Viscoelasticity is an important characteristic of many complex fluids such as polymer solutions and melts. Understanding the viscoelastic behavior of such complex fluids presents mathematical, modeling and computational challenges, particularly in the case of fluids affected by elastic turbulence at high Weissenberg number. A numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three-dimensional flow of viscoelastic fluids. The discrete elastic viscous split stress (DEVSS) formulation in cooperating with log-conformation formulation transformation is employed to improve computational stability at high Weissenberg number. The momentum equation is calculated after introducing an ellipticity factor and the constitutive equation is calculated based on the logarithm of the conformation tensor. The finite element-finite difference formulations of governing equations are derived. The planar contraction as a representative benchmark problem is used to test the robustness of the numerical method to predict real flow patterns of viscoelastic fluids at different Weissenberg numbers. The simulation results predicted with differential constitutive models based on the logarithm of the conformation tensor agree well with Quinzani’s experimental results. Both the stability and the accuracy are improved compared with traditional calculation method. The numerical methodology proposed in the study can well predict complex flow patterns of viscoelastic fluids at high Weissenberg number.

粘弹性是许多复杂流体（例如聚合物溶液和熔体）的重要特征。理解这种复杂流体的粘弹性行为提出了数学，建模和计算方面的挑战，特别是在高Weissenberg数下受弹性湍流影响的流体的情况下。该研究提出了一种基于惩罚有限元法和解耦算法的数值方法，以模拟粘弹性流体的三维流动。离散弹性粘滞分裂应力（DEVSS）公式与对数构象公式转换配合使用，以提高高Weissenberg数下的计算稳定性。在引入椭圆率因子后计算动量方程，并基于构象张量的对数计算本构方程。推导了控制方程的有限元-有限差分公式。以平面收缩为代表的基准问题用于测试数值方法的稳健性，以预测在不同魏森伯格数下的粘弹性流体的实际流动模式。基于构象张量对数的差分本构模型预测的模拟结果与Quinzani的实验结果非常吻合。与传统的计算方法相比，其稳定性和准确性都得到了提高。