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Pietro G. FRE

Alma Fluida Navier Stokes equations with periodic boundary conditions

Pietro G. FRE, University of Torino

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WOLFRAM MATERIALS for the ARTICLE:

Pietro G. Fré, Mario Trigiante, (2022).
Chaos from Symmetry: Navier Stokes equations, Beltrami fields and the Universal Classifying Crystallographic Group.
arXiv:2204.01037 [nlin.CD]
https://doi.org/10.48550/arXiv.2204.01037
Full article in PDF

enter image description here

Alma Fluida Navier Stokes equations with periodic boundary conditions
by Pietro G. Fré



& Mario Trigiante ⋆
 University of Torino, Department of Physics, Via P. Giuria, 1, I-10125 Torino, Italy. email: pietro.fre@unito.it
⋆ Politecnico di Torino, DISAT, C.so Duca degli Abruzzi,24, I-10129, Torino, Italy. email: mario.trigiante@polito.it

The calculation codes here presented are organized in a coherent system of WOLFRAM MATHEMATICA Notebooks and are devoted to Navier Stokes Equations with Periodic Boundary Conditions. The Alma Fluida NSP System is an instrument for the mathematical study of NS equations within the previously unknown Group Theoretical Framework provided by the notion of the Universal Classifying Group, introduced by P.Fré and A.S. Sorin in 2015 and further developed, extended, completed and systematized by the two present authors in a new publication on the ArXive entitled “Chaos from Symmetry: Navier Stokes equations, Beltrami fields and the Universal Classifying Crystallographic Group”:
https://arxiv.org/abs/2204.01037

Highlights of the Theory presented in Chaos from Symmetry

From the abstract

The general set up for the classification and construction of Arnold-Beltrami Flows on three dimensional torii is revised, reorganized and completed.The focal idea of a Universal Classifying Group UCG for crystallographic lattices is duly improved and extended. We constructed the missing UCG for the hexagonal lattice and mastering the cubic and hexagonal instances of this group, we can cover all cases.

TheUCGforthecubiclatticeis

1536

=solvablegroupwith1536elements,37conjugacyclassesand37irreps;

1536

∼

semidirect

(



)

TheUCGforthehexagonallatticeis

=solvablegroupwith72elements,24conjugacyclassesand24irreps;

∼



semidirect

(



)

The upgrading of Beltrami Flows to a special type of periodic solutions of the Navier Stokes equations is presented and the relation between the classification of Beltrami Flows with the classification of contact structures is enlightened. The recent developments in contact and symplectic geometry that allow the consideration of singular contact structures and Beltrami fields in the framework of b-manifolds with distinguished critical surfaces is reviewed and it is shown that the choice of the allowed critical surface for the b-deformation of a Beltrami field seems to be strongly related with its group-theoretical structure. The Landscape conception of spherical layers in the dual momentum lattice as main token for the construction of Beltrami fields is clarified and established on general grounds.

From the conclusions

The Beltrami operator is a chiral one. There are solutions with ±λ where λ is an admissible eigenvalue. Hence the conventional Fourier expansion of any periodic field can be subdivided in two parts the Beltrami plus and the Beltrami minus part (or anti-Beltrami). It was advocated in the following old paper

P. Constantin and A. Majda, “The beltrami spectrum for incompressible fluids”, Communications in Mathematical Physics, vol. 115, p. 864 - 896, 1988.

Beltrami fields weakly interact among themselves and one can define and index of Beltramicity per frequency that is approximately conserved in time. This viewpoint can be very fertile in dealing with the group theoretical construction of Beltrami fields. In the landscape approach we are free to sum exact solutions only in a definite spherical layer. If we mix different layers we obtain objects that are not exact solutions. Yet the weak interaction viewpoint suggests that such sums can approximate exact solutions in a nice and controllable way. This is a very much promising direction of research if the summation on layers is organized in irreps of the UCG group or suitable subgroups thereof.

From figures obtained with the Alma Fluids NSP system

In the following slot we show three illustrative figures. The first is a single chaotic trajectory in the fundamental cell of the cubic torus, produced by a Beltrami velocity field. The second is a set of spiraling trajectories produced by a Beltrami velocity field in the hexagonal lattice. The third is a length 6 orbit of the octahedral point group in the cubic momentum lattice.

Structure and installation of the Alma Fluida NSP system

The calculation system Alma Fluida NSP is composed of two Bakground NoteBooks, respectively dealing with the case of the Cubic Lattice and of the Hexagonal Lattice:

UniClasGroupCubicLatN10.nb

HexagBckgroundN6.nb

that contain all the routines and subroutines of the system plus a theoretical summary and a very much extended User Guide for each one of the two cases.
In addition the system comprises two libraries of data, one for the cubic case, one for the hexagonal one that contain essential information in order for either one of the two systems to work, as the list of elements organized in conjugacy classes of the two universal class groups, the list of all their irreps and the correspondence rules to obtain the matrix representation of each group element in each irrep, the character tables, the list of relevant subgroups and the directives to create branching rules.

In order to work with either one of the two Codes (Cubic or Hexagonal) the User should have downloaded on his/her computer the folder MathFiles2022 whose structure is displayed below

The full folder is available and you can get it freely from the Personal Google Drive Repository of Prof. Pietro Fré on the University of Torino Server, using the link:
https://drive.google.com/drive/folders/1hDz4dRmcTOYBrK7jKpUr2xmbc9GO4zUh?usp=sharing

Installation instructions

After you have obtained the entire MathFiles2022 folder you save it as it stands, without altering anything of its internal structure, in your favorite working directory
XXXXXXXXXXX

In each of the two background NoteBooks you find a preliminary section like the following one that applies to the cubic case

Working with the system

From your reading of the first section in each of the two background NoteBook you will learn all what you can do with each one. The important warning is that before Evaluating either one of the two background NoteBooks you should Quit Kernel. The two subsystems, for the Cubic Lattice and for the Hexagonal Lattice are self-consistent but interfere destructively with each other.

You can get a flavor of the possibilities by looking at one of the examples contained in the directory CubicLat, for instance the Execution NoteBook Illustrocubic.nb

Examples

We assume that you have evaluated the background NoteBook HexagBckgroundN6.nb with your working directory correctly assigned:
direcyours=”XXXXXXXXXXXXXXXXXXXXXXXXXXX\MathFiles2022\GruppoUniHex”;
and the instruction:
direc=direcyours

Classical solution of NS equation

We recall the classical solution of NS equation named after Poiseuille in a cylindrical tube.

The velocity field is of the form

f(x,y)

={0,0,f(x,y)}

The pressure field is of the form

p=p(z)=az+b

It easily follows that

U·∇U≡



[U]

dΩ

Hence the equation reduces to

f(x,y)-

∂p(z)

∂z

where

∂

is the two - dimensional Laplacian

Setting :

f(x,y)=

We find :

∂

f(x,y)

∂x∂x

∂

f(x,y)

∂y∂y

=-2

ν=a

Hence the complete solution of the steady Navier Stokes equations named after Poiseuille is the following

U=0,0,

;p=-2

νz+

WhereRistheradiusofthecirculartubeand

istheinitialpressure

Hexagonal Poiseuille

We show how to adapt that solution to a tube with a hexagonal section in the fundamental cell of the hexagonal lattice

The basis vector of the lattice are recalled here

In[]:=

Out[]=



,0,0,-

,0,0,0,



Routine hexagpoiseul


Running the routine hexagpoiseul

In[]:=

hexagpoiseul

=====================================

This is the routine hexagpoiseul that constructs the profile of a Poiseuille solution in a hexagonal chamber

This routine changes the coordinates from the standard orthogonal ones to those adapted to the hexagonal lattice {u,v,r}

and generates functions for the illustrative pictures

Then we solve the Laplace equation in the hexagon

We have determined the



invariant function solution of Laplace Equation in 2D

It is the following α(

+uv+

-β)

ΔΨ =

10α

Next we continue in 3D

We construct the solution profile in the fundamental cell

It is named robiola

Then we rotate it with Z6

Next we call the routine fundacellalung

==================================================================================

This is the routine fundacellalung which creates a double hexagonal fundamental chamber model of the tube

The fundamental polyhedric chamber (hexagonal) has been created and it is named fundchamberlung

It is a Graphics3D object and can be utilized in showing pictures

------------------------------------------------------------------------------------------------

The planar base of the fundamental chamber is named hexagonus and it a 2D Graphics object

It can be used as argument of RegionMember in constructing the Region Function of plots

Example: RegionFunctionFunction[{x,y,z},RegionMember[hexagonus,{x,y}]]

------------------------------------------------------------------------------------------------

=====================================

First we display inside the hexagonal tube the solution in the fundamental cell

=====================================

Next we show the solution repeated in the other three sectors

Out[]=

{Null}

Running initiomute

We initialize the background programme for Beltrami solutions

In[]:=

initiomute

================================================================================================

This is routine initiomute which starts all the background items for the hexagonal lattices and its point group

Dih

=======================================================================================================

I upload and/or generate all the background group theoretical items relative to the point group

Dih

=============================================================================

I construct all the items relative to the Dih6 group

=============================================================================

This is the routine generaDih6 which generates the Dih6 group in the defining three dimensional representation

First I introduce the generators AA, BB and their product UU=SS.TT

The generators are

AA =

1 2	3 2	0
- 3 2	1 2	0
0	0	1

BB =

-1	0	0
0	1	0
0	0	-1

Now I try to generate all elements

========================================

I try to generate all group elements of the Group from the generators you gave me

So far your group has number elements = 4

I try all their binary products

Now your group has number of elements = 11

I try to generate all group elements of the Group from the generators you gave me

So far your group has number elements = 11

I try all their binary products

Now your group has number of elements = 12

I try to generate all group elements of the Group from the generators you gave me

So far your group has number elements = 12

I try all their binary products

Now your group has number of elements = 12

-------------------------

I finished my task the list of 12 elements is encoded in the file Allgroup

=============================================================================

I finished my task the list of 12 elements is encoded in the file dihedro6

I order the elements of the group you gave me according to their order

You gave me the finite group gruppone

gruppone contains 12 elements

I finished ordering the elements

orders present in the group are in the file ordini

ordini = {1,2,3,6}

The order subdived group is named orderGG

I have reordered the group elements according to their order

The ordered group elements are in the array orderGroup

Now I redefine the set dihedro6, according to orders

The list of elements is reordered

=============================================================================

Now I generate all the conjugacy classes of group dihedro6

rr = 1

The number of group elements in loppo is 12

I found a conjugacy class of order 1

The number of group elements in loppo is 11

rr = 2

The number of group elements in loppo is 11

I found a conjugacy class of order 1

The number of group elements in loppo is 10

rr = 3

The number of group elements in loppo is 10

I found a conjugacy class of order 3

The number of group elements in loppo is 7

rr = 4

The number of group elements in loppo is 7

I found a conjugacy class of order 3

The number of group elements in loppo is 4

rr = 5

The number of group elements in loppo is 4

I found a conjugacy class of order 2

The number of group elements in loppo is 2

rr = 6

The number of group elements in loppo is 2

I found a conjugacy class of order 2

The number of group elements in loppo is 0

===============================================================================

Standard rearrangement of conjugacy classes according to Lia Vas Chemical Nomenclature

===============================================================================

{{EE,1},{A,6},{

,5},{

,2},{B,3},{BA,4}}

This shows that the result from the literature is correct and we can indeed use Vclass as our codification of group elements into conjugacy classes

=============================================================================

Final organization of the point group

=============================================================================

-----------------------------------------------------------------------------

Deriving the multiplication table

We have

g	XX	1 1	2 1	2 2	3 1	3 2	4 1	5 1	5 2	5 3	6 1	6 2	6 3
g	XX	XX	XX	XX	XX	XX	XX	XX	XX	XX	XX	XX	XX
1 1	XX	1 1	2 1	2 2	3 1	3 2	4 1	5 1	5 2	5 3	6 1	6 2	6 3
2 1	XX	2 1	3 1	1 1	4 1	2 2	3 2	6 3	6 1	6 2	5 1	5 2	5 3
2 2	XX	2 2	1 1	3 2	2 1	4 1	3 1	6 1	6 2	6 3	5 2	5 3	5 1
3 1	XX	3 1	4 1	2 1	3 2	1 1	2 2	5 3	5 1	5 2	6 3	6 1	6 2
3 2	XX	3 2	2 2	4 1	1 1	3 1	2 1	5 2	5 3	5 1	6 2	6 3	6 1
4 1	XX	4 1	3 2	3 1	2 2

Message Boards

Alma Fluida Navier Stokes equations with periodic boundary conditions

Highlights of the Theory presented in Chaos from Symmetry

From the abstract

From the conclusions

From figures obtained with the Alma Fluids NSP system

Structure and installation of the Alma Fluida NSP system

Installation instructions

Directory

Setting the Directory and the Library

Writing the user's addresses

Choosing the working directory

Working with the system

Classical solution of NS equation

Hexagonal Poiseuille

Routine hexagpoiseul


Running the routine hexagpoiseul

Running initiomute

Alma Fluida Navier Stokes equations with periodic boundary conditions

Highlights of the Theory presented in Chaos from Symmetry

From the abstract

From the conclusions

From figures obtained with the Alma Fluids NSP system

Structure and installation of the Alma Fluida NSP system

Installation instructions

Directory

Setting the Directory and the Library

Writing the user's addresses

Choosing the working directory

Working with the system

Classical solution of NS equation

Hexagonal Poiseuille

Routine hexagpoiseul

Running the routine hexagpoiseul

Running initiomute

Routine hexagpoiseul
