International Journal of Numerical Methods for Heat & Fluid Flow A novel and efficient approach to specifying Dirichlet far-field boundary condition of pressure Poisson equation Jun-Hyeok Lee, Seung-Jae Lee, Jung-chun Suh,
Article information: To cite this document: Jun-Hyeok Lee, Seung-Jae Lee, Jung-chun Suh, "A novel and efficient approach to specifying Dirichlet far-field boundary condition of pressure Poisson equation", International Journal of Numerical Methods for Heat & Fluid Flow, https:// doi.org/10.1108/HFF-02-2017-0060 Permanent link to this document: https://doi.org/10.1108/HFF-02-2017-0060
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1
Introduction
The vortex-in-cell method [7, 32, 33] combined with the penalization technique [1], herein referred to as the penalized vortex-in-cell (pVIC) method, is a state-of-the-art vortex particle method (refer to, for example, [10, 13, 26, 22]). In the pVIC method, both velocity and vorticity fields are computed on a grid using a particle approximation. On an Eulerian grid, finite differences can be applied to evaluate the evolution of both fields and the grid solution can then be interpolated back to the Lagrangian particles. Hybrid approaches such as this can achieve much higher performance than pure Lagrangian vortex methods. Because the pVIC method uses the vorticity formulation of the Navier-Stokes equations, no knowledge of the pressure is required to solve for the evolution of flows. The fact that the pressure-velocity coupling problem is completely eliminated is one of the advantages of vorticity-based methods such as pVIC, compared with conventional methods that use the primitive variables (u, p) as unknowns (for more on the benefits of the vorticity formulation, refer to [31, 30]). However, pressure is a dynamic flow property that plays an important role in understanding engineering and physics-related problems. Accordingly, a key motivation of this study is to explicitly determine the pressure from vorticity and velocity vector fields. The present study aims to identify infinite-domain boundary conditions for solving pressure Poisson equation (PPE). The pressure can be obtained by solving the PPE ∇2 H = ∇ · (u × ω), where the total pressure H is the sum of the static and dynamic pressures. Kim et al. [19] introduced an integral approach based on the Lagrangian formulation to obtain the pressure field by solving the PPE for H with a Neumann boundary condition at the solid walls. They utilized Green’s identity to derive a boundary integral formulation of the PPE. Their approach is well suited to treat solid bodies with complicated boundaries because the integral equation is formulated only on the boundaries of the body under consideration and it naturally incorporates a free-stream condition at infinity. However, their integral approach has practical disadvantages. The equation’s dense and asymmetric matrix is not straightforward to solve and may become computationally infeasible for large problems. Computation of the volume and surface integrals is also expensive, although this can be accelerated by fast algorithms such as the fast multipole method (FMM) [14, 4]. Over the past few decades, several kinds of fast methods for solving Poisson equations on uniform grids have been proposed, such as multigrid methods, iterative/relaxation methods, and direct methods based on Fourier transforms. Direct methods, which were first described in 1965 by Hockney [15], yield theoretically exact solutions rather than iteratively approximating the solution. Direct methods based on Fourier analysis are particularly attractive for regular domains such as Cartesian, cylindrical, or spherical regions. Lee et al. [21] solved a PPE on a uniform rectangular grid by employing an FFT-based Poisson solver. Their approach maintains numerical consistency in the vorticity transport equation (VTE) and PPE by adopting a penalization technique for both equations. By modifying the PPE with a penalty term, they obtained ∇2 H = ∇ · (u × ω) − ∇ · (λ0 χs u), where λ0 > 0 denotes a numerical parameter. The mask function χs (x) describes the shape of an obstacle, yielding 0 in fluid and 1 in solid regions. The penalty term imposes solid boundary conditions for obstacles inside a given computational domain. They investigated particular values for λ0 by numerical experimentation in 2D, showing that λ0 = O(1/∆t), independent of the grid size. Lee et al. [21] set H = 0 at the domain boundaries to solve the PPE in a bounded domain. Such boundary conditions would be reasonable if the computational domain was large enough to impose a homogeneous Dirichlet condition at its boundaries. In the pVIC method, however, the computational domain is determined by the size of a minimal box containing the fluid particles so as to reduce the computational costs. It is therefore an important numerical issue to correctly and quickly specify non-homogeneous boundary conditions for solving the PPE in a small (minimal) computational domain. In this study, we propose a highly effective numerical approach for such a task. This paper is structured as follows. The basic formulations are first described in Section 2. Section 3 describes how to quantify Dirichlet boundary conditions for solving the PPE, which is a challenging goal of this study. Numerical results are presented and discussed in Section 4, followed by concluding remarks in Section 5.
1
2
Methodology
Before discussing the boundary conditions for solving the PPE, showing how to evaluate the velocity and vorticity fields using the pVIC method is necessary. The penalized VTE and PPE are the basic equations of interest in the present study. The velocity and vorticity vectors are herein denoted by u and ω, respectively. We work with non-dimensional variables; therefore, ν = 1/Re, where Re is the Reynolds number and the fluid density is ρ = 1.
2.1
Basic equations
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For an incompressible viscous flow, the penalized Navier-Stokes equation [1] with no body forces can be expressed as 1 2 ∂u + (u · ∇)u = ∇ u − ∇p + λχs (us − u), (1) ∂t Re where λ is the penalization constant and us is the velocity of the solid body (if the body is at rest, then us = 0). The penalized VTE is derived by applying the curl operator to Equation (1), thus yielding 1 2 Dω = (ω · ∇)u + ∇ ω + ∇ × [λχs (us − u)]. Dt Re
(2)
Equation (2) governs the evolution of vorticity in a fluid flow. u can be self-consistently coupled with ω at all times via u = u∞ + ∇ × ψ and ∇2 ψ = −ω (or ∇2 u = −∇ × ω), where ψ is referred to as a stream function in 2D and a vector potential in 3D. By applying the divergence operator to Equation (1), the penalized PPE can be derived as 1 (3) ∇2 p + u · u = ∇ · (u × ω) + ∇ · [λ0 χs (us − u)] , 2 where λ0 is distinguished from λ by its order of magnitude [21, 24]. The second term on the right-hand side is equivalent to λ0 (us − u) · ∇χs , and this is confined to a narrow region adjacent to the fluid-solid interface. In both Equations (2) and (3), the penalty terms greatly simplify the imposition of solid boundary conditions.
2.2
The pVIC method
In the pVIC method, a uniform grid covering a computational domain Ωf ⊂ Rd is defined. Ωf denotes the domain containing all the fluid particles and d is the dimension of the space; d = 2 or d = 3. The solid body is immersed in a grid that does not conform to its surface. Both the vorticity and velocity fields are evaluated on the uniform rectangular grid by first interpolating the vorticity of the fluid particles onto the grid via ω(x) =
Np X
ω(xp )M40
p
x − xp h
,
(4)
where Np is the number of vorticity particles, h is the grid spacing, and the function M40 is the third order interpolation kernel [27]. The velocity can then be obtained from the Poisson equation ∇2 ψ = −ω with non-homogeneous Dirichlet boundary conditions. ψ at boundaries of a domain ∂Ωf can be approximated using a Green’s function approach, where it is given by Z ψ(xb ) = − G(xb ; xp )ω(xp ) dxp , (5) Ωf
where xb ∈ ∂Ωf and xp ∈ Ωf . The subscripts b and p denote boundary and particle quantities, respectively. The well-known Green’s function is expressed as 1 in 2D log|xb − xp | 2π G(xb ; xp ) = (6) 1 in 3D, − 4π|xb − xp | 2
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where |x − xp | denotes the distance between x and xp . Computation of the boundary condition for ψ can be greatly accelerated by using cubic spline approximation [23]. In this study, the Poisson equation for ψ was solved using an FFT solver based on the Fastest Fourier Transform in the West (FFTW) [12] open-source library. Given the calculated velocity and interpolated vorticity fields, the right-hand side of Equation (2) can be evaluated using appropriate finite difference schemes. In this study, the stretching term in Equation (2) was replaced with (ω · ∇T )u (the so-called transpose scheme), which helps to control spurious vorticity divergence. The diffusion term ∇2 ω was discretized using a 27-point isotropic Laplacian scheme, which is less sensitive to grid orientation than the standard central finite difference scheme [8]. In 2D, the diffusion term was computed with a 9-point finite difference scheme. The penalization term was evaluated using the implicit scheme λχs ∆ω =∇× (us − u) (7) ∆t 1 + λχs ∆t with a smoothed mask function [16] to reduce numerical error. To ensure that the divergence-free condition for vorticity (namely, ∇ · ω = 0) was satisfied, the vorticity field was corrected by adding ∇Fω , which was obtained by solving ∇2 Fω = −∇ · ω with Fω (= ∇ · ψ) = 0 at the boundaries [9]. Finally, both the vorticity and velocity values computed on the grid were transferred back to the fluid particles using the M40 kernel. The vorticity-carrying particles move independently in d-dimensional space with their own individual velocities during a unit time interval, ∆t. In this study, the particles were advanced in time using a midpoint predictor-corrector method.
2.3
Numerical implementation
A distributed-memory parallel system was employed to speed up the numerical computations. We used the open Message Passing Interface (Open MPI1 ) to implement parallelization across multiple processes. Brick-type domain decomposition along multiple directions is not suitable for FFT solvers because single-direction transforms are more efficient. Hence, the domain was split only along the x-direction in both 2D and 3D, and the size of each subdomain was kept constant. Before performing the FFT along the x-direction, the data was redistributed in a block-cyclic fashion over the processors as it was initially distributed across different processors. The FFTs along the y- and z- directions were performed independently on each processor because all the data was now local to the processor. Overlapping between the subdomains, whose size depended on the numerical schemes used, was necessary to minimize interprocessor communication. The distorted fluid particle locations were periodically regularized to guarantee particle overlap. In this redistribution step, particles with small vorticities were discarded to prevent unnecessary increase in particle population; in this study, for example, the cutoff threshold was 10−4 |ω|max in 2D and 10−5 |ω|max in 3D [24].
3
Boundary conditions for the penalized PPE
Equation (3) with appropriate boundary conditions can be solved using an FFT solver. Here we propose a novel numerical approach and present existing numerical approaches to specifying Dirichlet boundary conditions for solving the infinite-domain PPE within a bounded domain. The computational domain for pressure is denoted by Ω.
3.1
Direct evaluation via Green’s function
The total pressure can be defined so that H(x) → 0 as x → ∞: H = p − p∞ +
1 |u|2 − |u∞ |2 , 2
(8)
where p∞ and u∞ denote the reference pressure and velocity at infinity, respectively. This definition allows the use of Equation (6), which is often known as the free-space Green’s function, to estimate H at domain boundaries. Therefore, the Poisson equation for H can be expressed as ∇2 H = Q, 1 All
versions of the open MPI are available at https://www.open-mpi.org/.
3
(9)
where Q denotes the right-hand terms of Equation (3) (i.e., the source terms). The infinite-domain boundary condition for H can be computed using N0
Z G(xb ; x)Q(x) dx ≈
H(xb ) = Ωf
p X
G(xb ; xp )Q(xp )Vp ,
(10)
p
where Np0 is the number of nonzero Q elements and is slightly greater than Np due to the divergence operator, and Vp is the volume (or area in 2D) of a single element. This approach requires O(Nb Np0 ) operations, where Nb is the number of domain boundary nodes. Although the FMM and its variants can be used to speed up the computation, they require complex and hierarchical data structures, which is a drawback. If H is sufficiently spatially smooth at the domain boundaries, it can be approximated using an appropriate interpolation method such as spline interpolation [23]. In this case, the operation count becomes O(Mb Np0 ), where Mb is the number of equidistant points chosen as interpolation nodes and thus Mb < Nb . This still becomes cumbersome and expensive as the number of particles and boundary nodes increases.
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3.2
The James-Lackner algorithm
Following the approach described in [17] and [20], one is able to specify infinite-domain boundary conditions. The James-Lackner algorithm can be summarized as follows [3]. (a) Find the solution to the PPE with a homogeneous Dirichlet boundary condition: ∇2 H = Q in Ω and H = 0 at ∂Ω. (b) Define a charge q along boundaries of the interior domain via q ≡ n · ∇H at ∂Ωin , where n is the unit outward normal vector to ∂Ωin , and the closed rectangular domain Ωin lies completely inside Ω. (c) Approximate H at ∂Ω using the convolution of the free-space Green’s function, Z H(xb ) = G(xb ; y) q(y) dS, ∂Ωin
where xb ∈ ∂Ω and y ∈ ∂Ωin . (d) Solve the PPE with the infinite-domain boundary condition, H(xb ). Steps (b) to (d) are repeated until a satisfactory degree of convergence is achieved. The operation count for the overall process is O(kNb2 ), where k is the number of iterations.
3.3
Miller’s algorithm
Miller [25] introduced an iterative method for solving infinite-domain Poisson problems. The solution to the Poisson problem is decomposed as the sum of two fields, namely H = φ + Ψ. This algorithm results in the following iterative method. (a) Find the solution to the PPE with a homogeneous Dirichlet boundary condition: ∇2 φ = Q in Ω and φ = 0 at ∂Ω. (b) Define a charge q along the boundaries of the interior domain: q ≡ n · ∇φ at ∂Ωin . (c) Approximate φ at ∂Ω using the convolution of the free-space Green’s function, Z (0) φ (xb ) = G(xb ; y) q(y) dS, ∂Ωin
where xb ∈ ∂Ω and y ∈ ∂Ωin . (d) Solve the following Laplace problem: ∇2 Ψ(k) = 0 in Ω and Ψ = φ(k−1) at ∂Ω, where k is the iteration index. (e) Compute the charge on the interior boundary: q (k) = n · ∇Ψ(k) at ∂Ωin .
4
(f) Update φ at ∂Ω: φ(k) (xb ) = aφ(0) (xb ) + a
Z
G(xb ; y) q (k) (y) dS + (1 − a)φ(k−1) (xb ),
(11)
∂Ωin
where the parameter a is in 0 < a < 1. Steps (d) to (f) are repeated until a satisfactory degree of convergence is achieved. When φ converges at the boundaries, iteration ceases and H = φ + Ψ(k) . The interior boundary in Miller’s original algorithm denotes solid walls; however, in the present study, it is defined as the domain boundary of a small rectangular domain, ∂Ωin . As a result, this approach is similar to the JamesLackner algorithm. We have therefore focused on comparing its convergence to that of the JamesLackner algorithm.
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3.4
Non-iterative approach via contour integration
R For a conservative vector field F = ∇f , its line integral C F · dr depends only on the endpoints of the path and is independent of the path itself; thus, conservative vector fields are often referred to R as path-independent vector fields. If a vector field is path independent, the work integral C F · dr takes the form required by the fundamental line integral theorem Z ∇f · dr = f (x1 ) − f (x0 ), (12) C
where x0 and x1 are the starting point and endpoint, respectively. Note that the total pressure H is a scalar function and the gradient field ∇H can be given by ∇H = −
1 2 ∂u + (u × ω) + ∇ u. ∂t Re
Thus, one can easily obtain the following equation: Z Z ∂u 1 2 m−1 − + (u × ω) + ∇ u · dr = ∇H · dr = H(xm ), b ) − H(xb ∂t Re ∂Ω ∂Ω
(13)
(14)
where xm b (m = 1, · · · , Mb ) are nodes on the domain boundary ∂Ω. As a result, Equation (14) can be expressed as 1 2 ∂u(xm m−1 m m m m b ) H(xb ) = H(xb ) + ∆xb − + u(xb ) × ω(xb ) + ∇ u(xb ) · t , (15) ∂t Re where ∆xb is equal to the grid spacing h and t denotes the unit tangent vector along ∂Ω. To summarize, the two steps required to compute infinite-domain boundary conditions are as follows. (a) Compute H at the starting point x0b using the free-space Green’s function (Equation (6)): N0
H(x0b )
=
p X
G(x0b ; xp )Q(xp )Vp ,
(16)
p
where x0b ∈ ∂Ω and xp ∈ Ωf . (b) Approximate H(xb ) along consecutive boundary nodes using Equation (15). Figure 1 graphically shows the numerical strategies for approximating infinite-domain boundary conditions using Equations (15) and (16). Because the proposed approach includes the direct evaluation of infinite-domain boundary conditions using Green’s function, it requires O(Np0 ) operations for a single closed curve in 2D and O(Np0 Nc ) for multiple closed curves in 3D, where Nc is the number of closed curves and is proportional to the number of grid nodes in the x-direction. The number of operations required, regardless of dimension, is always less than the O(Nb2 ) per iteration required by the iterative approaches. Furthermore, the proposed approach does not require iteration. 5
4
Results and discussion
Here we compare the numerical methods for determining Dirichlet boundary conditions for solving the PPE, based on the simulations of impulsively started flows around a circular cylinder (2D) and a sphere (3D). In this study, the Reynolds number is defined as Re = u∞ D/ν, where D is the diameter of the cylinder or sphere, and the time is nondimensionalized by defining it based on the radius R as T = u∞ t/R. In both 2D and 3D, the penalization parameter λ was fixed at 108 and λp in the PPE was λp = 1/∆t.
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4.1
2D case
The numerical parameters used for the flow simulation were Re = 550, h/D = 0.002, and ∆t = 0.005. The time step was determined by the stability condition [29], ν∆t/h2 < 0.25. Figure 2 shows instantaneous vorticity iso-contours and streamlines for the flow over a circular cylinder at T = 3. A pair of symmetric vortices appears. The so-called bulge phenomenon [5, 6], that typically precedes the formation of substantial isolated vortices, is well captured by the numerical simulation. The pVIC method gave reliable results, both in terms of wake shape and induced forces, compared with previous experimental/computational studies. The details of the numerical solution can be found in [23]. At T = 3, the computational domain for which the vorticity and velocity fields were obtained had 512 × 512 grid nodes (the smallest box in Figure 3, which corresponds to Ω = [−1.0D, 1.56D] × [−1.28D, 1.28D]). Using different computational domains, as shown in Figure 3, we investigated the capability of the numerical approaches presented in Section 3 and the effect of domain size on the pressure field. To obtain vorticity and velocity fields for the different computational domains, the particle vorticity field was first reconstructed on the grid nodes of each domain and then the corresponding velocity field was recomputed. Figure 4 shows the distribution of the pressure coefficient Cp , which was defined and computed as 2(p − p∞ ) = 2H − |u|2 − 1 . (17) Cp = 2 |u∞ | Direct evaluation using Green’s function provided pressure fields that were independent of domain size, whereas the iterative methods (the James-Lackner approach and Miller’s approach) required relatively large domains to obtain domain-independent pressure fields. For both iterative methods, convergence was monitored by measuring the L2 norm of the difference between the previous and current Hs at the domain boundaries; in this study, we used L2 < 0.001 as the criterion for stopping iteration. Decreasing this threshold did not lead to better results. Compared with the James-Lackner approach, Miller’s approach exhibited a bad convergence property: its convergence was sensitive to the numerical parameter a in Equation (11) (we used a = 0.3 in this work). As can be seen from Figure 4(d), our proposed approach is less affected by computational domain size than the iterative methods. Figure 5 shows that the proposed approach worked well even for the smallest computational domain, whereas neither iterative methods provided acceptable pressure fields. The errors in the approximated domain boundary conditions and the computation times are presented in Table 1. The L2 error and L∞ error are defined as s 2 PMb Hidirect − Hi i L2 = , Mb L∞ = max |Hidirect − Hi | , where Hidirect denotes the value computed using Green’s function. The CPU times are the elapsed times taken to compute only the domain boundary condition and were measured using four CPUs (Intel Xeon64 3.3 GHz). The results presented in Table 1 indicate that our proposed method can correctly and quickly compute infinite-domain boundary conditions for the PPE. In the case of the domain with 512 × 512 nodes, our proposed approach is over 40 times faster than direct evaluation. The speedup significantly increases as the domain size increases. As can be seen in Figure 6(a), there is a small difference between the pressure fields produced by our proposed approach and direct evaluation using Green’s function. We believe that the accuracy can be improved by using higher-order numerical schemes to compute the second and 6
third terms of Equation (15). In this study, the time derivative on the right-hand side of Equation (15) was discretized using a first-order Euler scheme and the Laplacian term was discretized using a second-order central difference with the velocity given as first-order accuracy. In Figure 6(b), the Cp curves along the body surface look almost identical regardless of the numerical approach used because the pressure field near a solid body is dominated by the source term (the right-hand side of Equation (3)) rather than the far field boundary condition. Figure 7(b) shows the time evolution of Cp at the body surface. As experimentally measured by Norberg [28], the Cp = 0 point in the present simulation was located at approximately θ = 35◦ .
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4.2
3D case
To validate and assess our proposed approach in 3D, we considered viscous flow past a sphere at Re = 100. This flow is often chosen as a benchmark by virtue of its steady and axisymmetric wake structure. In [23], we demonstrated that the numerical solutions given by the pVIC method for this case compare well with the available data reported in the literature. The velocity and vorticity fields used as input to the pressure calculation in the present study were the same as those presented in [23]. In the grid refinement study of [23], the numerical solutions converged at h/D = 0.02 and the time increment was determined from ν∆t/h2 = 0.125 by considering diffusive stability. These parameters were retained here, and the flow simulation was conducted until T = 30 to obtain a steady-state solution. Figure 8 shows the vorticity contours and streamlines obtained in the xz-plane (y = 0) for the steady state. The pressure field was evaluated by explicitly solving the PPE for the given velocity and vorticity fields. Similar to the 2D case, the four numerical approaches were compared in terms of numerical error and computational cost for the following five different size regimes. • Domain I
: 2048 × 512 × 512, [−17.36D, 23.60D] × [−5.12D, 5.12D] × [−5.12D, 5.12D]
• Domain II
: 1024 × 256 × 256, [ −7.12D, 13.36D] × [−2.56D, 2.56D] × [−2.56D, 2.56D]
• Domain III : 512 × 128 × 128, [ −2.00D, 8.24D] × [−1.28D, 1.28D] × [−1.28D, 1.28D] • Domain IV : 256 × 128 × 128, [ −2.00D, 3.12D] × [−1.28D, 1.28D] × [−1.28D, 1.28D] • Domain V
: 128 × 128 × 128, [ −1.28D, 1.28D] × [−1.28D, 1.28D] × [−1.28D, 1.28D]
Here the grid spacing was kept fixed at h/D = 0.02. Figure 9 shows the configurations of the different domains. Domains I and II cover all the fluid particles, whereas the smaller domains do not. Using the velocity and vorticity fields computed for domain I, we evaluated the pressure fields using four CPUs (Intel Xeon64 3.4GHz). The convergence criterion was the same as that for the 2D cases, and the parameter a in Miller’s approach was set to 0.5 which was the optimal value for 3D. Table 2 shows the errors and computational costs of the four different approaches. For the two iterative methods, the number of iterations required decreased as the domain size increased. Miller’s approach converged faster than the James-Lackner approach, although the L2 and L∞ errors were of a similar magnitude. For the largest domain (I), however, the iterative methods took 20% to 40% longer than direct evaluation because Nb was not small enough compared with Np0 (Nb and Np0 were approximately 4.7 and 6.0 million, respectively). This means that the computational speedup depends on domain size as well as the number of iterations. The errors for the iterative methods increased significantly as the domain size decreased because the smaller domains did not cover all the fluid particles. Our proposed method outperformed the two iteration methods in terms of the computation time, and the speedups were even more remarkable in 3D than 2D. As shown in Table 2, our approach ran approximately 360 (domain V) to 1950 (domain I) times faster than direct evaluation. The error norms of our approach were less than those of the iterative methods in general. Particularly for the smaller domains, our approach has much better accuracy than the iterative methods. Figure 10 compares the Cp contours in the xz-plane (y/D = 0) for domains I, II, III, and IV. The contours of our approach are hardly affected by domain size, whereas those of the iterative methods show some discrepancies in the wake region near the domain boundary. These differences can be most clearly seen for the most compact domain, V. Figure 11 compares the Cp distributions in the xz-plane for domain V. By contrast with the undesirable results given by the iterative methods, the result of our approach agrees well with that of the Green’s approach even 7
for the smallest domain. Figure 12 presents all the Cp contours given by our approach for the smaller domains together, showing that they are almost identical regardless of domain size and in fair agreement with the numerical result of Johnson and Patel [18]. It is worth noting that our approach can be applied to desired computation domains regardless of whether the domain contains all the particles. Figure 13 shows the Cp distribution along the body surface interpolated from the pressure solution for the smallest domain V, again showing reasonable agreement with the literature data.
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5
Concluding remarks
We have presented a numerical approach based on contour integral representations toward specifying infinite-domain boundary conditions for solving the PPE in a bounded domain. We have also validated its numerical implementation through the simulations of impulsively started flows around a circular cylinder (2D) and a sphere (3D). Our non-iterative approach to specifying infinitedomain boundary conditions was compared with three other algorithms, namely, direct evaluation using Green’s function and two iterative methods (the James-Lackner algorithm and Miller’s algorithm). In terms of efficiency, our approach outperformed all the other methods tested. For the most compact domain, the results show that our approach was 40 times (2D) and 360 times (3D) faster than direct evaluation, and the speedup significantly increased as the domain became larger. Furthermore, our approach enables reasonable pressure field to be provided for much smaller computational domains than the iterative methods can offer. The presented approach can also be adopted to other situations such as spatial densities and magnetic fields.
8
References [1] Angot, P., Brunear, C.-H., Fabrie, P., 1999. A penalization method to take into account obstacles in incompressible viscous flows, Numerische Mathematik, 81(4), 497–520. [2] Bagchi, P. and Balachandar, S., 2002. Steady planar straining flow past a rigid sphere at moderate Reynolds number, Journal of Fluid Mechanics, 466(1), 365–407. [3] Balls, G.T. and Colella, P., 2002. A finite difference domain decomposition method using local corrections for the solution of Poisson’s equation, Journal of Computational Physics, 180(1), 25–53. [4] Beatson, R. and Greengard, L., 1997. A short course on fast multipole methods, In: Wavelets, Multilevel Methods and Elliptic PDEs, Oxford University Press, 1–37. [5] Bouard, R., and Coutanceau, M., 1980. The early stage of development of the wake behind an impulsively started circular cylinder for 40 < Re < 104 , Journal of Fluid Mechanics, 101(3), 586–607.
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[6] Chang, C.-C. and Chern, R.-L., 1991. a nerumical study of flow around an impulsively started circular cylinder by a determinestic vortex method, Journal of Fluid Mechanics, 233, 243–263. [7] Christiansen, J.P., 1973. Numerical Simulation of Hydrodynamics by the Method of Point Vortices, Journal of Computational Physics, 13(3), 363–379. [8] Cocle, R., Winckelmans, G. and Daeninck, G., 2008. Combining the vortex-in-cell and parallel fast multipole methods for efficient domain decomposition simulations, Journal of Computational Physics, 227, 9091-9120. [9] Cottet, G.-H. and Koumoutsakos, P., 2000. Vortex Methods: Theory and Practice. Cambridge University Press, Cambridge, UK. [10] El Ossmani, M. and Poncet, P., 2010. Efficiency of multiscale hybrid grid-particle vortex methods, Multiscale Modeling and Simulation, 8(5), 1671–1690. [11] Fornberg, B., 1988. Steady viscous flow past a sphere at high Reynolds numbers, Journal of Fluid Mechanics, 190, 471–489. [12] Frigo, M. and Johnson, S.G., 2005. The design and implementation of FFTW3, Proceedings of the IEEE, 93(2), 216–231. [13] Gazzola, M., Chatelain, P., van Rees, W.M., and Koumoutsakos, P., 2011. Simulations of single and multiple swimmers with non-divergence free deforming geometries, Journal of Computational Physics, 230(19), 7093–7114. [14] Greengard, L. and Rokhlin, V., 1987. A fast algorithm for particle simulations, Journal of Computational Physics, 73(2), 325-348. [15] Hockney, R.W., 1965. A fast direct solution of Poisson equation using Fourier analysis, Journal of the Association for Computing Machinery, 8, 95–113. [16] Iwakami, W., Yatagai, Y., Hatakeyama, N., and Hattori, Y., 2014. New approach for error reduction in the volume penalization method, Communications in Computational Physics, 16(5), 1181-1200. [17] James, R.A., 1977. The solution of Poisson’s equation for isolated source distribution, Journal of Computational Physics, 25(2), 71–93. [18] Johnson, T.A. and Patel, V.C., 1999. Flow past a sphere up to a Reynolds number of 300, Journal of Fluid Mechanics, 378, 19–70. [19] Kim, K.-S., Lee, S.-J., and Suh, J.-C., 2005. A combined vortex and panel method for numerical simulations of viscous flows: a comparative study of a vortex particle method and a finite volume method, International Journal for Numerical Methods in Fluids, 49(10), 1087-1110.
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[20] Lackner, K., 1976. Computation of ideal MHD equilibria, Computer Physics Communications, 12(1), 33–44. [21] Lee, S.-J., Lee, J.-H., and Suh, J.-C., 2014. Computation of pressure fields around a twodimensional circular cylinder using the vortex-in-cell and penalization methods, Modelling and Simulation in Engineering, Article ID 708372. [22] Lee, S.-J., Lee, J.-H., and Suh, J.-C., 2015. Further validation of the hybrid particle-mesh method for vortex shedding flow simulations, International Journal of Naval Architecture & Ocean engineering, 7, 1034–1043. [23] Lee, S.-J. and Suh, J.-C., 2017. Fast computation of domain boundary conditions using splines in penalized VIC Method, International Journal of Computational Methods, 14(2), 1750076 (16 pages).
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[24] Lee, S.-J., 2017. Numerical simulation of vortex-dominated flows using the penalized VIC Method, Vortex Dynamics and Optical Vortices (Ed. Hector Perez-de-Tejada), InTech, Croatia. [25] Miller, G.H., 2008. An iterative boundary potential method for the infinite domain Poisson problem with interior Dirichlet boundaries, Journal of Computational Physics, 227, 7917– 7928. [26] Mimeau, C., Gallizio, F., Cottet, G.-H., and Mortazavi, I., 2015. Vortex penalization method for bluff body flows, International Journal for Numerical Methods in Fluids, 79(2), 55–83. [27] Monaghan, J.J., 1985. Extrapolating B splines for interpolation, Journal of Computational Physics, 60(2), 253–262. [28] Norberg, C., 2002. Pressure distributions around a circular cylinder in cross-flow, Proceedings of the Symposium on Bluff Body Wakes and Vortex-Induced Vibrations (BBVIC-3 ’02), Queensland, Austrailia, 1–4. [29] Rasmussen, J.T., Cottet, G.H., and Walther, J.H., 2011. A multiresolution remeshed VortexIn-Cell algorithm using patches, Journal of Computational Physics, 230(17), 6742–6755. [30] Sarpkaya, T., 1989. Computational methods with vortices-the 1988 freeman scholar lecture, Journal of Fluids Engineering. 111(1), 5–52. [31] Speziale, C.G., 1987. On the advantages of the vorticity-velocity formulation of the equations of fluid dynamics, Journal of Computational Physics, 73(2), 476–480. [32] Uchiyama, T., Yoshii, Y., and Hamada, H., 2013. Direct numerical simulation of a turbulent channel flow by an improved vortex in cell method, International Journal of Numerical Methods for Heat & Fluid Flow, 24(1), 103–123. [33] Wang, C., Sun, J., and Ba, Y., 2017. A semi-Lagrangian Vortex-In-Cell method and its application to high-Re lid driven cavity flow, International Journal of Numerical Methods for Heat & Fluid Flow, 27(6), doi:10.1108/HFF-08-2015-0320103–123.
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Table 1: Errors in Dirichlet boundary conditions and computational time (2D cases) Numerical approach
grid nodes 512 × 512 1024 × 1024 2048 × 2048 4096 × 4096
L2 error -
L∞ error -
James-Lackner’s approach
512 × 512 1024 × 1024 2048 × 2048 4096 × 4096
5.95 × 10−2 4.05 × 10−3 2.26 × 10−3 1.67 × 10−3
2.12 × 10−1 7.30 × 10−3 3.47 × 10−3 2.51 × 10−3
14.0 39.2 131.8 476.9
s s s s
18 13 11 10
Miller’s approach
512 × 512 1024 × 1024 2048 × 2048 4096 × 4096
4.12 × 10−2 1.50 × 10−2 9.77 × 10−3 5.49 × 10−3
1.23 × 10−1 2.09 × 10−2 1.41 × 10−2 8.47 × 10−3
11.5 27.3 111.6 432.6
s s s s
15 9 10 9
Presented method
512 × 512 1024 × 1024 2048 × 2048 4096 × 4096
4.07 × 10−3 2.22 × 10−3 1.12 × 10−3 5.61 × 10−4
5.23 × 10−3 4.00 × 10−3 1.74 × 10−3 9.52 × 10−4
0.3 1.3 5.0 20.5
s s s s
-
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Direct evaluation
CPU time 13.2 s 106.0 s 847.9 s 6784.3 s
Iterations -
Table 2: Errors in Dirichlet boundary conditions and computational time (3D cases); the domains are denoted by I(2048×5122 ), II (1024×2562 ), III (512×1282 ), IV (256×1282 ), and V (128×1282 ). Numerical approach Direct evaluation
James-Lackner’s approach
Miller’s approach
Presented method
Domain I II III IV V
L2 error -
L∞ error -
CPU time 46938.2 s 8468.5 s 2159.1 s 1199.8 s 719.6 s
Iterations -
I II III IV V
2.90 × 10−5 9.39 × 10−5 6.47 × 10−3 2.30 × 10−2 4.67 × 10−2
9.23 × 10−5 1.03 × 10−3 9.68 × 10−2 2.43 × 10−1 3.17 × 10−1
66892.9 5592.2 514.1 184.3 70.9
s s s s s
7 10 15 17 17
I II III IV V
2.89 × 10−5 1.83 × 10−4 9.57 × 10−3 3.41 × 10−2 7.06 × 10−2
9.25 × 10−5 2.01 × 10−3 1.42 × 10−1 3.57 × 10−1 4.80 × 10−1
57436.0 3344.6 240.0 75.3 33.1
s s s s s
6 6 7 7 8
I II III IV V
1.51 × 10−4 1.62 × 10−4 4.93 × 10−4 6.57 × 10−4 8.64 × 10−4
4.97 × 10−4 6.12 × 10−4 2.26 × 10−3 2.26 × 10−3 2.26 × 10−3
24.0 10.7 4.9 2.9 2.0
s s s s s
-
11
#4 #3
Global domain boundary
#2 Subdomain #1
Subdomain #1
#2
#3
#4
integration H(࢞ )
H(࢞ )
integration
(a) in 2D
(b) in 3D
1
1 12 9 7 5 3 0.1 -0.1 -3 -5 -7 -9 -12
0
-0.5
-1 -1
-0.5
0
0.5
1
1.5
0.5
y/D
0.5
y/D
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Figure 1: Numerical strategies to approximate Dirichlet boundary conditions. Note that the total pressure at the starting-point, H(x0b ), is directly computed using the Green’s function and then the others is sequentially integrated along boundary nodes of the global computational domain.
0
-0.5
-1 -1
2
x/D
-0.5
0
0.5
1
1.5
2
x/D
(a) vorticity contour
(b) streamlines
Figure 2: Instantaneous vorticity contour (right) and streamlines (left) for a flow past a circular cylinder at T = 3 (Re = 550); the figures were reprinted with permission from [24].
12
4096 × 4096
10
2048 × 2048
5
1024 × 1024
y/D
512 × 512
0
fluid particles
-5
-10
-5
0
5
10
x/D
Figure 3: Four computational domains used for the 2D comparative study
1
y/D
y/D
1
0
-1
0
-1 -1
0
x/D
1
2
-1
(a) direct evaluation
0
x/D
1
2
(b) James-Lackner’s approach
1
y/D
1
y/D
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-10
0
-1
0
-1 -1
0
x/D
1
2
-1
(c) Miller’s approach
0
x/D
1
2
(d) presented method
Figure 4: Instantaneous Cp contours around a circular cylinder at T = 3 (Re = 550); the red, green, and blue lines indicate the computational domain with 1024 × 1024, 2048 × 2048, and 4096 × 4096 grid nodes, respectively.
13
1 0.8 0.6 0.4 0.2 0.05 -0.05 -0.3 -0.6 -1 -1.5 -2
0
-1 -1
0
x/D
1
1 0.8 0.6 0.4 0.2 0.05 -0.05 -0.3 -0.6 -1 -1.5 -2
1
y/D
y/D
1
0
-1
2
-1
(a) direct evaluation
1 0.8 0.6 0.4 0.2 0.05 -0.05 -0.3 -0.6 -1 -1.5 -2
x/D
1
2
1 0.8 0.6 0.4 0.2 0.05 -0.05 -0.3 -0.6 -1 -1.5 -2
1
y/D
y/D
-1
0
-1
2
-1
0
(c) Miller’s approach
1
x/D
2
(d) presented method
Figure 5: Distributions of Cp computed on the computational domain with 512 × 512 grid nodes (Re = 550, T = 3).
1
1
direct evaluation present
CP
0
y/D
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0
0
1
x/D
(b) James-Lackner’s approach
1
-1
0
0
-1
-2 -1 -3 -1
0
x/D
1
2
(a)
0 Front
30
60
90
θ [degree]
120
150
180 Rear
(b)
Figure 6: Comparison of Cp computed by the direct evaluation (red) and the proposed approach (blue). Note that θ is a cylindrical coordinate defined such that the front stagnation point corresponds to θ = 0 and the computational domain has 512 × 512 grid nodes.
14
1
0
CP
0
CP
1
direct evaluation James-Lackner’s approach Miller’s approach present
-1
-2
-3
-1 T=1 T=3 T=5 T=7 T=10
-2
0 Front
30
60
90
θ [degree]
120
150
180 Rear
-3
0 Front
30
60
(a)
90
θ [degree]
120
150
180 Rear
(b)
0
-2 0
2
4
x/D
6
8
10
2 1.2 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1.2 -2
2
0
z/D
2
z/D
-2 0
2
4
(a) vorticity contour
x/D
6
8
10
(b) streamlines
Figure 8: Vorticity contour (right) and streamlines (left) for a flow past a sphere at steady state; the figures were reproduced with permission from [24].
-5 0
y/D 5 5
0
z/D
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Figure 7: (a) Comparison of numerical approaches for Cp at a solid wall and (b) time evolution of Cp computed using the proposed method. Note that θ is a cylindrical coordinate defined such that the front stagnation point corresponds to θ = 0 and the computational domain has 512 × 512 grid nodes.
-5
-10
10
0
20
x/D
Figure 9: Configuration of the computational domains and the distribution of given source field for the 3D cases; I (dashed black ), II (solid black ), III (solid red ), IV (solid green), and V (solid blue)
15
z/D
z/D
1
0
0
-1
-1
-2
-1
0
1
x/D
2
-2
(a) direct evaluation
1
x/D
2
1
z/D
z/D
0
(b) James-Lackner’s approach
1
0
0
-1
-1
-2
-1
0
1
x/D
2
-2
(c) Miller’s approach
-1
0
1
x/D
2
(d) presented method
Figure 10: Comparison of Cp contours (y/D = 0) computed on the different sized domains; the black, green, blue and red lines indicate the results of I, II, III and IV, respectively.
1 0.8 0.6 0.4 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6
0
-1
-2
-1
0
x/D
1
0
-1
2
-2
(a) direct evaluation
1 0.8 0.6 0.4 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6
-1
0
x/D
1
0
x/D
1
2
1 0.8 0.6 0.4 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6
1
z/D
0
-1
-1
(b) James-Lackner’s approach
1
-2
1 0.8 0.6 0.4 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6
1
z/D
z/D
1
z/D
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-1
0
-1
2
-2
(c) Miller’s approach
-1
0
x/D
1
2
(d) presented method
Figure 11: Cp distributions on y/D = 0 plane, computed with the domain V.
16
z/D
1
0
-1
-1
0
1
x/D
2
(a) present method
(b) Johnson and Patel (1999)
Figure 12: Cp distributions on y/D = 0 plane; (a) the results of the contour integration approach with III (green), IV (blue) and V (red) where the box represents the boundary of the domain V, (b) the numerical result of [18].
Bagchi and Balachandar (2002) Fornberg (1988) present
1 0.5
Cp
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-2
Domain V
0 -0.5
u∝ -1 0
θ
30
60
90
θ [degree]
120
150
180
Figure 13: Cp at the solid surface. Note that the result of present method was interpolated from the solution in domain (V) and the literature data were reproduced from [11] and [2].
17
