Yes, I do think so. 

When we extract the vorticity equation from the NS equation, certain information regarding the pressure is lost. Similarly, one may guess that the NS equation may be derived from a universal equation, and there is a loss of information through the reformulation...

In my opinion, there isnt anything missing in the Navier-Stokes equations, as far as the description of the fluid flow is concerned. In general the boundary conditions are the source of instabilities, which lead to turbulence. For instance, imagine an experiment with PERFECT slip wall boundary condition and also no other perturbations. The flow in this scenario should be laminar even at infinite Reynolds number. But this is not a practical case, since the flow is under the influence of unknown perturbations both in experiments and numerical simulations. If these perturbations dominate the viscous effect, they introduce instability leading towards the turbulent flow.  Now if we perform an experiment of a turbulent flow ( e.g. flow past a cylinder at high Reynolds number) and  Direct Numerical Simulation of the Navier-Stokes equations for the same configuration and compare the turbulence, there should be a good comparison (I am not sure about the source of the instabilities).

On the principle of independence of the boundary conditions of structure direction of turbulence

Lu Panming

Abstract

This is an introduction to the turbulence modeling theory of the present author, which can take the boundary conditions of turbulence structure into consideration ( the main contents have either not been published in normal journals or only published in chinese papers).   It is included that： (I)。Difficulty Problems Encountered in “Second-Moment-Closure” turbulence modelling ，i.e.  “Gao-Ge Anomaly” ;  (II)。The improvement  to the ”Second-Moment-Closure” turbulence modelling in order to allows the boundary conditions of turbulence structure direction could be prescribed;  (III)。An explanation to the principle of  independence  of  the boundary conditions of

turbulence structure direction ;  (IV)。Six suggestions to the possible future work directions;  (V)。 An explanation to the ”Gao-Ge Anomaly”.

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4. The principle of the independence of the boundary conditions is not only to the turbulence modelling, but also suitable to the other subject, because it is frequently meet the equations are not easy to solve, so there is often a need of modelling or predigestion.  For example, both the D‘Alembert anomaly (~1752) and the Stokes anomaly (~1856), are due to breach the  principle of independence  of the boundary conditions, according to today`s point of view .This is because, first, by using the inviscid potential flow`s dynamic equations it is not possible to affiliate the slip-less condition of the real fluid at the wall, and  second, using the Stokes equation it is not possible to affiliate the boundary conditions both at the wall of cylinder and at the infinite to the cylinder.  The first anomaly lead to the discovery of the Prandlt`s boundary layer theory（1904）, and the  second anomaly  was solved by Oseen（1910）with introducing a modification. All these reflect the scientific worthiness and broad latency of the present principle   of the independence of the boundary conditions.

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above is an example paragraph,for more details or need the full text, please go to the end of this letter, where will be an web address existed.

            http://staff.ustc.edu.cn/~pmlu/