We employ the lattice Boltzmann method (LBM) to perform flow simulations with the objective of evaluating the performance of the D3Q27 stencil in capturing complex flow structures. We used a specific variation of the method known as the Moment Representation Lattice Boltzmann Method (MR-LBM), which employs a second-order moment representation to regularize the distribution functions, thereby enhancing computational performance. The implementation incorporates the Bhatnagar-Gross-Krook (BGK) collision operator, which is embedded in the moment evaluation process. As benchmark cases, we considered the lid-driven cavity (LDC) flow and a turbulent jet, both of which require robust numerical treatment to handle boundary conditions and resolve small-scale flow features. Dirichlet boundary conditions are enforced at solid walls, and the implementation adopts the incompressible regularized boundary condition (IRBC), which reduces the number of constraints on the second-order moments, thereby improving the stability and efficiency of the simulations. Additionally, we incorporated a high-order regularization technique and enforced a zero-trace condition on the momentum flux tensor to further enhance numerical stability and accuracy. Numerical experiments were conducted for Reynolds numbers of 10,000, 15,000, 25,000, and 50,000, employing mesh sizes up to 256 lattice nodes per dimension, ensuring a comprehensive assessment of the performance of the method under different flow conditions. The results were evaluated based on root mean square (RMS) and mean velocity profiles across different mesh sizes and Reynolds numbers. The findings were validated with results from the literature and showed good agreement.