# A Particle Method Based on a Generalized Finite Difference Scheme to Solve Weakly Compressible Viscous Flow Problems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Finite Difference Particle Method for Weakly Compressible Flow

#### 2.1. Lagrangian Form of the Governing Equations for Weakly Compressible Viscous Flow

**u**is the particle velocity, p is the pressure, v

_{k}is the kinematic viscosity, and

**F**is an external body force, such as gravity. All these variables are related to the physical properties of fluid particles that can move in both space and time, rather than remain at a fixed position.

#### 2.2. Generalized Finite Difference Scheme

_{s}represents the radius of the support domain, which is called the smoothing length in the SPH method. Particles j are white circles around particle i, which is the orange circle, and Ω represents the computational domain. The closest nodes to particle i are selected as j particles, and these particles should be in the support domain at the same time.

_{i}and F

_{j}, respectively. This F can be the pressure, velocity or density of particles in the computation. Let us expand this term as the Taylor series of F for particle i:

_{j}= x

_{j}- x

_{i}, k

_{j}= y

_{j}− y

_{i}.

_{j}, k

_{j}, r

_{s}) is a kernel function in 2D and r

_{s}represents the size of the support domain. For W, different kernel functions, including Gaussian, cubic spline, and quintic spline functions, can be found in [13]. In the following equations, we use W for the kernel function.

**AD**=

**B**,

**D**. Thus, the spatial derivatives in Equations (1) and (2) can be obtained by solving a set of symmetric linear equations, and the material derivatives in the equations can be integrated using a time integration scheme.

#### 2.3. Particle Representation for Governing Equations

**D**and

**B**are denoted by D

_{m}(f

_{i}) and B

_{m}(f

_{i}), respectively, with m = 1, 2, …, 5. For example, ${D}_{2}\left({f}_{i}\right)=\frac{\partial {f}_{i}}{\partial y}$ (the second coefficient in Equation(13)), and ${B}_{2}\left({f}_{i}\right)={\sum}_{j=1}^{N}{(f}_{j}{-f}_{i}{)k}_{j}{W}^{2}$ (the second coefficient in Equation (14)). In addition, the symmetric matrix

**A**can be decomposed into the upper and lower triangular matrices

**A**=

**LL**. The coefficients of the matrix

^{T}**L**are denoted by L(m, n), with m and n = 1, 2, 3, 4, 5.

_{i}and v

_{i}are the velocity of particle i in two directions and

#### 2.4. Artificial Particle Displacement

**r**

_{i}is the position of particle i, α is a problem-dependent parameter that is usually set between 0.01 and 0.1, v

_{max}is the maximum velocity of all particles in the computational domain,

**r**

_{ij}=

**r**

_{i}−

**r**

_{j}which is the distance between particles i and j, and $\overline{{r}_{i}}$ is the average distance between the neighboring particles of particle i:

#### 2.5. Boundary Conditions

## 3. Applications of the Finite Difference Particle Method

#### 3.1. Fundamental Definition

_{RMS}) and the maximum errors (ε

_{MAX}), which are defined as

_{num}(k) and S

_{ana}(k) are the numerical and analytical results of variable k, respectively. k could be the velocity or pressure.

_{ERMS}) and the maximum error convergence rate (R

_{EMAX}) as follows:

#### 3.2. Unsteady Flow in a Pipeline

_{0}is the flow velocity distribution and x is the coordinate along the length of the pipeline. The coefficients (the unit can be obtained from dimensional analysis) C

_{1}= 30.0, C

_{2}= –1.0 × 10

^{6}, C

_{3}= 82571.0, and γ = 1.4; moreover, the initial time is 12.5 s, and the pipe length x is 700 m.

_{s}is 3.2 times Δx. The cubic spline kernel function is used in the calculations. Table 1 provides data for comparing the numerical velocity and theoretical solution of a particular particle at different times and positions. Notably, although the particle moves from x = 2.48 m to x = 25.64 m, the FDPM results agree well with the theoretical values, and this result verifies the algorithm.

_{s}is 3.2 times Δx in the computation.

_{ERMS}value of 1.7 and R

_{EMAX}value of 1.8.

_{s}conditions, as shown in Figure 5. The figure shows that the maximum error of the Gaussian kernel function is larger than that of the other methods. The errors of other types of functions are similar.

#### 3.3. Poisseuille Flow

_{k}= 10

^{−6}m

^{2}s

^{−1}, L = 10

^{−3}m, ρ = 10

^{3}kgm

^{−3}, and F = 10

^{−4}ms

^{−2}, so the maximum velocity is 1.25 × 10

^{−5}ms

^{−1}and the Reynold number is Re = 1.25 × 10

^{−2}. The plate boundaries at the upper and lower ends are established using rigid walls. One layer of boundary particles and three layers of virtual particles are used. The FDPM with second-order Taylor truncation is utilized to perform the computation. The speed of sound c is taken as 0.001 m/s, as suggested in [49], and the time step Δt is 3.0 × 10

^{−4}s. The initial particle spacings Δx and Δy are both set as 5 × 10

^{−5}m, r

_{s}is 3.2 times Δx, and the kernel function is selected as a cubic spline function.

_{s}remains 3.2 times Δx.

_{RMS}and ε

_{MAX}decrease as Δx decreases, which indicates that the numerical accuracy converges with the initial particle spacing at different times. ε

_{RMS}is on the order of 10

^{−3}, indicating that the computational results agree well with the theoretical solutions. For different error evaluation indexes, the R

_{ERMS}and R

_{EMAX}values of the FDPM method are approximately 1.7 and 1.8, respectively, with good convergence at t = 1.0 s. Since the second-order Taylor expansion-based FDPM is implemented in the test, the convergence rate is reasonable.

^{−3}, cubic spline, and quantic spline.

#### 3.4. Couette Flow

_{0}, the theoretical solution of the flow velocity over time in the direction perpendicular to the plate [48] is as follows:

_{k}= 10

^{6}m

^{2}s

^{−1}, L = 10

^{−3}m, ρ = 10

^{3}kgm

^{−3}, and u

_{0}= 1.25 × 10

^{−5}m/s.

_{0}. One layer of boundary particles and three layers of virtual particles are used. The FDPM with second-order Taylor truncation is utilized to perform the computation. The speed of sound c is taken as 0.001 m/s, and the time step Δt is 5.0 × 10

^{−5}s. The initial particle spacings Δx and Δy are both set as 2.5 × 10

^{−5}m, r

_{s}is 3.2 times Δx, and the kernel function selected is the cubic spline function. Figure 9 shows a comparison between the FDPM method and the theoretical solution for the flow velocity at different times.

_{RMS}and ε

_{MAX}) at different times and at different Δx values is used to evaluate the convergence of the FDPM, as shown in Figure 10.

_{RMS}is on the order of 10

^{−2}, indicating that the computational results agree well with the theoretical solution. When t = 1.0 s, the two errors result in an R

_{ERMS}value of 1.7 and R

_{EMAX}value of 1.8. Since the second-order Taylor expansion-based FDPM is implemented in the test, the convergence rate is reasonable.

_{s}/Δx values are applied, as shown in Figure 11. Different types of kernel functions can be used in the FDPM method, and the calculation error of the Gaussian kernel function is larger than that of the other methods.

#### 3.5. Flow in Porous Media

_{k}= 10

^{−6}m

^{2}s

^{−1}, the cylindrical radius R = 2 × 10

^{−2}m, the volume force F

_{0}= 1.5 × 10

^{−7}ms

^{−2}, and the speed of sound c = 5.77 × 10

^{−4}ms

^{−1}. Δx and Δy are 0.003 m, r

_{s}is 3.2 times Δx, Δt = 1.04 s with 2000 steps, and the coefficient of artificial particle displacement is 0.05. A rigid wall boundary is used for the cylindrical boundary, and a periodic boundary is used on the four sides of the computational domain. One layer of boundary particles and three layers of virtual particles are used. The FDPM with second-order Taylor truncation is utilized to perform the computations. The particle distribution and velocity contours at the initial time and the final steady state are shown in Figure 13.

#### 3.6. Lid-Driven Cavity Flow

_{0}. The size of the cavity L = 1.0 m, the kinematic viscosity v

_{k}= 0.01 m

^{2}s

^{−1}, the sliding velocity u

_{0}= 1.0 m/s, and the speed of sound c = 10.0 ms

^{−1}. Δx and Δy are 0.025 m, r

_{s}is 2.7 times Δx, Δt = 0.001 s with 3000 steps, the coefficient of artificial particle displacement is 0.05, and the kernel function selected is the cubic spline function. One layer of boundary particles and three layers of virtual particles are used. The FDPM with second-order Taylor truncation is utilized to perform the computations. The particle distribution and velocity contours at the initial time and the final steady state are shown in Figure 17.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Computational domain Ω, support domain of particle i (point circle line), radius of the support domain

**r**

_{s}, and fluid particles (circles).

**Figure 2.**A sketch of a simulation of an acoustic boundary using virtual particles. Fluid particles are inside the computational domain and boundary particles are fixed on the boundary.

**Figure 3.**A sketch of a simulation of an acoustic boundary with a curved surface using virtual particles. Fluid particles are inside the computational domain and boundary particles are fixed on the boundary.

**Figure 4.**Convergence curves of the FDPM method for 1D unsteady flow simulation: (

**a**) Root mean square error, see Equation (26), of FDPM simulation using different particle spacing Δx at different time; (

**b**) maximum error, see Equation (27), of FDPM simulation using different particle spacing Δx at different time.

**Figure 5.**Velocity errors of the computations with different kernel functions in pipe flow modeling: (

**a**) Root mean square error, see Equation (26), of FDPM simulation using different smoothing length and kernel functions; (

**b**) maximum error, see Equation (27), of FDPM simulation using different smoothing length and kernel functions.

**Figure 6.**Velocity profiles of the FDPM results (stars) and theoretical solutions (lines) at different times along the y direction (from the bottom plate to the top plate).

**Figure 7.**Convergence curves of the FDPM computation in Poisseuille flow modeling: (

**a**) Root mean square error, see Equation (26), of FDPM simulation using different particle spacing Δx at different time; (

**b**) maximum error, see Equation (27), of FDPM simulation using different particle spacing Δx at different time.

**Figure 8.**Velocity errors of the computations with different kernel functions in Poisseuille flow modeling: (

**a**) Root mean square error, see Equation (26), of FDPM simulation using different smoothing length and kernel functions; (

**b**) maximum error, see Equation (27), of FDPM simulation using different smoothing length and kernel functions.

**Figure 9.**Comparison of the FDPM result (stars) and the theoretical solution (lines) for the flow velocity at different times along the y direction (from the stationary plate to the sliding plate).

**Figure 10.**Convergence curves of the FDPM computations in Couette flow modeling: (

**a**) Root mean square error, see Equation (26), of FDPM simulation using different particle spacing Δx at different time; (

**b**) maximum error, see Equation (27), of FDPM simulation using different particle spacing Δx at different time.

**Figure 11.**Velocity errors of the computations with different kernel functions in Couette flow modeling: (

**a**) Root mean square error, see Equation (26), of FDPM simulation using different smoothing length and kernel functions; (

**b**) maximum error, see Equation (27), of FDPM simulation using different smoothing length and kernel functions.

**Figure 12.**Simplified model of porous media. The solid circle is a circular cylinder and four sides of the domain are periodic boundaries. L is the size of the computational domain, R is the cylindrical radius, and F

_{0}is the volume force.

**Figure 13.**Particle distribution (solid circles) and velocity contours from the initial time to the steady state: (

**a**) t = 0, (

**b**) t = 693 s, (

**c**) t =1386 s and (

**d**) t = 2080 s.

**Figure 14.**Velocity distributions along observation lines 1 and 2 (dotted-dashed lines in Figure 12): (

**a**) Observation line 1 and (

**b**) observation line 2. Lines are FEM results and solid points are FDPM results with different particle spacing Δx.

**Figure 15.**Comparison along observation lines 1 and 2 (dotted-dashed lines in Figure 12) between the FDPM results (solid points) with different kernel functions and the FEM results (lines): (

**a**) Observation line 1 and (

**b**) observation line 2.

**Figure 16.**Schematic of the lid-driven cavity flow. The solid circle is a circular cylinder and four sides of the domain are periodic boundaries. L is the size of the computational domain, R is the cylindrical radius, and F

_{0}is the volume force.

**Figure 17.**Particle distribution (solid circles) and velocity contours from the initial time to the steady state: (

**a**) t = 0, (

**b**) t = 1.0 s, (

**c**) t =2.0 s and (

**d**) t = 3.0 s.

**Figure 18.**Horizontal Velocity distributions along horizontal and vertical geometric centerlines at t = 3.0 s: (

**a**) Along vertical geometric centerlines and (

**b**) along horizontal geometric centerlines.

**Table 1.**FDPM results and theoretical solutions of the position and velocity of a particle (particle number: 70) at different times.

Time (s) | Particle Method: x (m) | Theoretical Solution: u (m/s) | Particle Method: u (m/s) | Error (10 ^{−8}) |
---|---|---|---|---|

0.25 | 2.48 | 30.16201462 | 30.16201544 | 2.74 |

0.50 | 10.08 | 30.64615858 | 30.64615942 | 2.75 |

0.75 | 17.80 | 31.11956025 | 31.11956110 | 2.75 |

1.00 | 25.64 | 31.58265402 | 31.58265470 | 2.13 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Xiong, A.
A Particle Method Based on a Generalized Finite Difference Scheme to Solve Weakly Compressible Viscous Flow Problems. *Symmetry* **2019**, *11*, 1086.
https://doi.org/10.3390/sym11091086

**AMA Style**

Zhang Y, Xiong A.
A Particle Method Based on a Generalized Finite Difference Scheme to Solve Weakly Compressible Viscous Flow Problems. *Symmetry*. 2019; 11(9):1086.
https://doi.org/10.3390/sym11091086

**Chicago/Turabian Style**

Zhang, Yongou, and Aokui Xiong.
2019. "A Particle Method Based on a Generalized Finite Difference Scheme to Solve Weakly Compressible Viscous Flow Problems" *Symmetry* 11, no. 9: 1086.
https://doi.org/10.3390/sym11091086