Cheng Lei, Li Yizeng, Grosh Karl
Department of Mechanical Engineering University of Michigan, Ann Arbor, MI 48109, USA.
J Comput Phys. 2013 Aug 15;247:248-261. doi: 10.1016/j.jcp.2013.03.063.
An approximate boundary condition is developed in this paper to model fluid shear viscosity at boundaries of coupled fluid-structure system. The effect of shear viscosity is approximated by a correction term to the inviscid boundary condition, written in terms of second order in-plane derivatives of pressure. Both thin and thick viscous boundary layer approximations are formulated; the latter subsumes the former. These approximations are used to develop a variational formation, upon which a viscous finite element method (FEM) model is based, requiring only minor modifications to the boundary integral contributions of an existing inviscid FEM model. Since this FEM formulation has only one degree of freedom for pressure, it holds a great computational advantage over the conventional viscous FEM formulation which requires discretization of the full set of linearized Navier-Stokes equations. The results from thick viscous boundary layer approximation are found to be in good agreement with the prediction from a Navier-Stokes model. When applicable, thin viscous boundary layer approximation also gives accurate results with computational simplicity compared to the thick boundary layer formulation. Direct comparison of simulation results using the boundary layer approximations and a full, linearized Navier-Stokes model are made and used to evaluate the accuracy of the approximate technique. Guidelines are given for the parameter ranges over which the accurate application of the thick and thin boundary approximations can be used for a fluid-structure interaction problem.
本文推导了一种近似边界条件，用于模拟流固耦合系统边界处的流体剪切粘性。剪切粘性的影响通过对无粘边界条件的修正项来近似，该修正项用压力的二阶面内导数表示。分别推导了薄粘性边界层近似和厚粘性边界层近似；后者包含前者。这些近似用于推导一个变分形式，在此基础上建立了粘性有限元法（FEM）模型，该模型只需要对现有的无粘有限元模型的边界积分贡献进行微小修改。由于该有限元公式对压力只有一个自由度，因此与需要对完整的线性化纳维 - 斯托克斯方程组进行离散化的传统粘性有限元公式相比，具有很大的计算优势。厚粘性边界层近似的结果与纳维 - 斯托克斯模型的预测结果吻合良好。在适用时，与厚边界层公式相比，薄粘性边界层近似也能给出计算简单的准确结果。对使用边界层近似和完整的线性化纳维 - 斯托克斯模型的模拟结果进行了直接比较，并用于评估近似技术的准确性。给出了关于厚、薄边界近似准确应用于流固相互作用问题时的参数范围指南。
