The Statistics of Navier Stokes Equations

[John Creighto]

I am curious about what insight people might have as to the statistics of Navier Stokes equation. I thought of the following way someone might try to calculate these.

1) Choose a bais (Basis A)
2) Pick a discrete number of points to constrain the solution of stokes equation.
3) Find the solution space of this basis when constrained by these points (This can be done by row reduction and will give you a new basis) (Basis B)
4) Relate the derivative of stokes equation, to this new basis via a matrix equation.
5) Set the derivative equal to zero at the constraint points chosen above.
6) Find the new solution space given this derivative constraint. (Basis C)
7) Orthogonalize this basis (use singular value decomposition) to get a new basis again. (Basis D)
8) Find the energy of each basis function from Basis D.
9) Assign the probability of each basis function based upon the energy of each basis function and the temperature of the region.
10) Knowing the probability of each basis function average over space and time to calculate the statistics.

One problem I see is that the energy of each basis function will likely depend upon which basis functions are occupied. The reason is that for instance the energy of wind is related to the cube of the velocity. However, at each estimate of the statistics the energy of each basis function could possibly be re-evaluated. Thus maybe any nonlinearities could be taken care of iteratively. Also note that the basis functions are chosen to be orthogonal, so perhaps this would simplify these nonlinarities somewhat.

 

[mmwave]

I find this approach interesting and potentially useful in understanding the statistics of Navier Stokes equation. However, there are some potential limitations and challenges that should be considered.

Firstly, the choice of basis A could greatly impact the results. Different basis functions may capture different features of the solution space, leading to different results. It would be important to carefully consider and justify the choice of basis A.

Additionally, the number of points chosen to constrain the solution (step 2) could also affect the results. A small number of points may not accurately represent the entire solution space, while a large number of points could lead to a computationally intensive process.

Furthermore, the use of row reduction in step 3 may be limited by the size and complexity of the problem. In high-dimensional systems, the solution space may be too large to accurately represent using row reduction. In such cases, alternative methods may need to be explored.

Moreover, the assumption of orthogonality in step 7 may not hold in all cases. Nonlinearities in the system may lead to non-orthogonal basis functions, which could impact the accuracy of the results.

Finally, the iterative approach proposed in step 10 may also be limited by the complexity of the problem. In highly nonlinear systems, it may be difficult to accurately estimate the energy of each basis function at each iteration, leading to potential errors in the final statistics.

In conclusion, while this approach may provide some insights into the statistics of Navier Stokes equation, it is important to carefully consider its limitations and potential challenges before applying it to a specific problem. Further research and development may be needed to refine and improve this method for more accurate and reliable results.