# A Fast Hybrid Pressure-Correction Algorithm for Simulating Incompressible Flows by Projection Methods

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## Abstract

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## 1. Introduction

## 2. Mathematical Modelling and Numerical Methods

#### 2.1. Projection Method

#### 2.2. Weighted Least-Squares Approximation

#### 2.3. Pressure Correction and Poisson Solvers

- Compute the coefficients of the right-hand side in the discrete Fourier modes using the forward one-dimensional discrete transforms sequentially on each separate dimension.
- Compute the coefficients of the solution in the discrete Fourier modes by dividing the coefficients obtained in Step 1 by the eigenvalues computed as the sum of one-dimensional eigenvalues.
- Transform the solution back to the basis of the grid point values using the backward one-dimensional discrete transforms sequentially on each separate dimension.

## 3. Results

#### 3.1. Flow over a Square

#### 3.2. Flow around a Cylinder

#### 3.3. Flow Past a Cylinder between Parallel Plates

#### 3.4. Flow over Periodic Triangular Hills

#### 3.5. Lid-Driven Polar Cavity

## 4. Conclusions and Future Work

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AGMG | Aggregation-based algebraic multigrid method |

CFD | Computational fluid dynamics |

CFL | Courant–Friedrichs–Lewy |

DNS | Direct numerical simulation |

FFT | Fast Fourier transform |

IBM | Immersed boundary method |

LES | Large-eddy simulation |

RK | Runge–Kutta |

SPH | Smoothed particle hydrodynamics |

WLS | Weighted least squares |

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**Figure 3.**Geometry of the flow over a square with periodic lateral boundary conditions. The side length of the square is $D=0.04$ m, and the side length of the square lattice is $L=0.1$ m. The two monitoring points 1 and 2 for comparing the time histories of numerical solutions are located at $(-L/2,-L/4)$ and $(-D/2,-D/2)$, respectively.

**Figure 4.**Comparison of numerical solutions obtained by the reference method (

**left**) and the hybrid method (

**right**) for the first test case at $t=4000$ s. Shown from top to bottom are the contours of the horizontal velocity component U, the vertical velocity component V, and the pressure P, respectively.

**Figure 5.**Comparison of the time histories of the horizontal and vertical velocity components at Monitoring Point 1 obtained by the reference method and the hybrid method for the first test case.

**Figure 6.**Comparison of the time histories of the pressure at the two monitoring points 1 and 2 obtained by the reference method and the hybrid method for the first test case.

**Figure 7.**Spatial error convergence for laminar flow over a square simulated by the reference (

**left**) and hybrid (

**right**) methods.

**Figure 8.**Temporal error convergence for laminar flow over a square simulated by the reference (

**left**) and hybrid (

**right**) methods.

**Figure 9.**Geometry of the flow around a cylinder with periodic lateral boundary conditions. The radius of the cylinder is $r=0.02$ m, and the side length of the square lattice is $L=0.1$ m. The three monitoring points 1, 2, and 3 for comparing the time histories of numerical solutions are located at $(-L/2,-L/4)$, $(-r,0)$, and $(r,0)$, respectively.

**Figure 10.**Comparison of numerical solutions obtained by the reference method (

**left**) and the hybrid method (

**right**) for the second test case at $t=300$ s. Shown from top to bottom are the contours of the horizontal velocity component U, the vertical velocity component V, and the pressure P, respectively.

**Figure 11.**Comparison of the time histories of the horizontal and vertical velocity components at Monitoring Point 1 obtained by the reference method and the hybrid method for the second test case.

**Figure 12.**Comparison of the time histories of the pressure at Monitoring Point 1 and the pressure difference between the two monitoring points 2 and 3 obtained by the reference method and the hybrid method for the second test case.

**Figure 13.**Comparison of numerical solutions obtained by the reference method (

**left**) and the hybrid method (

**right**) for the third test case at $t=300$ s. Shown from top to bottom are the contours of the horizontal velocity component U, the vertical velocity component V, and the pressure P, respectively.

**Figure 14.**Comparison of the time histories of the horizontal and vertical velocity components at Monitoring Point 1 obtained by the reference method and the hybrid method for the third test case.

**Figure 15.**Comparison of the time histories of the pressure at Monitoring Point 1 and the pressure difference between the two monitoring points 2 and 3 obtained by the reference method and the hybrid method for the third test case.

**Figure 16.**Geometry of the flow over periodic triangular hills in a channel. The height of the hill is $h=0.02$ m, and the width of the hill is $0.04$ m. The height of the channel is $L=0.1$ m. The two monitoring points 1 and 2 for comparing the time histories of numerical solutions are located at $(-L/2,-L/4)$ and $(0,-L/2+h)$, respectively.

**Figure 17.**Comparison of numerical solutions obtained by the reference method (

**left**) and the hybrid method (

**right**) for the fourth test case at the time when the horizontal velocity at Monitoring Point 1 reaches the median in the accelerating phase. Shown from top to down are the contours of the horizontal velocity component U, the vertical velocity component V, and the pressure P, respectively.

**Figure 18.**Comparison of time histories of the horizontal and vertical velocity components at Monitoring Point 1 obtained by the reference method and the hybrid method for the fourth test case.

**Figure 19.**Comparison of time histories of the pressure at the two monitoring points 1 and 2 obtained by the reference method and the hybrid method for the fourth test case.

**Figure 20.**Schematic of the geometry for the lid-driven polar cavity flow. The flow is driven by the moving wall at $r={R}_{i}$, and the polar cavity is defined by ${R}_{i}\le r\le {R}_{o}$ and $-\alpha /2\le \theta \le \alpha /2$. The red dashed rectangle indicates the extended computational domain for solving the pressure Poisson equation.

**Figure 21.**Comparison of the radial and azimuthal velocity components along four radial lines for the lid-driven polar cavity flow at different Reynolds numbers.

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**MDPI and ACS Style**

Fang, J.
A Fast Hybrid Pressure-Correction Algorithm for Simulating Incompressible Flows by Projection Methods. *Algorithms* **2023**, *16*, 287.
https://doi.org/10.3390/a16060287

**AMA Style**

Fang J.
A Fast Hybrid Pressure-Correction Algorithm for Simulating Incompressible Flows by Projection Methods. *Algorithms*. 2023; 16(6):287.
https://doi.org/10.3390/a16060287

**Chicago/Turabian Style**

Fang, Jiannong.
2023. "A Fast Hybrid Pressure-Correction Algorithm for Simulating Incompressible Flows by Projection Methods" *Algorithms* 16, no. 6: 287.
https://doi.org/10.3390/a16060287