An immersed boundary-lattice Boltzmann method

Applied Mathematical Modelling 55 (2018) 502–521

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

An immersed boundary-lattice Boltzmann method combined with a robust lattice spring model for solving flow–structure interaction problems B. Afra a, M. Nazari a,∗, M.H. Kayhani a, A. Amiri Delouei b, G. Ahmadi c a

Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran Department of Mechanical Engineering, University of Bojnord, Bojnord, Iran c Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA b

a r t i c l e

i n f o

Article history: Received 17 September 2016 Revised 22 September 2017 Accepted 9 October 2017 Available online 16 October 2017 Keywords: Immersed boundary method Lattice Boltzmann method Robust lattice spring model Deformable body Split-forcing algorithm

a b s t r a c t An immersed boundary (IB)-lattice Boltzmann method (LBM) combined with a robust lattice spring model (LSM) was developed for modeling fluid–elastic body interactions. To include the effects of viscous flow forces on the deformation of a flexible body, rotational invariant springs were connected regularly inside the deformable body with square lattices. Fluid–solid interactions were due to an additional force density in the lattice Boltzmann equation enhanced by the split-forcing approach. To check the validity and accuracy of the numerical method, the flow over a rigid plate and the deformation of a cantilever beam were investigated. To demonstrate the capability of the new method, different test cases were examined. The deformation of a two-dimensional flexible vertical plate in a laminar cross-flow stream at different conditions was analyzed. The simulations were performed for different boundary conditions imposed on the elastic plate, namely, fixed-end corners and fixed middle point. Different flow conditions such as “steady flow regime”, “vortex shedding flow regime”, and the limit of “rigid body motion” were examined using the new IB-LBM-LSM approach. A general formulation for evaluating the deformation of the elastic body was also introduced, in which the position of the LSM nodes (inside the body) was updated implicitly at each time step. Two dimensionless groups, namely capillary number (Ca) and Reynolds number (Re), were used for parametric study of the behavior of the flow around the deformable plate. It was found that for low Reynolds numbers (Re < 50) and when the middle of the plate was fixed, decreasing the capillary number led to a decrease in the drag coefficient. The fluctuation of the plate during the vortex shedding flow regime was also explored. It was found that when the middle of the plate was fixed, the critical Reynolds number for the initiation of vortex shedding increased. For Re > 100, the Strouhal number was observed to increase with the decrease in capillary number. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Deformation problems occur when objects are immersed in a viscous fluid. In this condition; one can refer to the motion of red blood cells [1], interactions between heart leaflets and body fluid in biological systems [2,3], beating cilia in airways ∗

Corresponding author. E-mail address: [email protected] (M. Nazari).

https://doi.org/10.1016/j.apm.2017.10.014 0307-904X/© 2017 Elsevier Inc. All rights reserved.

B. Afra et al. / Applied Mathematical Modelling 55 (2018) 502–521

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that causes pumping the mucus layer including entrapped pollutant particles [4,5], fish swimming, self-propel of slender in micro-organisms [6], and reconfiguration of creatures in nature to lessen drag [7]. Different models are available for simulating the deformable body motion under the effect of viscous flow with some success. For example, Buxton et al. [8] simulated the breathing-mode behavior of an elastic shell that was filled with fluid by using a combination of lattice Boltzmann and lattice spring models (LBM and LSM, respectively). MacMeccan et al. [9] coupled finite element analysis and LBM to describe the linear deformation of RBCs in shear flow. Wu and Aidun [10] applied the external body force method in LBM combined with LSM to track RBC deformation when suspended in fluids. They simulated 120 deformable RBCs at 47% volume fraction that led to significant changes in the effective viscosity of the suspension at a constant shear rate. In the hybrid models specified above, the fluid and solid motion were separately evaluated; however, there are conventional approaches that solve the two-phase flows concurrently [2,11]. In general, because of the two different types of computational domains, the study of fluid–structure interaction problems is considered complex and costly. Therefore, the development of a straightforward and robust strategy that efficiently solves the fundamental coupling problem and reduces numerical instabilities has been the objective of many researchers [2,12–16]. Non-body fitted mesh methods, such as the immersed boundary method, are one of the most attractive interface tracking schemes. In particular, in the case of hydrodynamics with a flexible boundary, the boundary evolves into complex and unknown shapes. Thus, the re-meshing process leads to a high computational cost. Peskin [2] was the first to develop the immersed boundary method for simulating blood flow in the heart. He expressed the flow and structural domains in the Eulerian and Lagrangian frames, respectively. He substituted the immersed solid with normal and angular springs, for which the total force of the springs at each Lagrangian node is scattered into the Eulerian nodes, and the momentum equations are resolved with an additional density force. In this case, the rigid boundary maps into highly stiff springs. This method is called the feedback IBM [17,18], which requires user-defined parameters for evaluating the spring constants for different conditions. In this approach, the simulation of deformable particle requires a constitutive formula that relates external stresses to body deformation. There are two main differences between the LSM and the feedback-forcing IBM for the simulation of particle deformation: (1) using the LSM requires setting a collection of springs in the entire body, while in the IBM, deformation is portrayed by the deflection of a collection of springs just on the solid outer face. (2) For the LSM, the spring constant is analytically related to solid rigidity, while there is no direct relationship between these two variables in the feedback IBM [17]. However, the boundary force is calculated from the velocity and position of the interface. The IBM has been coupled with all traditional fluid solvers such as FVM and FDM [19,20]. Furthermore, a simple Cartesian computational domain is typically used in IBM; therefore, the LBM can be conveniently used for solving the fluid velocity associated with the IBM. The LBM originates from the lattice gas automata (LGA) technique [21], which is based on the kinetic theory of gases. When a discrete particle number is used in this model, statistical noise will be generated. To remove this instability, instead of discrete Boolean variables, a distribution function is used [22]. Different types of lattice Boltzmann equation (LBE) have been developed, with many using the Bhatangar–Gross–Krook (BGK) model [23]. Collision and streaming operators are introduced for solving the LBE numerically. To recover the Navier–Stokes and continuity equations in mesoscale, the Chapman–Enskog expansion [24] is employed. A combination of LBM and IBM has significant advantage in solving the flow structure interaction (FSI) problems, which was first attempted by Feng and Michaelides [12]. They used the IBM proposed by Lai and Peskin [25] and the LBM as the fluid solver to analyze the particle sedimentation under the laminar regime. Mohd-Yusof [26] introduced a new type of IBM that does not require choosing arbitrary parameters and forcing terms. In that approach (the so-called direct-forcing method), a forcing term is added to the discretized equations that implicitly imposes the non-slip boundary condition in the immersed boundary method. In general, the direct forcing method has two different schemes for boundary force calculation, namely sharp and diffuse interface schemes. In the sharp interface scheme [27], the forcing points do not necessarily coincide with the interface, and they can be perched out of the boundary [19,20]. The boundary force on the forcing point is calculated by the linear interpolation from the boundary and fluid velocities in an arbitrary direction. It has been shown that the sharp scheme causes instability, especially for the moving boundary problem because of the interpolation [28,29]. In the diffuse interface scheme proposed by Silva et al. [30], the force density is calculated from the velocity difference between the desired velocity and the non-forcing velocity on the boundary points. Values of non-forcing velocity on the boundary points are obtained by interpolating the local fluid velocity on the boundary points. In addition, the calculated force on these points is distributed through the fluid neighboring points. Interpolation is applied by the discrete delta function, which was proposed by Peskin [31]. Depending on the extent of the interpolation area around each boundary point, the diffuse interface is divided into various types, namely two-point, four-point, and higher in which there are proportional delta functions [32]. For example, Delouei et al. [33] showed that for a stationary flow over a cylinder, the four-point diffuse interface has better agreement with experimental data. The application of IB-LBM based on the direct forcing method results in first-order accuracy because the momentum changes depend only on the last time step. Later, Guo et al. [34] proposed another IB-LBM based on the split forcing method in which the momentum exchange depends not only on the last time step but also on the force density at the present time. This approach increases the IB-LBM to second-order accuracy, which is important for non-uniform and unsteady problems. Although there are a number of earlier studies on fluid–elastic solid interactions [35,36], there remains a need for the development of more accurate and computationally more efficient approaches that allow detailed and fast simulations. Direct numerical simulation techniques such as IB-LBM can provide insight into the mechanisms of FSIs. To the best of our

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knowledge, the application of the IB-LBM in combination with a robust lattice spring configuration for a detailed analysis of flow-elastic solid interactions has not been reported in the literature. Therefore, in this study, a suitable IB-LBM along with a robust lattice spring arrangement has been developed and used to simulate the behavior of deformable vertical plate in a laminar cross-flow. To solve the flow field, the split forcing LBM is implemented to increase the numerical accuracy. In addition, the elastic plate is modeled by using a square lattice structure model (LSM). The main advantage of the LSM over alternative mechanical models is its computational efficiency and ease with which the deformation of highly heterogeneous or highly stretched systems can be simulated. In addition, the collection of small deformations of each spring link in the domain creates a large deformation in the entire solid. The LSM nodes are connected by stretching spring links, which introduce central interaction forces. The present model leads to a fixed Poisson ratio of 1/3. The governing equations (related to the interactions of fluid and solid) are solved numerically, and the solution of fluid domain is combined with that of solid domain by the direct forcing IB technique with four-point diffuse delta function. In the following sections, the details of IB-LBM based on split forcing algorithm are first presented. Then, the complete description of LSM is provided. This includes the discussion of the relationship of mechanical properties and spring stiffness in two common lattice arrangements. In the last part of the methodology section, a user-friendly technique to update the solid configuration that is fitted on square lattice, which reduces instabilities that occur in other LSM methods, is described. The proposed IB-LBM is then verified by comparing the model predictions with those of earlier works. The implicit LSM is also verified by the simulation of a cantilever beam deflection, which is under a static force. Finally, by solving the coupled FSI problem, the deformation of a plate in cross-flow under various conditions is analyzed. In addition to the Reynolds number, a capillary number, which is defined as the ratio of elastic force to the viscous force, is introduced. The drag, lift, and Strouhal numbers are evaluated for ranges of capillary number and Reynolds number. 2. Methodology The governing equations of a viscous and incompressible fluid and a linear deformable structure are as follows:

ρ

Du P + μ∇ 2 u + F = −∇ Dt

(1)

·u =0 ∇

(2)

σ = Eε

(3)

, P, F , and μ are the density, velocity vector, pressure, body force, and dynamic viscosity of the fluid. In addition, where ρ , u σ , E, and ε are the stress tensor, Young’s modulus, and strain tensor for the deformable structure. Eqs. (1) and (2) are the expressions of balance of momentum and continuity for the fluid phase, respectively. Eq. (3) is the simplified equation for stress-strain relationship for an elastic body [37]. In the present numerical study, the Navier–Stokes equation is recovered by the LBM, and the no-slip conditions for submerged boundary are satisfied by employing the IBM. Structural equation is described by IB along with a suitable LSM to reduce computational cost. 2.1. Split forcing LBM By using the BGK model [23], Eqs. (1) and (2) can be satisfied with the use of LB equation and proper use of the equilibrium distribution function. The LB equation has the following form:

− → fi r + ei δt, t + δt = fi (r, t ) − τ −1

fi (r, t ) − fieq (r, t ) + Fi (r, t )δt

(4)

− → where fi and Fi are the density distribution function and the force distribution, respectively, at position r and time t. Here, ei is the velocity vector in each nine directions in the two-dimensional (2D) structural lattice. For lattice arrangement, there are different models represented in terms of Dn Qm , where n and m indicate the dimension of problem and the velocity vector, respectively. In this study, the popular D2 Q9 model, which consists of nine directions for the velocity vector, is used. In Eq. (4), τ is the dimensionless relaxation time parameter related to the kinematic viscosity by ν = cs2 (τ − 12 )δt, where cs is the speed of sound, which is √13 for this model. δ x and δ t are the lattice size and time step, respectively. The velocity set is given as follows [38]:

⎧ (i − 1 )π (i − 1 )π ⎪ cos , sin ⎪ ⎪ ⎨ 2 2 − → ei = (i − 5 ) π (i − 5 ) π + , sin + ⎪ cos ⎪ 2 4 2 4 ⎪ ⎩ 0

c i = 1, 2, 3, 4 c i = 5, 6, 7, 8

(5)

i=9

where c = δ x/δ t and usually δ x and δ t are equal to 1. To properly map the Navier–Stokes equation in mesoscale in LBM, the equilibrium distribution function is expressed as follows [33]:

fieq = wi ρ 1 +

3 3 − 9 − → → 2 ei .u + 4 ei .u − 2 u. u 2 c 2c 2c

(6)


505

where wi is the weighting function, which is selected such that wi = 1/9 for i = 1, 2, 3, 4, wi = 1/36 for i = 5, 6, 7, 8, and w9 = 4/9. To evaluate the body force density in the right hand side of Eq. (4), the discrete forcing technique employed by Guo et al. [34] and simplified by Kang and Hassan [14] is used. Accordingly, the discrete force distribution function Fi (r, t ) is given as follows:

Fi (r, t ) = 1 −

− → − → 1 ei − u ( r , t ) ei .u ( r , t ) − → wi 3 + 9 e i . F (r , t ) 2τ c2 c4

(7)

where F (r, t ) is the body force and will be introduced in the next section. This force density term is mainly accounting for the coupling of the fluid and the elastic membrane. In the split forcing method, the momentum exchange is affected not only by forces exerted at time t but also by forces exerted at t + δ t. Collision and streaming steps in the LBM are modified by the following four steps [33]: First forcing:

ρ (r, t )u (r, t ) =

− δt → ei fi (r, t ) + F (r, t ) 2

(8)

i

Collision:

fi (r, t ) = fi (r, t ) − τ −1

fi (r, t ) − fieq (r, t )

(9)

Second forcing:

fi (r, t ) = fi (r, t ) + Fi (r, t )δt Streaming:

(10)

− → fi r + ei δt, t + δt = fi (r, t + δt )

(11)

The split forcing method can recover continuity and momentum equations with a second-order accuracy [33]. The macroscopic properties of the fluid are defined as follows:

ρ=

9

fi

(12)

i=1

p = ρ cs2

ρ u =

(13)

9 − → δt fi ei + F 2

(14)

i=1

2.2. Immersed boundary method The IBM is used to satisfy appropriate boundary conditions between fluid and flexible structure without moving mesh computational process. In fact, the interaction between the fluid and the structure is modeled by the interpolation of fluid velocity on the boundary points and distribution of body force (calculated from momentum exchange) onto the bulk fluid in the vicinity of boundary nodes to generate body force, as shown in Eq. (4). 2.2.1. Direct-forcing approach The direct-forcing method was first developed by Mohd-Yusof [26]. Accordingly, the interaction force is calculated as follows [33]:

F (rb , t + δt ) = ρ

desire (rb , t + δt ) − u bno f (rb , t + δt ) U b

δt/2

(15)

desire is the velocity at the boundary point b (b = 1, 2, …,n). Moreover, u bno f is the no-forced velocity and is given as where U b follows [33]:

bno f = u

n

u

no f

ri j , t D ri j − rb

(16)

b=1

where the subscripts i, j are the on-lattice (Eulerian) grid points in the x and y directions, respectively, and D is the Dirac delta function for matching the surrounded viscous fluid with the solid structure. The Dirac function transfers the fluid properties to the boundary nodes by the interpolation of velocity vector, which is calculated from the LB equation without the forcing term (Eq. (14)) at the same time step. A number of functions are available for this interpolation [27,32]; however, we use the common 2D decomposition formula [33], which is carried out on four on-lattice points in the x and y directions.

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Fig. 1. Schematic of fluid–structure interaction in the IBM technique.

These points are present around each off-lattice point as shown in Fig. 1. This figure also shows that the neighboring onlattice nodes that are located inside the influence circle are affected by the off-lattice node and vice versa. The Delta function is expressed as follows [33]:

D ri j − rb =

1 d xi j − xb d yi j − yb h2

(17)

where h = xlbm , and function d is defined as follows:

⎧ 1 2 ⎪ ⎪ ⎨ 8 3 − 2|r| + 1 + 4|r| − 4|r| d (r ) = 1 2 5 − 2 r − −7 + 12 r − 4 r | | | | | | ⎪8 ⎪ ⎩ 0

Finally, the total force [33]:

F total (t + δt ) = −

Ftotal

and the total torque

0 ≤ |r| < 1 1 ≤ |r| < 2

(18)

|r| ≥ 2 Ttotal

exerted on submerged boundary are given, respectively, as follows

F (rb , t + δt ) Sb

(19)

b

T total (t + δt ) = −

(rb − rc ) × F (rb , t + δt ) Sb

(20)

b

where Sb is the arc length of boundary division and rc is the position of center of mass. Finally, the density body force on the fluid domain is given by the distribution of boundary force onto the neighboring Eulerian nodes [33]:

F (r, t ) =

n

F (rb , t + δt ) D ri j − rb Sb

(21)

b=1

To evaluate the dynamic motion of a particle, Newton’s equations of motion are implemented. That is,

n+1 V P

Mf n n−1 n + 1 − −V =V F (rb ) sb + (M p − M f )g dt + V P P Mp Mp P b

ω

n+1 P

(22)

If n 1 + −ω Pn−1 =ω − ω (rb − rc ) × F (rb ) sb dt + IP Ip P n P

(23)

b

where MP is the mass of the solid particle calculated as

N i=1

ρs a2 . Mf , IP , and If are the mass of equivalent volume fluid and

P , and ω P represent the acceleration of gravity, particle velocity, and angular the corresponding moments of inertia. Here, g, V


507

Fig. 2. Two-dimensional (a) triangular lattice with six neighbors and (b) square lattice with four nearest (1, 2, 3, 4) and four next-nearest neighbors (5, 6, 7, 8).

velocity, respectively. Thus, the position of center of mass and the velocity vectors of the Lagrangian nodes (desired velocity) are obtained as follows:

− → rc

n+1

− → = rc

n

+

1 n+1 n V + VP dt 2 P

− → − → desired (rb ) = V n+1 + ω Pn+1 × ( rb − rc U b P

(24) (25)

Therefore, the distribution functions can be updated with the known velocity and density body force. Thus, the iterative procedure for solving the fluid domain is established. 2.3. Lattice spring model In this study, the solid elastic body is modeled by a collection of 2D rotationally invariant Hookean springs, which are joined together inside the body. The representation of an isotropic solid body by lattice springs requires specifying the appropriate mechanical properties. As shown in Fig. 2, there are two common lattice structures corresponding to isotropic elastic materials, namely triangular and square [39,40]. The relationship between the spring stiffness, Young’s modulus, and the Poisson ratio depends on the lattice spring structures considered. The deformation of the body is evaluated using the normal local displacement of each node rather than other neighbors, which satisfy rotationally invariant and linear elastic behavior. Moreover, this model uses a Poisson ratio of 1/3. 2.3.1. Spring stiffness Monette and Anderson [39] proposed the relationships between macroscopic mechanical properties and spring stiffness for two kinds of lattice springs based on energy conservation around the LSM nodes. Thus, the spring constants are related to the mechanical properties of the solid. In the next section, a robust numerical algorithm for the application of the IBLBM-LSM to evaluate the fluid-deformable solid interactions is presented. Triangular mesh. By considering a normal harmonic force between LSM nodes in the triangular lattice structure, the amount of energy in node j related to the neighbors is given as follows:

E=

6 1 eq 2 k ri j − rlsm 2

(26)

i=1

eq where k, ri j , and rlsm are the spring constant, the relative position vector between the neighboring nodes i and j, and the equilibrium length of ith LSM bond connected to node j, (Fig. 2(a)). Note that the equilibrium length of the lattice structure is considered as unity. By comparing the energy density per unit surface and the 2D elastic energy [41], the Lamé coefficients can be written as follows:

√ 3k μ=λ= 4

(27)

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Moreover, according to continuum mechanics, Young’s modulus and the Poisson ratio are given as follows [41]:

E=

4μ ( λ + μ ) 2k = √ λ + 2μ 3

(28)

υ=

λ 1 = λ+μ 3

(29)

For this simple network, the Poisson ratio is constant. The Poisson ratio can be modified by considering a bending interaction force, which explains the rotation of each of the six links made by node j and six of its neighbors at constant length. Square lattice. For rotational invariant square structure as shown in Fig. 2(b), each internal node is joined to four nearest and four next-nearest neighbors with different stiffness constants. The corresponding elastic energy is expressed as follows:

E=

4 8 2 2 1 1 kn ri j − rneq + kn−n ri j − rneq−n 2 2 i=1

(30)

i=5

where kn (and kn − n ), the force-free length a function of normal

rneq , and rneq−n are the spring stiffness, the force-free length of nearest links connected to node j, and of next-nearest links connected to node j, respectively. The mechanical properties can be obtained as spring constants as follows:

E=

8kn kn−n , kn = 3 2

(31)

υ=

1 3

(32)

√ where the original lengths of the nearest and next-nearest bonds are a0 and 2a0 , respectively. In this study, the square lattice is implemented because the square network is well adapted for the present computational model, in which the LSM boundary nodes can be matched directly to the Lagrangian points. The amount of force exerted on the jthth LSM node by the eight neighbors is given as follows:

Fispring = j

4

−kn ri j +

i=1

8

−kn−n ri j

(33)

i=5

√

− → − → a 2a where ri j = (1 − |r 0 | )ri j for nearest points and ri j = (1 − |r |0 )ri j for the next-nearest points with ri j = ri − r j . In this ij ij manner, the strain tensor, which is as a function of stress tensor in physical scales, is mapped to the displacement of LSM nodes in space. 2.4. Proposed IM-LBM-LSM In the IB-LBM, the interactions between the solid boundary and the surrounding fluid can be accounted for by both explicit and implicit schemes. The explicit scheme applies the deformation procedure in the dynamic frame. In other words, in order to calculate the new positions of LSM nodes, each lattice spring node translates according to Newton’s second law of motion. We utilized the Verlet algorithm to update the positions and velocities of each LS node at time t + δ t [8]. That is,

ri (t + δt ) = ri (t ) + vi (t )δt +

1 ai (t )δt 2 2

1 2

vi (t + δt/2) = vi (t ) + ai (t )δt ai (t + δt ) = vi (t ) +

Fi (t ) Mi

(34) (35) (36)

1 2

vi (t + δt ) = vi (t + δt/2) + ai (t + δt )δt

(37)

where vi and ai are the velocity and acceleration of each LS node, respectively. The explicit approach, however, faces potential instability problem when the LBM is used. In the explicit approach, the time step should be less than the time needed for elastic wave propagation through lattice elements. This leads to the condition that δt < Vδpx , where Vp =

(K+4G/3) is the P-wave velocity [42], where K and G are bulk and shear modulus constants. ρs

In the implicit scheme, to recover a new shape at each time step, the physical behavior of the flexible body (in the presence of fluid flow) is substituted by two virtual steps. First, the flexible body is moved as a rigid body by applying


509

boundary forces and external body forces, and the corresponding velocities are updated. In this step, the spring bonds are treated as being rigid. Then, the rigid body movement is stopped, and the deformation of the body is evaluated under quasi-equilibrium conditions. That is, the LSM nodes change their positions relative to the neighboring nodes under the quasi-equilibrium state. Note that in the proposed method, after applying the deformation under quasi-equilibrium state, the velocity vectors of the LSM nodes preserve their previous values (previous step), and only their positions are updated. This is a valid assumption due to the gradual and smooth deformations of LSM nodes in the proposed model. At each time step for all the LSM nodes, one would have

Fi jspring = − Fi jext + Fi jboundry

(38)

boundry where Fiext and Fi j are the body force (e.g., gravity) and boundary force, respectively. By comparing Eqs. (37) and (38), j a general formulation for the deformation of the body is obtained. That is, the position of the ith LSM node is changed as follows: 4

−kn ri j +

8

j=1

Or

(39)

j=5

1 ri = 4

−kn−n ri j = − Fi jext + Fi jboundry

Fi jext + Fi jboundry kn + kn−n

4 8 √ kn a0 kn−n 2a0 ri j + ri j + ri j ri j kn + kn−n kn + kn−n j=1 j=5

(40)

where ri denotes the updated position of LSM nodes at each time step. It is clear that the boundary nodes are affected by both spring forces (from some LSM nodes inside the body) and fluid interaction forces (from the Eulerian lattices). For the internal LSM nodes (in the absence of gravity), the first term on the RHS of Eq. (40) is zero. Then, the equilibrium condition is obtained by balancing the internal spring forces (inside the body). These forces originate from LSM nodes relative to the neighboring nodes. Because of the implicit nature of the proposed scheme, the computational cost increases. For iterative solution of this implicit derivation, successive over-relaxation or the Gauss–Seidel method can be implemented. However, there is no limitation on the selection of lattice size in the LSM model in contrast to the explicit scheme. In the explicit scheme, when δ x < δ tVp (in LBM), especially when δ x < 1, numerical instability may occur due to the generation of large fluctuations in the external fluid force at the boundary nodes. The IB-LBM-implicit LSM algorithm can be summarized as follows: Step 1. Implement LBM procedure with known fluid velocity and body force to calculate the velocity on Eulerian nodes using Eqs. (8)–(11). It is important to mention that the body force at the current time step comes from that at the previous time step. bno f and the density boundary force on the Lagrangian nodes using Eqs. (15) and Step 2. Obtain the unforced velocity u (16), respectively. The body force (at previous time step) is used for the evaluation of Eulerian velocity at the current time step; therefore, the Lagrangian velocity is called the “unforced velocity”. Step 3. Update the position and velocity of the particle as a rigid body (regardless of flexibility) by integrating Newton’s equation of motion (i.e., Eqs. (22)–(25)) with the total force and torque exerted on the boundary nodes. Step 4. Perform implicit LSM algorithm to calculate the deformation of the particle with a quasi-equilibrium state (zero velocity) using Eq. (40). Note that in this step, the positions of LSM nodes are alone updated and their velocities remain as those in step 3. Step 5. Distribute the boundary forces (calculated in step 2) to the neighboring Eulerian nodes by using Eq. (21). Step 6. Go back to step 1 and repeat if the desired accuracy is not obtained. 3. Verification study To assess the accuracy of the numerical method, two verification studies are performed: (1) simulation of flow over a rigid plate in an unconfined channel using the IB-LBM with diffuse scheme. (2) Deformation of a cantilever beam subject to a static force that is acting at the end of the beam using the LSM. 3.1. Test case I: flow over a rigid plate Newtonian fluid at moderate Reynolds numbers flowing over a small thin plate of length D is analyzed in this section. The plate is placed at the middle of the unconfined channel as shown in Fig. 3. The channel length is L and its height is H. To reduce the wall effects, the blockage ratios H/D = 16 and L/D = 30 are used. Additional details are shown in Fig. 3. The accuracy of the present IB-LBM method with diffuse interface for the simulation of fluid/structure interaction problems is assessed by comparing the model predictions with the earlier numerical and experimental results of Dennis et al. [43] and numerical results of In et al. [44]. The drag coefficients are evaluated for a range of Reynolds number (Re = U∞ D/ν ). Here, U∞ is the uniform inlet flow velocity and ν is the kinematic viscosity. The drag coefficient is based on the pressure force

510


Fig. 3. Schematic of geometry and boundary conditions for flow over an obstacle.

Fig. 4. Drag coefficient as a function of Reynolds number.

Fig. 5. Vortex length as a function of Reynolds number.

acting on the plate in cross flow. That is, CD =

∫D

PdL 0 1 ρU 2 D ∞ 2

, where P is the pressure difference across the plate thickness, and


511

Fig. 6. Schematic of a cantilever beam subjected to a static force.

Fig. 7. Non-dimensional deformation of free side of the cantilever beam for different values of non-dimensional acting force. The present results are compared with Buxton et al. [8] and analytical solutions.

the values of P are obtained as follows:

P=

⎧ p i 4 ⎪ ⎨ i=1 l ⎪ ⎩

4

0

i

i=1 li

i = neighbor nodes

(41)

i = neighbor nodes

where li is the distance between a Lagrangian node and its closest Eulerian node. It is important to note that the pressure drag cannot be directly obtained in the IB-LBM. That is, a suitable calculation method for the evaluation of the pressure on the Lagrangian nodes is needed. In the present study, a simple interpolation is used, and the values of pressure on the Lagrangian nodes are evaluated by averaging the pressure over the four neighboring Eulerian nodes. In addition, it should be noted that for a plate in cross flow, there is no viscous drag, and the total drag is due to the pressure drag. Figs. 4 and 5 show a comparison of the present IB-LBM predictions for the drag coefficient and vortex length with the earlier results of Dennis et al. [43] and In et al. [44]. These figures show that the predicted drag coefficients and vortex lengths behind the plate as a function of the Reynolds number are in good agreement with the earlier results reported in the literature. 3.2. Test case II: deformation of cantilever beam To verify the LSM, the deformation of a cantilever beam subjected to a concentrated load F, as shown in Fig. 6, is simulated. The beam studied has a length of L = 30, a width of w = 3, and a thickness of t = 3. The beam elastic modulus EY and 2 the value of exerted force are selected such that the non-dimensional deformation ratio, FEL I , varies in the range 0 to 0.4. Y

512


Fig. 8. Non-dimensional displacement of mid-point of the plate for Type 1 boundary condition versus non-dimensional time for different capillary numbers.

Fig. 9. Non-dimensional displacement of the plate as a function of capillary number for (a) Type 1 boundary condition and (b) Type 2 boundary condition.

According to the Euler–Bernoulli beam theory, the peak deformation of the cantilever beam is given as follows:

δb L

=

F L2 3EY I

(42)

The simulation results for non-dimensional deflections, δ b /L, as a function of the non-dimensional force, FL2 /EY I, are compared with the theoretical results, as shown in Fig. 7. In this figure, the triangular symbols represent theoretical values, and the circles represent the simulation results of the present study. In addition, the predicted deformations of Buxton et al. [8] are also shown in this figure with square symbols. It is seen that the present simulation results are in reasonable agreement with the theoretical model and also the results of Buxton et al. [8].


513

Table 1 Comparison between IB-LBM-LSM and COMSOL4.4 results for Type 2 boundary condition and Re = 10. Ca

δ /D (COMSOL 4.4)

δ /D (IB-LBM-LSM)

5E3 7E3 1E4 2E4

0.1387 0.1162 0.091 0.0549

0.1232 0.1043 0.085 0.0538

Table 2 Non-dimensional displacement of the plate for Type 1 boundary condition with three different lattice spring grid sizes. a0 /δ x

2.0

1.0

0.5

δ

4.87

5.53

5.69

4. Results and discussions In this section, deformations of a vertical elastic plate in the presence of cross flow under two different boundary conditions, as shown in Fig. 3, are studied. The boundary conditions are as follows: (Type 1) Both ends of the plate (points A, A’) are fixed. (Type 2) Middle of plate (point B) is fixed. The Newtonian fluid flow over a deformable plate is assumed to be in laminar regime. The computational domain and the boundary conditions are shown in Fig. 3. Here, a plate thickness of ε = 0.08D is assumed and the passage width is changed to H = 4D. Solid mesh and Eulerian grid points have 5 × 51 and 1501 × 201 lattice nodes, respectively. Lattice size in LSM arrangements is equivalent to the LB size. At the channel entrance, a uniform flow is assumed. To prevent the numerical instability in the early time steps, the uniform flow is gradually increased from zero to a steady value of U∞ at time T, as follows:

U∞ sin

U (t ) =

U∞

π 2

T

t

t 0.5375. The associated Mach number (Ma = U∞ /Cs ) is 0.085. The capillary number is defined as follows:

Ca =

ED

μU∞

(44)

The capillary number represents the ratio of the elastic forces to the viscous forces acting on the body. The Reynolds number, drag coefficient, and lift coefficient are given as follows:

Re = CD =

CL =

U∞ D

ν

Fx 1 2

ρU∞ 2 D Fy

1 2

ρU∞ 2 D

(45)

(46)

(47)

where Fx and Fy are the drag and lift forces, respectively. Physical domain and the computational domain (i.e., lattice unit) are equivalent if they have the same capillary and Reynolds numbers. In this study, we used the value of Cd as a criterion. The solution stops when this parameter reaches a steady state. The number of iterations varies depending on the physical parameters such as Re and rigidity.

514


Fig. 10. Deformations of the plate for three different grid sizes.

4.1. Test case II: steady-state condition For Type 1 boundary condition with the edges of the plate fixed, Fig. 8 shows the temporal evolution of the mid-point of the plate (point B). Here, the non-dimensional deviation from the equilibrium state, δ /D, is shown as a function of non-dimensional time t ∗ = tU D for Re = 20 for three different capillary numbers. Fig. 8 shows that as the capillary num-


515

Fig. 11. Streamlines of flow over a deformable plate for a capillary number of 104 and different Reynolds numbers. The plate is considered (a) rigid, (b) with Type 1 boundary condition and (c) with Type 2 boundary condition.

Table 3 Effect of capillary number on critical Reynolds number for Type 2 boundary condition. Ca

Recr

1E8 (Rigid) 2E4 1.5E4 1E4

35 39 43 50

516


Fig. 12. The variations of drag coefficient with Reynolds number for different capillary numbers for Type 2 plate boundary condition. Table 4 Drag coefficient as a function of capillary number for Type 1 boundary condition. Ca

CD Re = 10

Re = 20

Re = 30

2E4 5E4 2E5

5.317 5.306 5.314

4.117 4.12 4.113

3.881 3.887 3.88

ber decreases, the deformation of the plate increases. More specifically, when the capillary number decreases from 105 to 2 × 104 , the non-dimensional displacement of the elastic plate (δ /D) increases from 0.018 to 0.078. For numerical stability, in the case of low capillary numbers, the value of T in Eq. (43) is increased to T = 3 × 104 to avoid the numerical instability. For Type 1 boundary condition, variations in non-dimensional displacement of the plate center versus capillary number for different Reynolds numbers are shown in Fig. 9(a). Similarly, the steady-state deformations of the plate end for Type 2 boundary condition for a range of Ca and Re are shown in Fig. 9(b). These figures show that for small values of capillary number, the non-dimensional displacement of the plate increases, which is due to the decrease in plate rigidity. Moreover, as the Reynolds number increases, the plate deformation increases as the flow momentum increases. By comparing Fig. 9(a) and (b), it can be seen that for capillary number 5 × 104 and for Reynolds numbers of 10, 20, and 30, the non-dimensional displacement of the plate for Type 2 boundary condition is about two times higher than that for Type 1 boundary condition. To verify the IB-LBM-LSM in this study, we also recheck the results for Type 2 boundary condition using COMSOL Multiphysics 4.4 software. For a Reynolds number of 10, the non-dimensional displacement is compared with the results obtained by the present numerical model, as shown in Table 1. It is seen that there is a good agreement between these results. As mentioned in Sections 2–4, in the proposed implicit scheme, the grid size does not play a critical role in the deformation analysis phase as it appears only in the rigid body movement phase. Therefore, the time step (grid size) can be set without a serious limitation. To justify this point, simulations were performed for three different lattice spring sizes, and the predicted plate deformations are shown in Table 2 and Fig. 10. Here, the capillary number and the Reynolds number are 104 and 30, respectively. It can be seen that by decreasing (or increasing) the lattice spring size (rather than LB size), the numerical solution does not vary significantly. In this study, from the above grid study, a lattice spring size of 1.0 was used. The corresponding deformations of the plate for different conditions are shown in Fig. 10 (a)–(c). Fig. 11 shows the flow streamlines around the plate for Ca = 104 and different Reynolds numbers. The interaction of elastic plate and fluid flow can be seen from this figure. Fig. 11(a) shows that for a rigid plate, the critical Reynolds number for the start of vortex shedding is Re = 35. For Reynolds numbers below 35, two symmetric vortices behind the obstacle are formed. Fig. 11(b) shows that when both ends of the plate are fixed, the critical Reynolds number remains at about Re = 35, and only the shape of vortices changes in comparison to that of the rigid plate. Finally, Fig. 11(c) shows the bending of the plate with the center fixed boundary condition and the corresponding streamlines in the channel. It is seen that the center fixed boundary condition leads to an increase in the value of critical Reynolds number to Re = 50 for the start of shedding flow. That is, for the center fixed boundary condition, the flow is steady for Re < 50, and vortex shedding is observed for Re > 50. For this case, the flow regime remains steady for Re = 35. This is because for Type 2 boundary condition (fixed plate center), the shape of the deformed plate becomes more aerodynamic leading to a decrease in the horizontal force acting on the obstacle. The effects of capillary number on the critical Reynolds number (for starting the unsteady vortex shedding) are shown in Table 3. Furthermore, Fig. 11(c) shows that for small capillary numbers and Type 2 boundary condition, the two symmetric wake vortices are replaced by four vortices. In this case, the left vortices are rotating in an opposite direction to that of right vortices.


517

Fig. 13. Temporal evolutions of (a) drag coefficient, (b) lift coefficient, and (c) non-dimensional displacement of the plate for different capillary numbers when both the ends of the plate are fixed.

Fig. 12 shows the variation in drag coefficient with Reynolds number for different values of capillary number for Type 2 plate boundary condition with the center of the plate fixed. It is seen that the drag coefficient has a decreasing trend with Reynolds number. Moreover, the drag coefficient decreases with the decrease in the capillary number from 105 to 104 . The amount of decrease in the drag coefficient increases for smaller values of capillary number. For very high capillary numbers, the drag coefficient curves, which are functions of Reynolds number, approach the limit of rigid plate case. Table 4 shows that for Type 1 boundary condition, the reduction in plate flexibility does not considerably affect the drag coefficient. The drag is typically composed of pressure drag and viscous drag. In the present case, because the thin plate is

518


Fig. 14. Temporal evolutions of (a) drag coefficient, (b) lift coefficient, and (c) non-dimensional displacement of the plate for different capillary numbers when the mid-point of the plate is fixed.

perpendicular to the flow direction and its deformation is small, the dominant source of the drag is the pressure drag. In this condition, the projected area remains constant roughly the same as the rigid plate, as shown in Fig. 11(b). Therefore, the total drag is equal to the pressure drag. The lift coefficient is zero in this case also because the flow field is symmetric.

4.2. Test case III: vortex shedding flows In this section, the numerical simulation is carried out for a Reynolds number of 120 for both Type 1 and Type 2 boundary conditions. As noted in the previous section, as the Reynolds number increases beyond a critical value, the vortex shed-


519

Fig. 15. Variation in Strouhal number for Type 2 plate boundary condition as a function of Reynolds number for different capillary numbers.

ding flow in the wake of the body is generated, which induces oscillation in the deformable plate. Under this condition, the flow is unsteady, and vortices are formed and detached periodically from both sides of the plate. The response of the deformable plate varies with changes in the capillary number in the vortex shedding flow regime. As expressed in Sections 1–4, for Type 2 boundary condition, the frontal area of the plate decreases because of its deformation, and the shape becomes more aerodynamic, which leads to a decrease in the drag coefficient. The temporal evolutions of drag coefficient, lift coefficient, and the non-dimensional plate displacement are plotted in Fig. 13 for different capillary numbers for Type 1 boundary condition, when both ends of the plate are fixed. It is seen that the deformation of the center of the plate and the drag and the lift coefficients follow an oscillatory pattern due to the vortex shedding flow. Similar to the case of moderate Reynolds numbers with Re < 35, Fig. 13(a) shows that the average drag coefficient remains roughly constant for different values of capillary number for Type 1 boundary condition. Fig. 13(b), however, shows that the amplitude of lift coefficient deceases slightly when Ca increases. The inherent nature of the vortex shedding flow causes oscillation of the deformable plate; thus, the vibration of the deformable plate can be seen from Fig. 13(c), which displays the temporal evolution of the non-dimensional displacement of the middle of the plate for capillary numbers 2 × 104 and 4 × 104 . It is seen that when the capillary number decreases, the amplitude of induced vibration increases. This is because of the increase in plate flexibility. However, when vortex shedding occurs, a large pressure difference is generated at the middle of the plate between the front and back of the plate. Therefore, the greatest pressure fluctuation is observed at this point of the body, which produces vibrations of large amplitude for Type 1 boundary condition. Fig. 14 shows the temporal evolutions of drag coefficient, lift coefficient, and the non-dimensional plate displacement for different capillary numbers for Type 2 boundary condition, when the middle of the plate is fixed. Fig. 14(a) shows that for Type 2 boundary condition, the drag coefficient markedly decreases when the capillary number decreases. As noted before, for smaller capillary number, the frontal area of the deformed plate decreases and the shape becomes more aerodynamic, which leads to a decrease in the average drag coefficient. In addition, Fig. 14(b) shows that the amplitude of the lift coefficient decreases slightly with the decrease in Ca. Fig. 14(c) shows that under this condition, the deformable plate with Type 2 boundary condition experiences smaller fluctuations than Type 1 boundary condition. As described above, for Reynolds numbers higher than 50, the vortex shedding flow is generated that leads to oscillations in the drag and lift coefficients. The Strouhal number is defined as a non-dimensional frequency given as follows:

St =

fD U∞

(48)

where f is the vortex shedding frequency. Variation in Strouhal number as a function of Reynolds number for different capillary numbers for Type 2 boundary condition is plotted in Fig. 15. It is seen that when Re > 100, the Strouhal number has an increasing trend, and the rate of increase depends on the flexibility of the plate for Type 2 boundary condition. The increase in flexibility of the plate (decrease in Ca) leads to a sharper increase in St with Reynolds number. For the case of rigid plate, the Strouhal number varies only slightly with Reynolds number. Fig. 15 also shows that the curves of St for different Ca reach a value of about 0.25 for a Reynolds number of 100. 5. Conclusions In this study, the IB-LBM based on the split forcing scheme with a robust LSM was developed for the simulation of flow around deformable bodies. The present IBM relies on a direct momentum forcing. The force is distributed on the Eulerian nodes (in the fluid domain) by a simple interpolation based on the four-point diffuse delta functions. By sing this approach, the required boundary conditions on the Lagrangian nodes at the boundary of the deformable immersed body were satisfied. To compute the deformation of a solid body due to the boundary forces, a set of springs inside the body

520


was used. The deformed configuration of the elastic body at each time step was generated by the movements of spring links. A general formulation for the deformation of the elastic body was also suggested in the present model in which the position of the LSM nodes (inside the body) could be updated implicitly at each time step. The hybrid model was verified by examining (a) the simulation of flow over a rigid body and (b) deflection of the free side of a cantilever beam subjected to a normal force. The accuracy of the proposed method was checked by solving several test cases including steady cross flow around a deformable vertical plate and unsteady vortex shedding flow regime in the wake of a deformable plate. It was found that for constant Reynolds number and Type 2 boundary condition, decreasing solid rigidity leads to a decrease in the drag coefficient and also a delay in the initiation of the unsteady flow condition. 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