# Morphing Numerical Simulation of Incompressible Flows Using Seamless Immersed Boundary Method

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## Abstract

In this paper, we proposed the morphing simulation method on the Cartesian grid in order to realize flow simulations for shape optimization with lower cost and versatility. In conventional morphing simulations, a simulation is performed while deforming a model shape and the computational grid using the boundary fitting grid. However, it is necessary to deform the computational grid each time, and it is difficult to apply to a model with complicated shape. The present method does not require grid regeneration or deformation. In order to apply the present method to models with various shapes on the Cartesian grid, the seamless immersed boundary method (SIBM) is used. Normally, when the SIBM is applied to a deformed object, the velocity condition on the boundary is imposed by the moving velocity of the boundary. In the present method, the velocity condition is imposed by zero velocity even if the object is deformed because the purpose of the present morphing simulation is to obtain simulation results for a stationary object. In order to verify the present method, two-dimensional simulations for the flow around an object were performed. In order to obtain drag coefficients of multiple models, the object was deformed in turn from the initial model to each model in the present morphing simulation. By using the present method, the drag coefficients for some models could be obtained by one simulation. It is concluded that the flow simulation for shape optimization can be performed very easily by using the present morphing simulation method.

## Keywords

Computational fluid dynamics Morphing simulation method Immersed boundary method Incompressible flow Shape optimization## 1 Introduction

There are many products around us that are closely related to the flow phenomenon. Improvements in the performance of these products are always expected. On the other hand, reducing the time and cost required to develop these products is also an important issue. Shape optimization through flow simulations at the stage of design is one of these efforts. By determining the optimum shape from many candidate product shapes (candidate models) at the early stage of product development, the effort of the redesign is reduced. As a result, development costs are reduced. Conventionally, flow simulations have been performed for each of these many candidate models. However, in recent years, the cost required for flow simulations has increased because the number of candidate models has increased in order to develop higher performance products. In order to reduce the number of these simulations, simulations are performed while deforming the model shape and the computational grid in shape optimization using flow simulations [1]. In this method, the number of flow simulations for shape optimization can be reduced, and the optimum shape can be determined in the flow simulation because results for many models can be obtained in one simulation. However, it is necessary to deform the computational grid each time, and it is difficult to apply to a model with complicated shape. In addition, the simulation on the boundary fitted grid can be expected to have high computational accuracy, however, the computational efficiency is inferior to the simulation on the Cartesian grid. In this paper, in order to realize flow simulations for shape optimization with lower cost and versatility, a method is proposed to perform simulation while deforming a model on the Cartesian grid that does not require grid regeneration or deformation. We call this method the morphing simulation method.

In order to apply the present method to models with various shapes on the Cartesian grid, the seamless immersed boundary method (SIBM) [2], which is an improved method of the immersed boundary method (IBM) [3] is used. In the IBM, additional force terms are added to the momentum equations to satisfy the velocity conditions on the virtual boundary points where the computational grid and the boundary of the object intersect. In order to apply the IBM to an object with arbitrary shape, it is only necessary to know the position of the virtual boundary on the grid. Therefore, the IBM can be easily applied to an object with a complicated shape. As for the estimation of the additional forcing term, there are mainly two methods, that is, the feedback [4, 5] and direct [6] forcing term estimations. Generally, the direct forcing term estimation is adopted because of the simplicity of the algorithm. However, the conventional IBM with the direct forcing term estimation generates the unphysical pressure oscillations near the virtual boundary because of the pressure jump between inside and outside of the virtual boundary. The SIBM was proposed in order to remove these unphysical pressure oscillations. In the past study, the SIBM was applied not only to stationary objects but also to moving or scaling objects [7, 8]. Therefore, it is possible to use the SIBM in the morphing simulation method proposed in this paper. Normally, when the SIBM is applied to a moving or scaling object, the velocity condition in the estimation of the additional forcing term is determined by the moving velocity of the object. In the present method, the additional forcing term is estimated under the condition that the velocity is zero even if the object is deformed because the purpose of the present morphing simulation is to obtain simulation results for a stationary object.

In this paper, the morphing simulation by the present method is performed for some models and compared with the conventional static SIBM where simulation is performed for each model and the effectiveness of the present method is discussed.

## 2 Morphing Numerical Simulation Using Seamless Immersed Boundary Method

### 2.1 Governing Equations

*Re*denotes the Reynolds number defined by \(Re=L_0U_0/\nu _0\). \(U_0\), \(L_0\) and \(\nu _0\) are the reference velocity, the reference length and the kinematic viscosity, respectively. \(u_i=(u, v)\) and

*p*are the velocity components and the pressure. \(G_i\) in Eq. 2 denotes the additional forcing term for the SIBM. \(F_i\) denotes the convective and diffusion terms.

### 2.2 Numerical Method

*v*is the

*y*component of velocity

*I*,

*J*are the grid index and \(\varDelta \) is grid spacing. The velocity at the midpoint (for example, \(J+\frac{1}{2}\)) of the grid is calculated by linear interpolation. The diffusive and pressure terms are discretized by the conventional second order centered finite difference method. For the time integration, the fractional step approach [12] based on the forward Euler method is applied. For the incompressible Navier-Stokes equations in the SIBM, the fractional step approach can be written by

### 2.3 Seamless Immersed Boundary Method

### 2.4 Morphing Numerical Simulation

## 3 Application to Two-Dimensional Model

*O*denotes the region to which the forcing term is added in the SIBM. \(\rho _0\) and \(U_0\) denote the reference density and velocity of the flow. The drag coefficient by the conventional static SIBM is in good agreement with the reference result in each model. Therefore, these drag coefficients are used as reference results for verifying the present morphing simulation method.

Non-dimensional time for a process in each condition.

Total time for a process | ||
---|---|---|

Deformation time | Downtime | |

Case 1 | 1 | – |

Case 2 | 2 | – |

Case 3 | 4 | – |

Case 4 | 8 | – |

Case 5 | 16 | – |

Case 6 | 1 | 1 |

Drag coefficient of each model.

Model 2 | Model 3 | Model 4 | Model 5 | |
---|---|---|---|---|

Case 1 | 1.175 | 1.288 | 1.802 | 2.327 |

Case 2 | 1.342 | 1.461 | 1.677 | 2.048 |

Case 3 | 1.428 | 1.567 | 1.615 | 1.908 |

Case 4 | 1.487 | 1.617 | 1.587 | 1.840 |

Case 5 | 1.525 | 1.638 | 1.577 | 1.801 |

Case 6 | 1.500 | 1.607 | 1.527 | 1.760 |

Static SIBM | 1.568 | 1.631 | 1.568 | 1.728 |

Rate of computational time of each morphing process.

Process 1 | Process 2 | Process 3 | Process 4 | |
---|---|---|---|---|

Case 1 | 1.00 | 0.67 | 1.04 | 1.88 |

Case 2 | 1.46 | 1.19 | 1.54 | 2.19 |

Case 3 | 2.02 | 2.03 | 2.22 | 2.55 |

Case 4 | 2.77 | 3.05 | 3.13 | 3.20 |

Case 5 | 3.68 | 4.50 | 4.49 | 4.03 |

Case 6 | 1.25 | 1.01 | 1.46 | 2.10 |

## 4 Conclusions

In this paper, we proposed the morphing simulation method on the Cartesian grid in order to realize flow simulations for shape optimization with lower cost and versatility. By using SIBM that is the Cartesian grid approach, the present method could be applied very easily to an object with arbitrary shape. In order to verify the present method, the two-dimensional simulations for the flow around an object were performed. In order to obtain drag coefficients of multiple models, the object was deformed in turn from the initial model to each model in the present morphing simulation. By using the present method, the drag coefficients for some models could be obtained by one simulation. These drag coefficients became closer to the reference values by decreasing the deformation speed of the model. Furthermore, by setting the downtime after the deformation, drag coefficients close to the reference values were obtained even when the deformation speed was high. Therefore, it can be concluded that the flow simulation for shape optimization can be performed very easily and the number of times of flow simulation for many models can be significantly reduced by using the present morphing simulation method.

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