The Navier-Stokes equations are a group of math equations that explain how thick fluids, like water or oil, move. These equations are very important in understanding fluid mechanics, but using them in real-life situations through a method called Computational Fluid Dynamics (CFD) can be tough. Instead of making things easier, these methods can actually add more problems.

The Complexity of the Navier-Stokes Equations

1. Nonlinearity:
   The Navier-Stokes equations are complicated because they include nonlinear parts, which makes them hard to solve. This is how the equation looks:

   $$\rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$$

   Here, $\mathbf{u}$ is how fast the fluid is moving, $\rho$ is the fluid's density, $p$ stands for pressure, $\mu$ is a measure of how thick or slippery the fluid is, and $\mathbf{f}$ shows the forces acting on the fluid (like gravity).

2. Turbulence:
   A big challenge is understanding turbulence, which is the chaotic and irregular movement of fluid. Turbulence happens on many different scales and can be hard to represent using regular models. Trying to use methods like Large Eddy Simulation (LES) or Reynolds-Averaged Navier-Stokes (RANS) often creates errors, especially when dealing with fast-moving flows.

3. Boundary Conditions:
   It’s essential to set the right boundary conditions, but this can be tricky in real-world applications. If the conditions are wrong or too simple, it can lead to unrealistic results or confusing answers, making the whole process of simulation harder.

Numerical Methods and Solutions

Because the Navier-Stokes equations are so complicated, many people use numerical methods to solve them, but this also brings more challenges:

• Discretization:
  This means breaking down the equations into smaller parts so that computers can process them. Using methods like Finite Volume or Finite Element requires a lot of computer power and knowledge. Mistakes in this process can cause the simulations to become unstable. To keep things stable, certain conditions need to be met, which adds more complexity.

• Grid Dependency:
  The type of grid used for computations can greatly change the results. If you use a fine grid, the results are more accurate but need more computer power. On the other hand, a coarse grid could oversimplify things and miss important details about the flow.

The Difficult Existence and Smoothness Problem

There’s a big question in math and fluid dynamics called the Navier-Stokes existence and smoothness problem. It asks whether certain solutions really exist under specific starting conditions. This unanswered question creates doubt about how reliable CFD simulations are.

Path to Solutions

Even with these tough challenges, there are some strategies that can help deal with the Navier-Stokes equations in CFD:

1. Improved Numerical Schemes:
   Researchers are working on better numerical methods that can make solutions more stable and accurate, like adaptive mesh refinement and higher-order methods.

2. Turbulence Modeling:
   Continued work on modeling turbulence, including using machine learning and data-driven techniques, offers hope for better understanding complex turbulent flows.

3. Hybrid Methods:
   Using hybrid methods that combine different simulation techniques, like coupling RANS with LES, can give us better insights into complicated flow behaviors.

In summary, while the Navier-Stokes equations are very important in understanding fluid movement, solving them numerically comes with many challenges. However, ongoing improvements in modeling and more powerful computers are helping to overcome some of these issues, bringing us closer between theory and real-world applications.