Calculation of the unit normal vector for wall shear stress in the lattice Boltzmann model

Highlights

• A method for the calculation of unit normal vector needed in wall shear stress analysis is proposed for lattice Boltzmann model.

• This method can be completely incorporated in the voxelization process of computational domain.

• This method requires fewer input parameters and has a higher computational efficiency than the existing methods.

Abstract

The pathology of aortic diseases is related to wall shear stress, which in the lattice Boltzmann method can be easily calculated using information for the unit normal vector of computational geometry. This paper proposes a method to calculate the unit normal vector. In comparison with the existing method, this approach requires a smaller number of input parameters and significantly reduces the computational cost. Tests on several geometries indicate that this method is a better choice from the viewpoint of accuracy and operability. Finally, the distribution of wall shear stress in a realistic aortic arch is obtained and found to agree with the conclusions of a previous study.

Introduction

Aortic aneurysm and dissection have high morbidity and mortality rates [1,2]. Their pathologies are thought to be related to vessel morphology, vessel mechanical properties, and hemodynamic forces [3]. Among the hemodynamic forces, wall shear stress (WSS) is a research emphasis. This stress affects the endothelium function, and a large WSS will induce the initiation of aortic aneurysm and dissection [4], [5], [6], [7]. The in vivo measurement of WSS is relatively difficult owing to the complexity of vessel geometry and the disturbance of other organs [8,9]. However, the distribution of WSS can be obtained by computational fluid dynamics (CFD) [10], [11], [12].

In traditional CFD methods such as the finite element and the finite volume methods, mesh generation for a given computational geometry is a time-consuming task [8,[13], [14], [15]. In the lattice Boltzmann method (LBM), the computational geometry is discretized to equal size voxels; thus, mesh generation can be completed entirely automatically. Easy implementation of the bounce-back boundary condition allows LBM to handle complex geometries, and the parallel nature of LBM makes it possible to solve flow problems at relatively large scales. Besides, WSS in LBM can be computed locally and does require calculation of the velocity gradient.

In LBM, the main problem in the calculation of WSS is obtaining the unit normal vector of computational geometry. Stahl et al. recommended calculating the unit normal vector as the vector that is perpendicular to the local velocity near wall [16]; this method has been adopted widely [17], [18], [19]. However, the capacity of this method in turbulent flow is questionable; moreover, its calculation procedure is relatively complex and does not provide the direction of the unit normal vector [15]. To address these problems, Matyka et al. proposed a further method in which the unit normal vector is calculated using only computational geometry information and is ‘geometric normal’ [15].

Herein, we propose a method to calculate the unit normal vector using a lower number of input parameters and significantly reduced computational costs. In the following, a brief introduction to the LBM and WSS is given in Section 2. An introduction to the proposed method, together with its comparison with the geometric normal in simple geometries, is given in Section 3. Numerical simulations for a realistic aortic arch are given Section 4. Discussion and conclusions are given in Section 5.