Development of immersed boundary methods for isothermal and thermal flows 2

  Chapter Governing Equations and Boundary Condition-Enforced IBM Chapter Governing Equations and Boundary Condition-Enforced Immersed Boundary Method A new version of boundary condition-enforced IBM, which is given under the framework of NS solver in primitive variable form, is presented in this chapter. It aims at extending the LBM solver-based IBM of Wu & Shu (2009) to the NS solver-based IBM for an accurate evaluation of the body force. The present boundary condition-enforced IBM is established based on the fractional step technique while the body force in the modified momentum equation is implicitly determined in a way that the no-slip condition on the immersed boundary is accurately satisfied. The performance of the new version of IBM is carefully examined, firstly through the classical problem of flow over a single stationary circular cylinder, and then the flow interference between two side-by-side circular cylinders. Results from moving boundary problems such as vortex-structure interaction around a transversely oscillating cylinder and vortex-induced-vibration of an elastically mounted circular cylinder are also provided as a further validation. 29      Chapter Governing Equations and Boundary Condition-Enforced IBM 2.1 Governing equations Let us begin by stating the mathematical expression of the present IBM. Consider an incompressible viscous flow in a two-dimensional domain Ω which contains an immersed object in the form of a closed curve Γ , as shown in Fig. 2.1. With the use of the IBM, the immersed object is modeled as localized body forces acting on the surrounding fluid. As a result, the IBM formulation for the incompressible viscous flow involving immersed objects/boundaries is expressed in the primitive variable form as ρ( ∂u + (u ⋅∇)u) = −∇p + μ∇2u + f ∂t ∇ ⋅u = (2.1) (2.2) subject to the no-slip boundary condition (2.3) on Γ u(X(s), t ) = UB . (2.3) The fluid pressure p and velocity vector u are the dominating flow variables. U B is the prescribed velocity of the immersed boundary Γ . ρ and μ are the fluid density and viscosity. Note that a forcing term f is added to the right hand side (RHS) of the momentum equation (2.1) to represent the effect of immersed object Γ . The forcing term f is the localized body force density at the fluid (Eulerian mesh) point, which is distributed from the surface force density F at the immersed boundary (Lagrangian) point and can be expressed as f ( x, t ) = ∫ F ( X ( s ), t )δ ( x − X( s, t )) ds (2.4) Γ Here x and X denote the Eulerian and Lagrangian coordinates describing 30      Chapter Governing Equations and Boundary Condition-Enforced IBM the fluid domain and immersed boundary respectively. δ ( x − X ( s , t )) is the Dirac delta function responsible for the interaction between fluid and the immersed boundary. In addition, the velocity at the immersed boundary (Lagrangian) point in IBM can be interpolated from the velocity at the fluid (Eulerian mesh) points as u ( X( s, t )) = ∫ u ( x)δ ( x − X( s, t )) dV . (2.5) Ω In summary, Eqs. (2.1)-(2.5) build the complete set of governing equations for an incompressible objects/boundaries, viscous among flow which system Eqs. involving (2.1)-(2.2) are immersed the familiar Navier-Stokes equations and Eqs. (2.4)-(2.5) represent the interaction between the fluid and the immersed objects/boundaries. 2.2 Solution procedure In IBM, the solution to Eqs. (2.1)-(2.5) is frequently accomplished by making a good use of the fractional step algorithm. In a time-discrete form, the fractional step procedure is written as: (1) Predictor step: Solve the normal Navier-Stokes equation for a predicted velocity field u* by disregarding the body force terms in Eq. (2.1), ρ u* − u n + ρ [ (u ⋅ ∇ )u ] = −∇ p + μ∇ 2u . Δt (2.6) (2) Corrector step: Take the effect of body forces into consideration and update the predicted 31      Chapter Governing Equations and Boundary Condition-Enforced IBM velocity field to the physical one ρ u n +1 − u* = f n +1 Δt (2.7) which satisfies the no-slip boundary condition (2.8) U nB+1 ( X ) = ∫ u n +1δ ( x − X ) dV . (2.8) Ω In the predictor step, Eq.(2.6) is advanced to the predicted velocity field u* under the divergence-free constraint (2.2) which couples the velocity and pressure. This constraint is the major difficulty in solving the incompressible Navier-Stokes equations and could be successfully overcome by the popular and well-established projection method. The details on the projection method and its implementation will be given in Section 2.3. The corrector step, as shown in Eq. (2.7), involves evaluating the unknown body force f n+1 and updating the velocity field u* to the desired one un+1 . Therefore, the evaluation of body force poses as a crucial issue and may embody the unique feature of the IBM. The corresponding technique to determine the body force will be illustrated in details in Section 2.4. 2.3 Calculation of Predicted velocity field – Projection method Projection method was introduced decades ago by Chorin (Chorin 1968) and later independently by Temam (1969), as an efficient numerical device to compute incompressible Navier-Stokes equations in primitive variable formulation where the pressure is only present as a Lagrangian multiplier for 32      Chapter Governing Equations and Boundary Condition-Enforced IBM the incompressibility/ divergence-free constraint (2.2). Based on the Hodge decomposition, projection method efficiently decouples the computation of velocity and pressure in a time-splitting scheme and avoids solving the momentum equation (2.6) and incompressibility constraint (2.2) simultaneously. Projection method proceeds in the first step to compute an intermediate velocity field u   by using the momentum equation (2.6) and ignoring the pressure gradient term and the incompressibility constraint (2.2). In the second step, the intermediate velocity field u is projected onto the space of incompressibility field to obtain the pressure and divergence-free velocity field. To be specific, its implementation is as follows: Firstly, solve for the intermediate velocity u through Eq. (2.9) ρ u − un μ n +1/2 + ρ [ ( u ⋅ ∇ )u ] = ∇ (u + un ) Δt (2.9) by approximating Eq. (2.6) using the trapezoidal rule and dropping the pressure gradient term. The convective term, which appears in Eq. (2.9), can be approximated using the 2nd-order explicit Adams-Bashforth formula [(u ⋅∇)u] n +1/2 = n n −1 [(u ⋅∇)u] − [(u ⋅∇)u] 2 (2.10) * Then, the immediate velocity u is corrected to the predicted velocity u through ρ u* − u = −∇ p n +1 Δt (2.11) Before the use of Eq. (2.11), the pressure field should be calculated first by solving an elliptical Poisson equation (2.12), which is deduced by taking the 33      Chapter Governing Equations and Boundary Condition-Enforced IBM * divergence on both sides of Eq. (2.11) and letting u be subjected to the continuity constraint ∇ p n +1 = ρ ∇ ⋅u Δt (2.12) Substituting the solution of pressure equation (2.12) into Eq. (2.11) will finally * produce the predicted velocity field u . 2.4 Evaluation of Body force The evaluation of the body force has long been the key issue for the IBM and a number of notable strategies have been developed. 2.4.1 The Conventional IBM Early remarkable methods to calculate the body force include the well-known penalty force scheme, feedback forcing scheme and direct forcing scheme which are generally known as “conventional IBM”. These conventional IBMs have played an important role in the early and current development of the IBM. 2.4.1.1 Penalty force scheme The penalty force scheme was originally proposed by Peskin (1972) to deal with elastic boundaries on the basis of Hooke’s law and was later utilized by Lai & Peskin (2000) to calculate the singular Lagrangian force density on solid objects. In the penalty force scheme, it is assumed that the boundary 34      Chapter Governing Equations and Boundary Condition-Enforced IBM points X of the immersed object are being attached to their equilibrium positions X e by a spring with high stiffness κ . When the boundary moves and deviates from its equilibrium location, a restoring force F will be generated according to the Hook’s law F(X, t n+1 ) = κ (X(t n+1 ) − Xe (t n+1 )) (2.13) so that the boundary points will stay close to their target boundary positions. To impose the no-slip condition on the immersed boundary accurately, a large value of stiffness κ is often required which, unfortunately, would render a stiff system of equations and lead to a severe stability constraint. However, if a lower value of κ is utilized, the spurious elastic effects such as an excessive deviation from the equilibrium location may arise. 2.4.1.2 Feedback forcing scheme Goldstein et al. (1993) generalized the penalty force model and provided a two-mode feedback forcing scheme F( X, t n +1 ) = α spring t n+1 ∫ [u(X, t ′) − U B ( X, t ′) ] dt ′ +β damp ⎡⎣u( X, t n +1 ) − U B ( X, t n +1 ) ⎤⎦ (2.14) which involves a spring constant α spring and a damping constant βdamp for the control of velocity condition at the immersed boundary. This forcing term is a reflection of the velocity difference between the desired boundary value U B and the interpolated u , and behaves in a feedback loop such that the boundary velocity remains close to the desired one. In general, the method is 35      Chapter Governing Equations and Boundary Condition-Enforced IBM successful for low Reynolds number flows but is confronted with similar difficulties as the penalty force model in enforcing the boundary conditions. Firstly, accurate satisfying of the boundary condition requires large values of the spring and damping constants, which can result in numerical instability. Secondly, these two constants are flow-dependent and have to be tuned in a semi-empirical way. There is no general rule for their determination, thus making the application of the method expensive (Fadlun et al. 2000). 2.4.1.3 Direct forcing scheme To remove the annoying empirical constants, Mohd-Yusof (1997) suggested a forcing evaluation approach in which the body force was directly derived from the transformed momentum equation f ( X, t n +1 ) = ρ U B ( X , t n +1 ) − u ( X , t n ) − RHS Δt (2.15) with RHS including convective and viscous terms as well as the pressure gradient RHS= − ρ u i( ∇u ) − ∇ p + μ∇ u . (2.16) n The term RHS together with u(X, t ) constitute the interpolated velocity u(X, t n+1 ) which is expressed as u ( X, t n +1 ) = u ( X, t n ) + Δt ρ RHS . (2.17) This method is frequently termed the direct forcing method. Essentially, the discretized momentum equation is transformed such that the forcing term is 36      Chapter Governing Equations and Boundary Condition-Enforced IBM calculated by compensating the difference between the interpolated velocities u(X, t n+1 ) and the desired physical velocities U B ( X, t n+1 ) on the boundary points. In this way, the method is free from empirical parameters and no longer suffers from the numerical stability limitation, thus showing substantial improvements as compared to previous formulations. Although it was initially suggested in a sharp interface method, the direct forcing scheme has been successfully generalized into Peskin’s immersed boundary method by Uhlmann (2005), who incorporated the regularized delta function into the force calculation and spreading process. This strategy allows for a straightforward and smoother transfer between Eulerian and Lagrangian representations, therefore making the scheme more stable and easier to implement. However, these conventional IBMs generally compute f n+1 explicitly using the information at time level n , and flow penetration to the surface of the immersed object frequently occurs, i.e., the velocity condition on Γ is only approximately satisfied. Therefore, special effort is required for an accurate evaluation of the body force. 2.4.2 Boundary condition-enforced IBM Recently, Wu & Shu (2009) proposed a novel velocity correction scheme within the framework of LBM which is proven to be effective in guaranteeing 37      Chapter Governing Equations and Boundary Condition-Enforced IBM the no-slip condition on the immersed boundary. They suggested that introducing the body force f n+1 was equivalent to making a velocity correction which should be determined implicitly in a way that the velocity u( X(s), t ) at the boundary (Lagrangian) point interpolated from the physical velocity u at the Eulerian points equals to the given boundary velocity U B . Therefore the basic idea of their velocity correction scheme may provide an effective and accurate way to evaluate the body force. However, their velocity correction procedure is proposed within the framework of LBM, and it would be worthwhile to extend it into the framework of NS solver. Following the idea in Wu & Shu (2009), the body force term f n+1 should be controlled by Eq. (2.18) U nB+1 ( X n +1 ) = ∫ (u* + Δt Ω f n +1 ρ )δ (x − X n +1 )dV , (2.18) which is derived by substituting Eq. (2.7) into Eq. (2.8). Note that the force density f at the Eulerian point, as shown in Eq. (2.4), is distributed from the boundary force F through the Dirac delta function δ (x − X( s, t )) interpolation, Eq. (2.18) can be reformulated to be U nB+1 ( X n +1 ) = ∫ (u* + Δt ∫F Γ n +1 ( X n +1 )δ (x − X n +1 )ds ρ Ω )δ (x − X n +1 )dV . (2.19) As a result, the correlation between U nB+1 and f n+1 is now converted to the correlation between U nB+1 and F n+1 , and the primary concentration in the following would become the evaluation of the boundary force F n+1 . Eq. (2.19) 38      Chapter Governing Equations and Boundary Condition-Enforced IBM move upstream than the value x = 0.651 in Yang & Stern (2012). In Fig. 2.16, the center of the “figure-of-eight” plot has been artificially shifted to the origin (0, 0) for an easy comparison. The phase plots between the cylinder location and velocity are also displayed (Fig. 2.17). They match well with the reference results from Yang & Stern (2012) except that our maximum vibrating amplitude in the y -direction is slightly smaller than theirs. The instantaneous vorticity field is presented in Fig. 2.18 and a “2S” pattern (two single vortices are shed alternatively in one shedding period), which is also reported by Yang & Stern (2012), is clearly established by the free vibrating cylinder. 2.7 Conclusions In this chapter, a new boundary condition-enforced IBM for incompressible viscous flows has been presented within the framework of NS solver in the primitive variable form. The specific idea is to extend the previously proposed velocity correction procedure to the traditional NS solver such that the body force in the momentum equation is accurately evaluated and the no-slip condition on the immersed boundary is strictly enforced. In this way, the critical issue which always plagues the NS solver-based conventional IBM is successfully addressed. The performance of the new boundary condition-enforced immersed boundary solver is well verified by several classical complex geometry and moving boundary problems, from which excellent agreements between the present numerical results with the 55      Chapter Governing Equations and Boundary Condition-Enforced IBM benchmark ones have been achieved. 56      Chapter Governing Equations and Boundary Condition-Enforced IBM Table 2.1 Domain independence study 24 D ×16 D 48 D × 32 D 72 D × 48 D Re = 40 1.566 1.549 1.546 Re = 100 1.455 1.389 1.385 Size CD Table 2.2 Mesh independence study for Re = 40 and 100 h CD D / 20 D / 30 D / 40 D / 50 D / 60 Re = 40 1.579 1.567 1.554 1.549 1.547 Re = 100 1.434 1.420 1.413 1.389 1.385 Table 2.3 Time independence study for Re = 100 Δt ×10−3 1×10−3 ×10−4 CD 1.395 1.385 1.383 CL 0.340 0.343 0.344 St 0.164 0.164 0.164 57      Chapter Governing Equations and Boundary Condition-Enforced IBM Table 2.4 Comparison of drag coefficient CD and recirculation length Lw / D for flow over a stationary circular cylinder Case Re = 40 Source CD Lw / D Present 1.56 2.32 Dennis & Chang (1970) 1.52 2.35 Shukla et al. (2007) 1.55 2.34 Russell & Wang (2003) 1.60 2.29 Lima E. Silva et al. (2003) 1.54 2.55 Le et al. (2008) 1.58 2.59 58      Chapter Governing Equations and Boundary Condition-Enforced IBM Table 2.5 Comparison of drag coefficient CD , lift coefficient CL and Strouhal number St for flow over a stationary circular cylinder Case Re = 100 Source CD CL St Present 1.389 ± 0.012 0.349 0.165 Li et al. (1991) 1.330 ± 0.025 0.360 0.163 Liu et al. (1998) 1.350 ± 0.012 0.339 0.165 Lai & Peskin 1.447 0.330 0.165 Uhlmann (2005) 1.453 ± 0.011 0.339 0.169 Ji et al. (2012) 1.402 ± 0.010 0.349 0.167 (2000) Table 2.6 Comparison of drag coefficient CD , lift coefficient CL and Strouhal number St for two side-by-side circular cylinders Case CD CL St Upper 1.556 ± 0.038 0.125 ± 0.253 0.181 Lower 1.556 ± 0.038 −0.125 ± 0.253 0.181 Chang & Upper 1.533 ± 0.04 0.108 ± 0.31 0.18 Song Lower 1.533 ± 0.04 −0.108 ± 0.31 0.18 Ding et al. Upper 1.560 ± 0.038 0.131 ± 0.253 0.182 (2007) Lower 1.560 ± 0.038 −0.131 ± 0.253 0.182 Source Present G =3 (1990) 59      Chapter Governing Equations and Boundary Condition-Enforced IBM Fig. 2.1 A two-dimensional domain Ω containing an immersed object in the form of a closed curve Γ U∞ uu==(0, (0,0) 0) 32D 16D 32D Fig. 2.2 Schematic view of flow over a stationary circular cylinder 60      Chapter Governing Equations and Boundary Condition-Enforced IBM (a) Streamlines (b) Vorticity contours Fig. 2.3 Steady-state streamlines and vorticity patterns for flow over a stationary circular cylinder at Re = 40 (a) Streamlines 61      Chapter Governing Equations and Boundary Condition-Enforced IBM (b) Vorticity contours Fig. 2.4 Instantaneous streamlines and vorticity patterns for flow over a stationary circular cylinder at Re = 100 Fig. 2.5 Instantaneous streamlines for flow over an isolated stationary circular cylinder at Re = 100 obtained using the conventional IBM 62      Chapter Governing Equations and Boundary Condition-Enforced IBM Fig. 2.6 Time evolution of drag and lift coefficients for an isolated circular cylinder in a free-stream at Re = 100 20D U∞ G y = A sin(2π f c t ) 20D 15D 35D Fig. 2.7 Schematic diagram of flow over a pair of side-by-side circular cylinders 63      Chapter Governing Equations and Boundary Condition-Enforced IBM Fig. 2.8 Instantaneous streamlines (left) and vorticity pattern (right) for flow around a pair of side-by-side circular cylinders at G / D = Fig. 2.9 Time histories of drag and lift coefficients on two side-by-side arranged circular cylinders at G / D = (a) G / D = 1.2 64      Chapter Governing Equations and Boundary Condition-Enforced IBM (b) G / D = 1.7 (c) G / D = 2.5 (d) G / D = Fig. 2.10 Instantaneous streamlines (left column) and vorticity patterns (right column) for flow around a pair of side-by-side circular cylinders at different gap ratios 65      Chapter Governing Equations and Boundary Condition-Enforced IBM (a) G / D = 1.2 (b) G / D = 1.7 (c) G / D = 2.5 (d) G / D = Fig. 2.11 Time histories of drag and lift coefficients on the two side-by-side circular cylinders at different gap ratios 66      Chapter Governing Equations and Boundary Condition-Enforced IBM (a) fc / f s = 0.8 (b) fc / f s = 0.9 (c) fc / f s = 1.0 (d) fc / f s = 1.1 (e) fc / fs = 1.12 (f) fc / f s = 1.2 Fig. 2.12 Time history of drag and lift coefficients for one transversely oscillating cylinder at Re = 185 67      Chapter Governing Equations and Boundary Condition-Enforced IBM Fig. 2.13 Comparison of time-mean drag and r.m.s of drag and lift coefficients for one transversely oscillating cylinder at Re = 185 (a) (c) (e) fc / fs = 0.8 fc / f s = 0.9 fc / f s = 0.9 (b) fc / f s = 0.9 (d) fc / f s = 0.9 (f) fc / f s = 0.9 Fig. 2.14 Instantaneous vorticity patterns for flow around a transversely oscillating cylinder at various oscillation frequencies 68      Chapter Governing Equations and Boundary Condition-Enforced IBM ∂u = 0, v = ∂y U∞ ∂u =0 ∂x ∂v =0 ∂x 32D ∂u = 0, v = ∂y 36D 12D Fig. 2.15 Schematic diagram of an elastically mounted circular cylinder in a free-stream Fig. 2.16 Trajectory of the cylinder center during its free vibration 69      Chapter Governing Equations and Boundary Condition-Enforced IBM Fig. 2.17 Phase plots between the cylinder location and its velocity Fig. 2.18 Instantaneous vorticity pattern for a freely vibrating circular cylinder in a free-stream 70    [...]... n+1 Eq (2. 23) form a well-defined equation system for variables Fi (i = 1, M) Particularly, the equation system (2. 23) for the boundary force can be written in the following matrix form as [ A F ][F ] = [B F ] (2. 24) where 11 ⎛ Dh ⎜ 21 Δt 2 ⎜ Dh [AF ] = h ⎜ ρ ⎜ M1 ⎜D ⎝ h 1 11 Dh N ⎞ ⎛ Dh Δs1 2 ⎟⎜ 21 Dh N ⎟ ⎜ Dh Δs1 ⎟⎜ ⎟⎜ MN ⎟ ⎜ Dh ⎠ ⎝ DhN 1Δs1 12 Dh 22 Dh DhM 2 12 Dh Δs2 22 Dh Δs2 DhN 2 Δs2 1 DhM... Chapter 2 Governing Equations and Boundary Condition-Enforced IBM Table 2. 4 Comparison of drag coefficient CD and recirculation length Lw / D for flow over a stationary circular cylinder CD Lw / D 1.56 2. 32 Dennis & Chang (1970) 1. 52 2.35 Shukla et al (20 07) 1.55 2. 34 Russell & Wang (20 03) 1.60 2. 29 Lima E Silva et al (20 03) 1.54 2. 55 Le et al (20 08) Re = 40 Source Present Case 1.58 2. 59 58      Chapter 2. .. ⎟ 2 Dh M ΔsM ⎟ ⎟ ⎟ NM Dh ΔsM ⎟ ⎠ (2. 25) ⎛ U n +1 B ,1 ⎜ n +1 U [B F ] = ⎜ B ,2 ⎜ ⎜ n +1 ⎜U ⎝ B ,M 11 ⎞ ⎛ Dh ⎟ ⎜ 21 ⎟ − h 2 ⎜ Dh ⎟ ⎜ ⎟ ⎜ M1 ⎜D ⎟ ⎝ h ⎠ 1 * Dh N ⎞ ⎛ u1 ⎞ ⎟ 2 ⎟⎜ Dh N ⎟ ⎜ u* ⎟ 2 ⎟⎜ ⎟ ⎟⎜ ⎟ DhMN ⎟ ⎜ u* ⎟ ⎠⎝ N ⎠ 12 Dh Dh 22 DhM 2 ⎛ F1n +1 ⎞ ⎜ n +1 ⎟ F [F] = ⎜ 2 ⎟ ⎜ ⎟ ⎜ n +1 ⎟ ⎜F ⎟ ⎝ M ⎠ and U n +i1 (i = 1, B, p n+1 ( j = 1, 2, j ,M) (2. 26) (2. 27) , n+1 Fi (i = 1, M) , u *j ( j = 1, , N) and. .. IBM 62     Chapter 2 Governing Equations and Boundary Condition-Enforced IBM Fig 2. 6 Time evolution of drag and lift coefficients for an isolated circular cylinder in a free-stream at Re = 100 20 D U∞ G y = A sin (2 f c t ) 20 D 15D 35D Fig 2. 7 Schematic diagram of flow over a pair of side-by-side circular cylinders 63      Chapter 2 Governing Equations and Boundary Condition-Enforced IBM Fig 2. 8 Instantaneous... 0.038 −0.131 ± 0 .25 3 0.1 82 Source Present G =3 (1990) 59      Chapter 2 Governing Equations and Boundary Condition-Enforced IBM Fig 2. 1 A two-dimensional domain Ω containing an immersed object in the form of a closed curve Γ U∞ u = (0, 0) u = (0, 0) 32D 16D 32D Fig 2. 2 Schematic view of flow over a stationary circular cylinder 60      Chapter 2 Governing Equations and Boundary Condition-Enforced IBM (a)... integration, Eqs (2. 4) and (2. 19) can be approximated as ij f n +1 ( x j ) = ∑ F n +1 ( X in +1 ) Dh Δ si (i = 1, M ; j = 1, 2, ,N) (2. 22) i and ij U n +1 ( Xin +1 ) = ∑ u* (x j ) Dh h 2 B j + ∑∑ j (i = 1, k M ; j = 1, n F n +1 ( X k +1 )Δt ρ N ; k = 1, ,M) 39    D Δsk D h kj h ij h (2. 23) 2   Chapter 2 Governing Equations and Boundary Condition-Enforced IBM where Δsi is the length of the i th boundary segment... al (20 02) , Lee et al (20 09) and Zhou & Shou (20 11), as shown in Fig 2. 13 It is observed that the time-mean drag peaks at fc / f s = 1.0 and then decreases as fc / f s increases The same behavior is also noted for the r.m.s value of the drag The r.m.s of 52     Chapter 2 Governing Equations and Boundary Condition-Enforced IBM lift, on the other hand, shows a monotonic increase with increasing of fc... the elements of matrix [ A F ] based on Eq. (2. 25); n+1 3) Solve equation system (2. 24) to obtain the boundary force Fi ( i = 1, M ) at all Lagrangian points and then substitute them into Eq (2. 22) to get the body force f n+1 at Eulerian points 4) Update the predicted velocity u* to the physical velocity u n+1 using Eq (2. 7); 41      Chapter 2 Governing Equations and Boundary Condition-Enforced IBM Now... (for unsteady case), Strouhal number (for unsteady case), recirculation length Lw / D (for steady case) behind the cylinder are then calculated and compared with published results in the literature The drag and lift coefficients are calculated based on the widely-used definitions CD = FD (2. 28) 1 2 ρU ∞ D 2 43      Chapter 2 Governing Equations and Boundary Condition-Enforced IBM CL = FL (2. 29) 1 2. .. let ( j = 1 ,2, , N) δ (x − X( s, t )) with be mesh smoothly approximated by a continuous kernel distribution ij Dh = Dh ( x j − X i ) = x − Xi y j − Yi 1 )δ h ( ) δ ( j 2 h h h h (2. 20) where δh (r) was proposed by Lai & Peskin (20 00) as ⎧1 2 ⎪ 8 (3 − 2 | r | + 1 + 4 | r | −4r ) ⎪ ⎪1 δ h (r ) = ⎨ (5 − 2 | r | + −7 + 12 | r | −4r 2 ) ⎪8 ⎪0 ⎪ ⎩ |r | ≤ 1 1 < |r | ≤ 2 (2. 21) |r | > 2 After performing the . the following matrix form as [ ] [ ] [ ] FF =AF B (2. 24) where [] 11 12 1 11 12 1 12 21 22 2 21 22 2 2 12 12 1 2 12 NM hh h h h hM NM hh h h h hM F MM MNN N NM hh h h h hM DD D DsDs Ds DD.  Chapter 2 Governing Equations and Boundary Condition-Enforced IBM 29   Chapter 2 Governing Equations and Boundary Condition-Enforced Immersed Boundary Method A new version of boundary. ⎟ ΔΔ Δ Δ ⎜⎟⎜ ⎟ = ⎜⎟⎜ ⎟ ⎜⎟⎜ ⎟ ⎜⎟⎜ ⎟ ΔΔ Δ ⎝⎠⎝ ⎠ A       (2. 25) [] 1 11 12 1 * ,1 1 1 21 22 2 * ,2 2 2 1 12 * , n N B hh h n N B hh h F n MM MN BM hh h N DD D DD D h DD D + + + ⎛⎞ ⎛⎞⎛⎞ ⎜⎟ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟⎜⎟ =− ⎜⎟ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎜⎟⎜⎟ ⎜⎟ ⎝⎠⎝⎠ ⎝⎠ U u U u B U u     