The short answer is no. The Lorenz models are derived after a number of substantial simplifications, which makes them extremely important from a conceptual viewpoint, but one cannot rely on them for realistic CFD applications. On the other hand, it is possible to perform simplifications similar to those considered by Lorenz and derive conceptual models for applications other than meteorology. These models could be used to analyze predictability issues for the full equations they are derived from. Truncated models derived along the lines of the Lorenz ones for the Navier Stokes equations are described e.g. in the book

G. Gallavotti, The elements of mechanics, Springer

I agree, the simplifications introduces have a very strong impact to be assumed also for practical application in rotating flows.

At present, owing to the available computational power, turbine flows problem are solved using the full NS equations.

Dear Profs. Bonaventura and Denaro, many thanks for your interest. My hope of a translation of NS pde into an ODE system is based also in works like that of James Serrin, Swirling Vortex, 1972, where NS equations are translate in an (unique) integro-differential one-variable equation in the context of a special and limited class of possible solutions. The aim is that of bypassing the strong limit of high Reynolds number (104 - 105) arising on the matter of fans-pumps simulation (I have found that results with commercial cfd softwares are not very good).  Gianluca

Dear Gianluca,

the simplification of NSE in a set of ODE is an hystorical process adopted years ago in fluid dynamics. For example, the classical von Karman equation for the integral representation of the BL equations is an ODE. The lack of computational power as well as the requirement of rapid estimation of some main parameters were the reason for approaching such a simplified approach to the NSE. Of course, the results are strongly affected by such simplifications.

At present, several formulation can be adopted for simulating turbulent flow problems starting from RANS, DES, LES, DNS. Depending on your computational power, a Re number of 10^4-10^5 can be considered moderate and DES/LES could be possible. Otherwise, RANS can be a candidate but that depends also on the particolar problem you want to solve.

If you are interested in rotating flows, concepts and methods developed in numerical weather forecasting and large scale atmospheric dynamics (Rossby number, geostrophic equilibrium, geostrophic adjustment and so on) could be important in your applications. For example, it is known that standard low order finite volume/finite element methods perform poorly when applied to rotating flows on geophysical scales. The suggestion is to study the NS equations in dimensionless form and see how important the rotation term is in your context.

The Lorenz system was designed with a single purpose to obtain a simplest possible but not trivial model of an atmospheric flow. Despite of the striking simplicity of the equations derived by Lorenz they advanced tremendously our understanding of how the atmosphere really works. It is also worth to mention that the study of the deceptively simple equations led to the rediscovery of "chaos" which was originally investigated by H. Poincare, albeit in much more complicated manner in the context of the celestial mechanics.

Lorentz contributed to the analysis of a rotating flow using his ingenious approach at the early stage of his work

http://eaps4.mit.edu/research/Lorenz/Simplified_rotating_62.pdf

The question is: can we apply this technique for the analysis of the engineering flow?

The answer is probably yes if use the appropriate coordinate system and the certain assumptions based on the symmetry of the problem.

The answers obtained from such an engineering version of the Lorenz system should not be considered as the replacement for complex Computational Dynamics codes but as a version of a Back of the Envelope calculations proposed in the past by Enrico Fermi.

https://en.m.wikipedia.org/wiki/Back-of-the-envelope_calculation