Chapter Stream Function-Vorticity Formulation-based IBM Chapter Stream Function-Vorticity Formulation-based Immersed Boundary Method1 In this chapter, a novel and efficient stream function-vorticity formulation-based immersed boundary method is presented. The main idea of the method is to accurately satisfy both the governing equation and boundary condition, which is realized through velocity correction and vorticity correction procedures. Since the physical boundary condition is usually for velocity, we enforce the no-slip boundary condition through a velocity correction process. For the vorticity correction, unlike adding complicated source terms into the vorticity transport equation as suggested by Wang et al. (2009), we not add any source term in the vorticity transport equation. Instead, the vorticity correction is directly evaluated from the first order derivatives of velocity correction. In this work, two ways are proposed to evaluate the vorticity correction. One is based on finite difference approximation of velocity-correction derivatives, and the other is based on derivative expressions of Dirac delta function and velocity correction. These                                                               Parts of materials in this chapter have been published in W.W. Ren, J. Wu, C. Shu, W.M. Yang, Int. J. Numer. Meth. Fluids, 70 (2012) 627-645.  71      Chapter Stream Function-Vorticity Formulation-based IBM two ways and the whole solver are tested by their applications to simulate flows over a circular cylinder for both steady and unsteady states. Examples of moving boundary problems, like flow over a left moving circular cylinder, flow over an inline oscillating circular cylinder, sedimentation of a circular particle between two parallel walls, are also provided for further validation. Compared to the previously proposed stream function-vorticity formulation-based immersed boundary method, the present method is more efficient and attractive for two-dimensional (2D) incompressible flows. 3. Methodology 3.1.1 Governing equations The two-dimensional computational configuration shown in Fig. 2.1 is recalled here. To take into account the effect of immersed boundary on the flow field, an additional term, which plays the role of vorticity correction, is introduced in the stream function formulation in this work. As a result, the governing equations describing the fluid motion with the use of IBM can be expressed in the stream function-vorticity formulation as ρ( ∂ω ∂ω ∂ω ∂ 2ω ∂ 2ω +u +v ) = μ( + ) ∂x ∂y ∂x ∂y ∂t ∂ 2ϕ ∂ 2ϕ + = ω +Θ ∂x ∂y (3.1) (3.2) which is subject to the no-slip boundary condition (2.3), where x and y denote the components of the Eulerian coordinate x , 72    u and v correspond   Chapter Stream Function-Vorticity Formulation-based IBM to the components of the velocity vector u in the x and y directions respectively. ϕ and ω denote the stream function and vorticity. The additional term Θ is the vorticity source. In addition, the vorticity is defined as ω+Θ = ∂u ∂v − ∂y ∂x (3.3) and the stream function is related to the velocity by u= ∂ϕ ∂y v=− (3.4) ∂ϕ ∂x (3.5) Note that ω + Θ in Eq. (3.3) is the physical vorticity. Eqs. (3.1) to (3.5) together with Eqs. (2.3) and (2.5) provide a complete description of stream function-vorticity formulation for incompressible viscous flows in the entire computational domain Ω involving the immersed boundary Γ . Note that the vorticity source Θ in Eq. (3.2) is to consider the effect of the immersed boundary Γ . The solution of equation systems (3.1)-(3.2) can be obtained by the following two steps. In the first step, the normal stream function-vorticity formulation without the vorticity source term (that is, the effect of immersed boundary is ignored) is solved, ρ( ∂ω ∂ω ∂ω ∂ 2ω ∂ 2ω +u +v ) = μ( + ) ∂y ∂x ∂y ∂t ∂x ∂ 2ϕ ∂ 2ϕ + =ω ∂x ∂y (3.7) 73    (3.6)   Chapter Stream Function-Vorticity Formulation-based IBM By solving the above equations, the predicted vorticity ω * and stream * function ϕ can be obtained. Then from Eqs. (3.4) and (3.5), the predicted velocity field u* is calculated. The basic solution procedure is described as follows. At first, we apply the Euler forward scheme to discretize the time derivative in Eq. (3.6) and obtain the predicted vorticity ω * by ρ ω* − ω n Δt = − ρ (u n ⋅ ∇ )ω n + μ∇ 2ω n (3.8) * Then the predicted stream function field ϕ is solved from the following Poisson equation ∇2ϕ * = ω* (3.9) For the spatial derivatives, the well-known central difference scheme is employed for numerical discretization. In the second step, the velocity field should be corrected so that the no-slip condition (2.3) on the immersed boundary Γ can be satisfied. It should be indicated that although the stream function-vorticity form is used, we still prefer to make velocity correction. This is because the physical condition at the immersed boundary is usually the condition for velocity rather than for stream function. In addition, for some multiply-connected domain problems, the value of stream function at the surface of immersed body is an unknown constant. This unknown constant may vary with time, and must be determined and updated at each time step during the computation. This brings difficulty in its numerical implementation. To make velocity correction, the corrected 74      Chapter Stream Function-Vorticity Formulation-based IBM velocity field u is written as u = u* + Δu (3.10) where Δu is the velocity correction, and u* is the predicted velocity field derived in the predictor step. Correspondingly, the corrected vorticity can be written as ω = ω* + Θ (3.11) Here, ω* is the predicted vorticity field, and Θ is the vorticity correction. ω* in Eq. (3.11) is actually the same as ω in Eq. (3.2) since they are from the same governing equation (vorticity transport equation without considering the effect of immersed boundary). It is only for convenience that we note the solution of Predictor step as ω* . In this sense, Θ in Eq. (3.11) is indeed the vorticity source in Eq. (3.2). Substituting Eqs. (3.10) and (3.11) into Eq. (3.3) and using Eq. (3.9) gives Θ= ∂ ( Δu ) ∂ ( Δ v ) − ∂y ∂x (3.12) It should be noted that the key step in the proposed stream function-vorticity formulation-based IBM is the calculation of velocity correction. Once it is determined, the corrected velocity and vorticity can be calculated from Eqs. (3.10) and (3.11) respectively. 3.1.2 Velocity correction procedure The velocity correction aims to enforce the no-slip boundary condition (2.3). The basic idea is that the velocity u(X(s, t ), t ) at the boundary (Lagrangian) 75      Chapter Stream Function-Vorticity Formulation-based IBM point interpolated from the corrected velocity u(x, t ) at the surrounding fluid (Eulerian) points should be equal to the given boundary velocity UB (X(s, t ), t ) , i.e., U B ( X( s, t ), t ) = ∫ u ( x, t )δ ( x − X( s, t )) dV (3.13) Ω Noticing that the corrected velocity u is contributed from the predicted velocity u* and the velocity correction Δu , Eq. (3.13) can be reformulated as U B ( X( s, t ), t ) = ∫ ( u* (x, t ) + Δu(x, t ) ) δ (x − X( s, t ))dV (3.14) Ω It is known that the velocity correction is introduced due to the presence of the immersed boundary, so it is reasonable to assume that the velocity correction stems from the virtual boundary flux ΔP through the delta function interpolation Δu( x, t ) = ∫ ΔP ( X( s, t ), t )δ ( x − X( s, t )) ds (3.15) Γ Employing a similar Eulerian and Lagrangian mesh discretization as those in Section 2.4.2, Eq. (3.14) and (3.15) can be approximated in their spatial discrete forms U B ( X i , t ) = ∑ ( u* ( x j ) + Δu ( x j ) ) Dhij h (3.16) j and Δ u ( x j , t ) = ∑ Δ P ( X i , t ) Dhij Δ si (i = 1, 2, " , M ; j = 1, 2, " , N ) (3.17) i Substituting Eq. (3.17) into Eq. (3.16), an equation system for the virtual boundary flux ΔP would be formed and written in the matrix form as [ A P ][ X P ] = [B P ] (3.18) 76      Chapter Stream Function-Vorticity Formulation-based IBM where ⎛ Dh11 ⎜ 21 ⎜ Dh [AP ] = h ⎜ # ⎜⎜ N ⎝ Dh " Dh1M " Dh2 M % # " DhNM Dh12 Dh22 # DhN ⎞ ⎛ Dh11Δs1 ⎟ ⎜ 21 ⎟ ⎜ Dh Δs1 ⎟⎜ # ⎟⎟ ⎜⎜ M ⎠ ⎝ Dh Δs1 Dh12 Δs2 Dh22 Δs2 # M2 Dh Δs2 " Dh1N ΔsM " Dh2 N ΔsM % # MN " Dh ΔsM ⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠ (3.19) ⎛ U1B ⎜ U [B P ] = ⎜⎜ B # ⎜⎜ M ⎝ UB ⎛ Dh11 ⎞ ⎜ 12 ⎟ ⎟ − h ⎜ Dh ⎜ # ⎟ ⎜⎜ 1M ⎟⎟ ⎠ ⎝ Dh ⎛ ΔP1 ⎜ ΔP [ X P ] = ⎜⎜ # ⎜ ⎝ ΔPM Dh21 Dh22 # Dh2 M DhN ⎞ ⎛ u1* ⎞ ⎟⎜ ⎟ " DhN ⎟ ⎜ u*2 ⎟ % # ⎟⎜ # ⎟ ⎟⎜ ⎟ " DhNM ⎟⎠ ⎜⎝ u*N ⎟⎠ " ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (3.20) (3.21) and ΔPi (i = 1,", M ) are the abbreviations for ΔP(Xi , t ) . Similar to [ A F ] , the elements of coefficient matrix [ A P ] are only related to the coordinates of Lagrangian boundary points and their adjacent Eulerian points. After solving the equation system (3.18) and obtaining ΔPi at all Lagrangian points, they are substituted into Eq. (3.17) to obtain the velocity correction Δu j , which are further substituted into Eq. (3.10) to get the corrected velocity u j ( j = 1,", N ) . 3.1.3 Vorticity correction procedure The vorticity correction procedure is straightforward and easier as compared to that of velocity correction. As shown in Eq. (3.12), once the velocity correction has been obtained, the vorticity correction can be calculated from 77      Chapter Stream Function-Vorticity Formulation-based IBM the first order derivatives of velocity correction. Then from Eq. (3.11), the corrected vorticity can be easily computed. In the vorticity correction procedure, the key step is to approximate the derivatives of ∂ ( Δu ) and ∂y ∂(Δv) at Eulerian points. In this work, we propose two ways to the ∂x approximation. They are denoted as Method and Method 2. Method is very simple. It directly approximates ∂(Δv) ∂ ( Δu ) and by finite difference ∂x ∂y schemes. Method avoids numerical approximation of derivatives. As shown in Eq. (3.15), the velocity correction Δu is a function of physical location x, and the Dirac delta function δ (x − X( s, t )) is also the function of x. Therefore, we have ∂ ( Δu (x, t )) ∂δ (x − X( s , t )) ds = ∫ ΔPx (X( s , t ), t ) ∂y ∂y Γ (3.22a) ∂ ( Δv(x, t )) ∂δ (x − X( s , t )) ds = ∫ ΔPy (X( s , t ), t ) ∂x ∂x Γ (3.22b) where ( Δ u , Δ v ) are components of Δu , and ( ΔPx , ΔPy ) are components of ΔP . Substituting Eq. (3.22) into Eq. (3.12) and approximating the integral by summation, we can obtain, Θ= − ∂ ⎛ ⎞ ΔPx ( Xi , t ) Dh (x − Xi )Δsi ⎟ ∑ ⎜ ∂y ⎝ i ⎠ ∂ ⎛ ⎞ ΔPy ( Xi , t ) Dh (x − Xi )Δsi ⎟ ∑ ⎜ ∂x ⎝ i ⎠ which can be further written as 78    (3.23)   Chapter Stream Function-Vorticity Formulation-based IBM ⎛ x − Xi ⎛ ∂ y − Yi ⎞ ⎞ ) ⎜ δh ( ) ⎟ Δsi ⎟ Θ = ⎜ ∑ ΔPx ( Xi , t ) δ h ( h h y h ∂ i ⎝ ⎠ ⎠ ⎝ x − Xi ⎞ y − Yi ⎛ ⎞ ⎛ ∂ ) ⎟δh ( )Δsi ⎟ − ⎜ ∑ ΔPy ( Xi , t ) ⎜ δ h ( h ⎝ ∂x h h ⎠ ⎝ i ⎠ (3.24) In Eq. (3.24), the continuous kernel functions take the form of (2.21), whose derivatives can then be analytically given as sgn( x − X i ) 2( x − X i ) ⎧ − ⎪ sgn( x − X i ) h h2 − + ⎪ 4h x − Xi x − Xi ⎪ 1+ | | −4 | | ⎪ h h ⎪ 3sgn( x − X i ) 2( x − X i ) ⎪⎪ − x − Xi ∂ sgn( ) x − X ) = ⎨− δh ( h h2 i + ∂x h ⎪ 4h x − Xi x − Xi | −4 | | −7 + 12 | ⎪ h h ⎪ ⎪ ⎪ ⎪0 ⎪⎩ ≤| x − Xi |≤ h h (3.25) sgn( y − Yi ) 2( y − Yi ) ⎧ − ⎪ sgn( y − Yi ) h h2 − + ⎪ 4h y − Yi y − Yi ⎪ 1+ | | −4 | | ⎪ h h ⎪ 3sgn( y − Yi ) 2( y − Yi ) ⎪⎪ − y − Yi ∂ ) = ⎨− sgn( y − Yi ) + δh ( h h2 ∂y h ⎪ 4h y − Yi y − Yi | −4 | | −7 + 12 | ⎪ h h ⎪ ⎪ ⎪ ⎪0 ⎪⎩ ≤| y − Yi |≤ h h (3.26) With above formulations, Method evaluates the vorticity correction directly from the velocity correction without numerical approximation of derivatives. 79      Chapter Stream Function-Vorticity Formulation-based IBM 3.1.4 Computational sequence The computational sequence of the present solver can be summarized as below. To march solution from time level n to n + , 1) Use u n and ω n as initial flow field to solve Eqs. (3.8) and (3.9) and obtain predicted vorticity ω * and stream function ϕ * ; 2) Substitute ϕ * to Eqs. (3.4) and (3.5) to calculate the predicted velocity * field u ; 3) Compute the elements of matrix [ A P ] ; 4) Use equation system (3.18) to calculate the virtual boundary fluxes ΔPi ( i =1,", M ) at all Lagrangian points and then substitute them into Eq. (3.17) to get the velocity correction Δu ; 5) Correct the fluid velocity at Eulerian points using Eq. (3.10); 6) Use either Method or Method described in Section 3.1.3 to calculate the vorticity correction Θ ; 7) Correct the vorticity at Eulerian points using Eq. (3.11); 8) Use the corrected vorticity and velocity as the initial conditions, and repeat steps to for the computation of next time level. The process continues until a converged solution is achieved (steady case) or the given time is reached (unsteady case). 3.2 Results and Discussion The proposed stream function-vorticity formulation-based IBM, using velocity 80      Chapter Stream Function-Vorticity Formulation-based IBM geometry as in the above example is used, except that the cylinder starts to move at a distance of 12 to the right boundary. The boundary conditions are imposed as ϕ= ∂ω =0 ∂x on the left boundary ϕ =ω = on the top and bottom boundary ∂ϕ ∂ω = =0 ∂x ∂x on the right boundary Note that the only difference between this left moving cylinder case and the previous stationary cylinder case is the adjustment of reference frame. These two problems should provide equivalent results. Fig. 3.4 shows the adjusted streamlines at a non-dimensional time of 16 . Fig. 3.5 presents the vorticity patterns in the vicinity of the cylinder for both stationary case and left moving cylinder case at the same moment. A comparison between histories of drag coefficient and vorticity distribution along the cylinder surface for both cases is also plotted in Fig. 3.6 and Fig. 3.7 respectively. It can be clearly seen that there are very little discrepancies between the results for two cases. 3.2.3 Flow over an inline oscillating circular cylinder in a fluid at rest Flow over an inline oscillating circular cylinder in a fluid at rest has been investigated both experimentally (Dütsch et al. 1998) and numerically (Wang 84      Chapter Stream Function-Vorticity Formulation-based IBM et al. 2009; Yang & Balaras 2006; Choi et al. 2007; Guilmineau & Queutey 2002; Lee et al. 2011). Herein, it is studied to validate the proposed method for solving moving boundary problems. The inline motion of the cylinder is given by the harmonic oscillation x(t ) = − A sin(2π fct ) (3.27) where x(t ) is location of the cylinder center in its oscillation direction (x-direction), A and f c are the oscillating amplitude and frequency of the cylinder, respectively. The two key parameters characterizing this flow are Reynolds number Re = ρU max D ( U max is the maximum velocity of the μ cylinder) and Keulegan-Carpenter number KC = U max , which are set as fc D Re = 100 and KC = in the present investigation, according to the experimental result of Dütsch et al. (1998). A dimensionless computational domain of size 24 × 24 is chosen, with the cylinder initially located at the center of the domain. A uniform mesh of resolution h = 1/ 40 with a time step size of Δt = 0.002 is used for the simulation. On all four boundaries, the natural boundary condition (Neumann type) is applied for both vorticity and velocity. Fig. 3.8 shows the velocity profiles in the oscillation and transverse direction at four different x locations ( x = −0.6 , , 0.6 , 1.2 ) and three different phase angles ( φ = 2π ft = 180° , 210° , 330° ). The corresponding experimental 85      Chapter Stream Function-Vorticity Formulation-based IBM results of Dütsch et al.(1998) and numerical results of Wang et al. (2009) are also displayed in these figures for comparison. A good agreement can be observed from the comparison. Furthermore, a time evolution of the in-line force Fx acting on the cylinder surface in one period is presented in Fig. 3.9. The experimental result of Dutsch et al. (1998) is also included in Fig. 3.9 for comparison. Once again, a good agreement is achieved. It is noted that for all the cases tested, the present solver does not show any spurious force oscillation. This may be attributed to the fact that the no-slip boundary condition is enforced in the present approach and the velocity correction, which is related to the boundary force, at all boundary points is obtained simultaneously from the algebraic equation system (3.18). 3.2.4 Sedimentation of a single circular particle inside a box Another problem we choose to further test the capability of present method in solving moving boundary problems is the sedimentation of a single circular particle inside a box. A single particle which settles inside a box has been studied by several researchers (Hu 1996; Naury 1999; Hu et al. 2001; Glowinski et al. 1999; Glowinski et al. 1999; Wan & Turek 2006). Specially, Feng et al. (2004; 2005), Uhlmann (2005), Luo et al. (2007), and Wu & Shu (2010) have used various versions of immersed boundary method to study this problem. 86      Chapter Stream Function-Vorticity Formulation-based IBM In the present simulation, a domain with width cm and height cm (Fig. 3.10) is taken. The density ρ and viscosity μ of incompressible viscous fluid filled in the domain are 1.0 g / cm3 and 0.1g /(cm⋅ s) , respectively. The circular particle is rigid with density of ρ p = 1.25 g / cm and diameter of d p = 0.25cm . Initially, the particle is released at (1cm,4cm) in static fluid, and then falls down due to the gravity force. A uniform mesh with resolution h = d p / 25 and time step Δt = 0.005 is used for the present simulation. The boundaries of the box are solid walls and can be treated as a single streamline. The instantaneous vorticity patterns at different moments of t = 0.2 s , .4 s , .6 s , .8 s are displayed in Fig. 3.11. The variation of flow structure can be clearly observed. The time evolutions of longitudinal coordinate Y , longitudinal velocity V , Reynolds number Re of particle center as well as the translational kinetic energy ET are plotted in Figs. 3.12-3.15. Here, Re and ET ρ pd p U + V are defined as Re = μ and ET = 0.5m p (U + V ) , respectively, where U and V are velocity components of particle center, and m p is the mass of the particle. The results of Wan et al. (2006) and Wu et al. (2010) are also included in the figures for comparison. We can see clearly that present results match well with the benchmark solutions. This shows that the present solver can be effectively used to simulate moving boundary flow problems. 87      Chapter Stream Function-Vorticity Formulation-based IBM 3.3 Conclusions In this chapter, a simple and efficient stream function-vorticity formulation-based IBM is developed for simulating 2D incompressible viscous flows. The effect of boundary on the flow field is considered through velocity correction and vorticity correction. The velocity correction is determined implicitly in such a way that the velocity at the immersed boundary interpolated from the corrected velocity field satisfies the physical boundary condition. Unlike adding complicated source terms into the vorticity transport equation in the literature, in this work, the vorticity correction is made through the stream function formulation. It is evaluated simply from the first order derivatives of velocity correction. Furthermore, two ways are presented to approximate velocity-correction derivatives. One is direct approximation by the finite difference scheme. The other is based on derivative expressions of the Dirac delta function and velocity correction. This way does not involve numerical approximation of derivatives. Numerical experiments show that both ways work very well, and the second way seems to perform better. The efficiency and capability of present method and two ways to evaluate the vorticity correction are tested by their application to simulate both stationary boundary and moving boundary problems. Numerical results show good agreement with available data in the literature. It seems that the present 88      Chapter Stream Function-Vorticity Formulation-based IBM method has a promising potential for solving 2D incompressible viscous flows with curved boundaries. 89      Chapter Stream Function-Vorticity Formulation-based IBM Table 3.1 Comparison of drag coefficient CD and recirculation length Lw / D References CD Lw / D Dennis et al. (1970) 1.52 2.35 Fornberg et al. (1980) 1.50 2.24 He et al. (1997) 1.499 2.245 Niu et al. (2006) 1.589 2.26 Wu & Shu (2009) 1.554 2.3 Method 1.552 2.37 Method 1.526 2.368 Cases Re=40 Present Table 3.2 Comparison of consumed CPU time Method CPU time ( s ) Stream function-vorticity formulation-based IBM 4669.2 Pressure-velocity formulation-based IBM 5529.18 90      Chapter Stream Function-Vorticity Formulation-based IBM Table 3.3 Comparison of drag coefficient CD , lift coefficient CL , and Strouhal number St Cases Re=100 References CD CL St Braza et al. (1986) 1.325 ± 0.008 ± 0.28 0.164 Liu et al. (1998) 1.350 ± 0.012 ± 0.339 0.164 Ding et al. (2004) 1.364 ± 0.015 ± 0.25 0.160 Method 1.383 ± 0.012 ± 0.373 0.164 Method 1.335 ± 0.011 ± 0.356 0.164 Present 91      Chapter Stream Function-Vorticity Formulation-based IBM (a) Streamlines obtained by Method (b) Streamlines obtained by Method (c) Vorticity contours obtained by Method (d) Vorticity contours obtained by Method Fig. 3.1 Streamlines and vorticity patterns in the vicinity of circular cylinder at Re = 40 92      Chapter Stream Function-Vorticity Formulation-based IBM (a) Streamlines (b) Vorticity contours Fig. 3.2 Streamlines and vorticity patterns in the vicinity of circular cylinder at Re = 100 (a) Method 93      Chapter Stream Function-Vorticity Formulation-based IBM (b) Method Fig. 3.3 Time evolution of drag and lift coefficients at Re = 100 Fig. 3.4 Adjusted streamlines for flow over a left moving circular cylinder ( Re = 40 ) Fig. 3.5 Vorticity patterns for flow over a left moving/stationary circular cylinder at Re = 40 (solid line represents the result for left moving case, and dashed line represents the result for stationary case) 94      Chapter Stream Function-Vorticity Formulation-based IBM Fig. 3.6 Vorticity distribution on the surface of cylinder at Re = 40 for stationary and left moving cylinder cases Fig. 3.7 Evolution of drag coefficient at Re = 40 for stationary and left moving cylinder cases 95      Chapter Stream Function-Vorticity Formulation-based IBM (a) φ = 2π ft = 180° (b) φ = 2π ft = 210° (c) φ = 2π ft = 330° Fig. 3.8 Comparison of velocity profiles ( u -component in the left column, v -component in the right column) at four different x locations and three phase angles of φ = 2π ft = 180°, 210°,330° . (Lines are the present results, empty symbols are the experimental results of Dütsch et al. (1998), and filled symbols represent numerical results of Wang et al. (2009)) 96      Chapter Stream Function-Vorticity Formulation-based IBM Fig. 3.9 Comparison of time evolution of inline force Fx in one period Fig. 3.10 Schematic view of sedimentation of a single particle between two parallel walls 97      Chapter Stream Function-Vorticity Formulation-based IBM Fig. 3.11 Instantaneous flow structures (vorticity) at different times of t = .2 s , . s , . s , . s Fig. 3.12 Time evolution of translational kinetic energy ET 98      Chapter Stream Function-Vorticity Formulation-based IBM Fig. 3.13 Time evolution of longitudinal coordinate Y of particle center Fig. 3.14 Time evolution of longitudinal velocity V of particle center Fig. 3.15 Time evolution of Reynolds number Re   99    [...]... and three phase angles of φ = 2π ft = 180°, 210° ,33 0° (Lines are the present results, empty symbols are the experimental results of Dütsch et al (1998), and filled symbols represent numerical results of Wang et al (2009)) 96      Chapter 3 Stream Function-Vorticity Formulation-based IBM Fig 3. 9 Comparison of time evolution of inline force Fx in one period Fig 3. 10 Schematic view of sedimentation of. .. Formulation-based IBM Fig 3. 6 Vorticity distribution on the surface of cylinder at Re = 40 for stationary and left moving cylinder cases Fig 3. 7 Evolution of drag coefficient at Re = 40 for stationary and left moving cylinder cases 95      Chapter 3 Stream Function-Vorticity Formulation-based IBM (a) φ = 2π ft = 180° (b) φ = 2π ft = 210° (c) φ = 2π ft = 33 0° Fig 3. 8 Comparison of velocity profiles ( u -component... Fig 3. 1 Streamlines and vorticity patterns in the vicinity of circular cylinder at Re = 40 92      Chapter 3 Stream Function-Vorticity Formulation-based IBM (a) Streamlines (b) Vorticity contours Fig 3. 2 Streamlines and vorticity patterns in the vicinity of circular cylinder at Re = 100 (a) Method 1 93     Chapter 3 Stream Function-Vorticity Formulation-based IBM (b) Method 2 Fig 3. 3 Time evolution of. .. the instantaneous streamlines and vorticity contours in the vicinity of the cylinder for Re = 100 It is clear that the Karman vortex street has been successfully revealed in both plots, which results in regular periodic variations of drag and lift coefficients presented in Fig 3. 3 Table 3. 3 provides the drag and lift coefficients (in the form of average value and magnitude of variation) C D , CL , Strouhal... 2 .3 Method 1 1.552 2 .37 Method 2 1.526 2 .36 8 Cases Re=40 Present Table 3. 2 Comparison of consumed CPU time Method CPU time ( s ) Stream function-vorticity formulation-based IBM 4669.2 Pressure-velocity formulation-based IBM 5529.18 90      Chapter 3 Stream Function-Vorticity Formulation-based IBM Table 3. 3 Comparison of drag coefficient CD , lift coefficient CL , and Strouhal number St CD CL St 1 .32 5...   Chapter 3 Stream Function-Vorticity Formulation-based IBM Fig 3. 11 Instantaneous flow structures (vorticity) at different times of t = 0 2 s , 0 4 s , 0 6 s , 0 8 s Fig 3. 12 Time evolution of translational kinetic energy ET 98      Chapter 3 Stream Function-Vorticity Formulation-based IBM Fig 3. 13 Time evolution of longitudinal coordinate Y of particle center Fig 3. 14 Time evolution of longitudinal... (2007), and Wu & Shu (2010) have used various versions of immersed boundary method to study this problem 86      Chapter 3 Stream Function-Vorticity Formulation-based IBM In the present simulation, a domain with width 2 cm and height 6 cm (Fig 3. 10) is taken The density ρ and viscosity μ of incompressible viscous fluid filled in the domain are 1.0 g / cm3 and 0.1g /(cm⋅ s) , respectively The 3 circular... , and Strouhal number St CD CL St 1 .32 5 ± 0.008 ± 0.28 0.164 Liu et al (1998) 1 .35 0 ± 0.012 ± 0 .33 9 0.164 Ding et al (2004) 1 .36 4 ± 0.015 ± 0.25 0.160 Method 1 1 .38 3 ± 0.012 ± 0 .37 3 0.164 Method 2 Re=100 References Braza et al (1986) Cases 1 .33 5 ± 0.011 ± 0 .35 6 0.164 Present 91      Chapter 3 Stream Function-Vorticity Formulation-based IBM (a) Streamlines obtained by Method 1 (b) Streamlines obtained... outflow boundary, the homogeneous Neumann boundary conditions of ∂ϕ ∂ω = 0 are applied Non-uniform meshes with a fine = ∂x ∂x resolution of Δx = Δy = h = D / 50 near the cylinder have been used for the 81      Chapter 3 Stream Function-Vorticity Formulation-based IBM following discussions For the unsteady case Re = 100 , a time step size of Δt = 0.001 is used To illustrate the capability of proposed two methods. .. Function-Vorticity Formulation-based IBM 3. 3 Conclusions In this chapter, a simple and efficient stream function-vorticity formulation-based IBM is developed for simulating 2D incompressible viscous flows The effect of boundary on the flow field is considered through velocity correction and vorticity correction The velocity correction is determined implicitly in such a way that the velocity at the immersed boundary . − ∂∂ (3. 3) and the stream function is related to the velocity by y u ∂ ∂ = ϕ (3. 4) x v ∂ ∂ −= ϕ (3. 5) Note that ω +Θ in Eq. (3. 3) is the physical vorticity. Eqs. (3. 1) to (3. 5) together. Eqs. (3. 10) and (3. 11) into Eq. (3. 3) and using Eq. (3. 9) gives () ()uv y x ∂Δ ∂Δ Θ= − ∂ ∂ (3. 12) It should be noted that the key step in the proposed stream function-vorticity formulation-based. 1) Use n u and n ω as initial flow field to solve Eqs. (3. 8) and (3. 9) and obtain predicted vorticity * ω and stream function * ϕ ; 2) Substitute * ϕ to Eqs. (3. 4) and (3. 5) to calculate