The modeling of turbulent flows is relevant in many engineering applications and, is an active field of research on numerical methods. Convergence and stability of proposed formulations are crucial to predict transitional flows from laminar to turbulent flows. In this work, a recently developed stabilized finite element formulation is used as a powerful tool to describe such kind of problems. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual based formulations, the introduction of two velocity subscale components. The theoretical rate of convergence of the method is verified numerically using linear and quadratic equal-order finite element discretizations. To this end, a standard convergence test of $L2$-norm is presented where the computed solutions are compared when manufactured solutions are imposed at Gauss-point level. Moreover, the Hopf bifurcation is studied for two well-known benchmark problems: flow past a cylinder and the three-dimensional lid-driven cavity flow. For the flow past a cylinder case, the Hopf bifurcation is verified using dynamic subscales and is assessed so that they do not disturb the solution. In particular, a dominant convective problem ($\mathrm{Re}=4000$) is solved using both the quasistatic and dynamic versions of the method, evaluating the performance of each one in the quality of the solution and in the CPU time needed to obtain a converged solution. For the 3D lid-driven cavity flow problem, the Hopf bifurcation is determined using two different boundary conditions, analyzing their effect on the dynamics of the problem and on the thickness and shape of the boundary layer. The final test case is the turbulent 3D lid-driven cavity problem ($\mathrm{Re}=12000$), where velocity profiles are compared with experimental, LES and DNS reference solutions. Additionally, pressure and velocity spectra are shown at certain representative points of the domain, as well as phase diagrams, correlation function graphs, the Poincaré map, and the Lyapunov exponent, typical mathematical tools used in dynamical system analysis. From the results, the method is robust and accurate for all the numerical tests both in viscous dominant problems as in dominant convective ones.

湍流的建模与许多工程应用有关，是数值方法研究的活跃领域。拟议制剂的收敛性和稳定性对于预测从层流到湍流的过渡流动至关重要。在这项工作中，最近开发的稳定有限元公式被用作描述此类问题的有力工具。公式的要点是子量表的时间依赖性，并且与基于残差的公式相反，引入了两个速度子量表分量。使用线性和二次等阶有限元离散化，对方法的理论收敛速度进行了数值验证。为此，进行了标准收敛测试$\mathrm{大号}2$-范数表示在以高斯点级别施加制造的解决方案时比较计算出的解决方案的情况。此外，针对两个众所周知的基准问题研究了霍夫夫分叉：流过圆柱体的流和三维盖子驱动的腔体流。对于通过圆柱壳的流量，使用动态子标尺对Hopf分叉进行验证，并对其进行评估，以使它们不会干扰解决方案。特别是主要的对流问题（$\mathrm{回覆}=4000$）是使用该方法的准静态和动态版本来解决的，评估解决方案的质量和获得融合解决方案所需的CPU时间中每个解决方案的性能。对于3D盖子驱动的腔体流动问题，使用两个不同的边界条件确定Hopf分支，并分析它们对问题动力学以及边界层厚度和形状的影响。最终的测试案例是湍流的3D盖子驱动型腔问题（$\mathrm{回覆}=12000$），将速度剖面与实验，LES和DNS参考解决方案进行比较。此外，在该域的某些代表点上还显示了压力和速度谱，以及相图，相关函数图，庞加莱图和Lyapunov指数，这是用于动力学系统分析的典型数学工具。从结果来看，该方法对于粘性主导问题和主导对流问题中的所有数值测试都是鲁棒且准确的。