Non-Newtonian fluid flows, especially in three dimensions (3D), arise in numerous settings of interest to physics. Prior studies using the lattice Boltzmann method (LBM) of such flows have so far been limited mainly to two dimensions and used less robust collision models. In this paper, we develop a new 3D cascaded LBM based on central moments and multiple relaxation times (MRT) on a three-dimensional, nineteen velocity (D3Q19) lattice for simulation of generalized Newtonian (power law) fluid flows. The relaxation times of the second order moments are varied locally based on the local shear rate and parameterized by the consistency coefficient and the power law index of the nonlinear constitutive relation of the power law fluid. Numerical validation study of the 3D cascaded LBM for various benchmark problems, including the complex 3D non-Newtonian flow in a cubic cavity at different Reynolds numbers and power law index magnitudes encompassing shear thinning and shear thickening fluids, are presented. Furthermore, in order to demonstrate the advantages of the proposed 3D cascaded LBM based on central moments, numerical stability comparisons against the LBMs based on a single relaxation time model and a MRT model using raw moments are made. Numerical results demonstrate the accuracy, second order grid convergence and significant improvements in numerical stability of the 3D cascaded LBM for simulation of 3D non-Newtonian flows of power law fluids.

非牛顿流体流，特别是在三维（3D）中，在许多物理感兴趣的环境中产生。迄今为止，使用此类流的格子玻尔兹曼方法（LBM）进行的先前研究主要限于二维，并且使用的鲁棒性较小的碰撞模型。在本文中，我们基于三维矩，十九速度（D3Q19）格上的中心矩和多重弛豫时间（MRT），开发了一种新的3D级联LBM，用于模拟广义牛顿（幂定律）流体流动。二阶矩的弛豫时间根据局部剪切速率而局部变化，并由幂律流体的非线性本构关系的稠度系数和幂律指数进行参数化。针对各种基准问题的3D级联LBM的数值验证研究，包括在不同雷诺数下的立方腔体中的复杂3D非牛顿流，以及包含剪切稀化和剪切增稠流体的幂律指数幅度。此外，为了证明所提出的基于中心矩的3D级联LBM的优势，与基于单张弛豫时间模型和使用原始矩的MRT模型的LBM进行了数值稳定性比较。数值结果证明了用于幂律流体的3D非牛顿流模拟的3D级联LBM的准确性，二阶网格收敛性和数字稳定性的显着提高。为了证明所提出的基于中心矩的3D级联LBM的优势，与基于单张弛豫时间模型和使用原始矩的MRT模型的LBM进行了数值稳定性比较。数值结果证明了用于幂律流体的3D非牛顿流模拟的3D级联LBM的准确性，二阶网格收敛性和数字稳定性的显着提高。为了证明所提出的基于中心矩的3D级联LBM的优势，与基于单张弛豫时间模型和使用原始矩的MRT模型的LBM进行了数值稳定性比较。数值结果证明了用于幂律流体的3D非牛顿流模拟的3D级联LBM的准确性，二阶网格收敛性和数字稳定性的显着提高。