Is it possible to solve the Navier-Stokes equation in a triangular coordinate system and wouldn't this be more accurate?

[MasterOgon]

The Navier-Stokes equation is solved in a vector grid in a Cartesian coordinate system. That is, rectangular. But does a rectangular mesh relate to what happens in a gas or liquid, and is it better to use a triangular mesh?

Undoubtedly, it is incredibly difficult to take into account all the factors even in a triangular or tetrider coordinate system, which is difficult even for visual perception. And in its direct form such a solution is impossible. But it is precisely this system that allows the logical formation of figures that we can see in water - a ring vortex or torus, similar to a figure eight (infinity) and a hexagon, similar to a snowflake or polar vortexes of gas giants.

Let's imagine a homogeneous medium that consists of individual particles. The only possible position of the particles relative to each other, at which absolute homogeneity is achieved, is a tetrider, or for simplicity, a triangular lattice in one plane, at the intersections of which the particles are located. Thus, all distances between particles are the same. Particles interact with each other by being attracted at a distance and repelled upon collision, which is caused by the forces of molecular attraction and repulsion.

Now suppose one particle received an impulse and moved from its place in the direction of the other two. If we considered particles as billiard balls, then we could assume that the momentum would be divided into two. But in this case we have forces of molecular attraction and repulsion, which allow us to regard further interaction as a chain reaction similar to the domino principle, where momentum is transmitted indefinitely due to the force of gravity. Having logically followed the trajectory of the particles, we will see that the impulse in a circle on both sides, forming a figure eight, returned to the first part, which caused the action, which will lead to an endless repetition of the process. It is precisely this mechanism that underlies the ring vortex, which under ideal conditions, according to viscous friction, can exist endlessly dissipating its energy.

Edited February 6 by MasterOgon

[sethoflagos]

29 minutes ago, MasterOgon said:

The Navier-Stokes equation is solved in a vector grid in a Cartesian coordinate system. That is, rectangular.

... or cylindrical (commonest in my line of work) or even spherical from time to time.

31 minutes ago, MasterOgon said:

But does a rectangular mesh relate to what happens in a gas or liquid, and is it better to use a triangular mesh?

False premise (see above) and no. If coordinate axes are not mutually perpendicular, how do you even separate the vectors into independent axial components?

52 minutes ago, MasterOgon said:

Undoubtedly, it is incredibly difficult to take into account all the factors even in a triangular or tetrider coordinate system, which is difficult even for visual perception.

Difficult for verbal perception too if you're going to use made up words.

53 minutes ago, MasterOgon said:

And in its direct form such a solution is impossible.

Then what use is it?

56 minutes ago, MasterOgon said:

But it is precisely this system that allows the logical formation of figures that we can see in water - a ring vortex or torus,

Where are the triangles in a torus?  

1 hour ago, MasterOgon said:

snowflake

Fluid?

1 hour ago, MasterOgon said:

Let's imagine a homogeneous medium that consists of individual particles.

Navier-Stokes is a continuum model, not a particle model.

This is a rather fundamental distinction and renders the rest of your post somewhat off-topic as far as solving Navier-Stokes is concerned.

[Genady]

7 hours ago, MasterOgon said:

Let's imagine a homogeneous medium that consists of individual particles.

Unimaginable. If it consists of individual particles, it is not homogeneous.

[MigL]

I had never heard of triangular co-ordinates.
They are not defined by an axis, nor angle, and would seem to be relational to other objects rather than to a frame.

Would like to hear @studiot's take on this ...

[swansont]

There are finite-element solutions that use triangular meshes; I used one such program for magnetic fields.

The mention of rectangular mesh suggests some confusion between this and coordinate systems.

[MigL]

4 hours ago, swansont said:

There are finite-element solutions that use triangular meshes; I used one such program for magnetic fields.

Ahh, Ok.
The FEA program that models fluid flow around aeronautical structures ( with an add-on; otherwise only structural ), that I'm familiar with ( but have never used ) is CATIA. originally developed by Dassault Systems ( part of Dassault Aviation, one of France's two major aerospace companies ), and now in use by many aerospace companies around the world.
But I do know that there are other FEA programs that do just that; I believe NASA even developed their own software, but I'm not aware if it supports fluid flow analysis.
The OP might look into that.

[MasterOgon]

I admit that my question is incorrect. I confused the coordinate system with the particle behavior model due to my incompetence.
My drawings perhaps depict not the behavior of individual particles, but the generalized behavior of a liquid or gas, which explains some controversial phenomena, such as the oscillatory motion of a body in a viscous medium. My topic has already been closed twice, so I will not repeat it, but in reality this question is still open since there is no consensus and comprehensive theory. Nobody tested my experiment and so I’m trying to figure out how to explain it mathematically.