overline{K}}_f$ of the previous case are applied, whereas the bending rigidity is set to ${\overline{K}}_b=2.0\times {10}^{-3}$. In addition, the wall boundary condition is applied to the four sides of the computational domain. The fluid is initially at rest, and the filament middle point impulsively begins to move along the positive X direction with velocity $U_{drive}=0.01$. At the initial stage of the simulation, the filament is bent into a “⊃” shape. As the simulation continues, the filament remains in this shape and only minor fluctuations can be detected at both wings. Using both the explicit and AWI methods to simulate this case, with identical sets of parameters, the results show that when other parameters remain unchanged, even if the fluid viscosity is set to $\nu =1.0\times {10}^{-1}$, the explicit method rapidly diverges because of numerical oscillations. However, when the fluid viscosity is set to $\nu =5.5\times {10}^{-5}$ (with an instantaneous maximum Reynolds number of Re = 21,600), the AWI method maintains numerical stability. Figure 15 shows the velocity field and its vorticity contour at $t=8.6\times {10}^3$. Both wings flap back and forth during the simulation and three pairs of symmetrical vortices are formed in the wake zone, indicating that the proposed method can improve the stability of the original IB-LB coupling scheme when the fluid viscosity is small.

5. Conclusions

To improve the stability of the immersed boundary–lattice Boltzmann (IB-LB) coupling scheme, we proposed effective and efficient approaches which are based on iterative correction to the external force term. Several typical cases are simulated to verify and validate the scheme. The main conclusions are as follows: (1) Directly introducing the iterative process to the IB-LB framework will induce numerical fluctuation around an analytical solution, which will not improve the stability. By introducing the AWI approach, most of the fluctuation can be eliminated, and both numerical stability and accuracy are improved. (2) The proposed method can significantly improve the band of material parameters in the framework of the IB-LB coupling scheme. The dimensionless stretch coefficient, fastening coefficient, and bending rigidity can be improved by 3, 12, and 8000 times, respectively. There is qualitative improvement in the choice of bending rigidity. (3) Because of the stability improvement of the proposed method, the inertial force can be directly discretized, which helps in avoiding the user-defined parameter. (4) Even when the fluid viscosity is notably small, stability can be maintained by the proposed method, which makes this method even more versatile.