What is the currect research status of solving Navier-Stokes Equation, any up-to-date review/good paper on this topic?
Or direct numerical simulation is still the best way to understand the complexity of turbulence?
Navier-Stokes Equation (NSE) is nonlinear PDE and numerical methods are applied for finding weak solution to that equation. One of the famous methods for solving NSE is finite element and finite volume methods.
Generally, tow areas of applied mathematics develop nonlinear PDEs solvers, Mesh generation methods, and solvers of systems of equations.
Recently mesh generation methods were developed (you can search and see). let me do not suggest any specific paper.
second for solvers of systems of equations (which is applied in wide area). you can use surveys.
Recently parallel methods is developed for solving nonlinear PDEs.
also there are meshless methods for solving nonlinear PDEs.
I suggest search in large scale PDE solvers.