### SIMULATION AND CALCULATION OF VARIOUS PARAMETERS OF A LIQUID FLOW IN CONDUITS WITH COMPLEX SECTION, BY THE FINITE ELEMENT METHOD

#### Résumé

Abstract:

The present study, gives to the engineer who is face to the installation or the use of the

conduits, the knowledge which will allow him in first part, to understand the studies

realized by the specialists to which he can be exposed, and in others parts, to use these

conduits correctly by knowing the phenomena and the constraints which made it possible to

realize them, as well as the practical methods of study of the conduits according to the

conditions of flow of the fluids transported.

Thus, our principal object is the study of the viscous, laminar and incompressible flow in

conduits of complex section, by the finite element method, by basing on the generalization

of the Poiseuille flow witch is theoretically study by Stocks for circular and plane sections,

in order to determine numerically the fields of the thermodynamic and dynamic parameters

through these sections.

Indeed, the complexity of the sections of the studied conduits induced that the analytical

solution (exact) does not exist, and obviously, our interest is directed towards the search for

the approximate numerical solutions.

The equations which control these flows in the general case, are complex and nolinear, but

with suitable assumptions for our study, these equations are reduced to the differential

equation of Poisson with second constant member, the solution of the equation represents

the distribution of the axial velocity through the section of control, and the boundary

conditions of the studied problem are of Dirichlet type, considering the presence of the

viscosity of the fluid and the forces of friction with walls.

The finite element method was applied successfully since it adapts for any section chosen,

while being based on an iterative representation, which uses the method of Khaletski for the

resolution of system.

The solution of the system gives us the distribution of velocity through the section chosen,

and once this speed is given, we can deduce from them the thermodynamic and dynamic

parameters such as the pressure, the speed max, the volume flow, the force friction to exert on

the wall, as well as the shear stress.

However, once this work is realized for a section quite selected such as the circular section, we can contribute with complex sections, and, since the analytical solution exists for some

geometry such as the rectangular and elliptic sections, a validation will make the object of a

comparison between these solutions and the results obtained by our developed dataprocessing

program, in order to pass to other more complex simulations.

These results allow, for example for engines constructor or another type of installation, and

in particular when they are interested in modifications of conduits, to have the results and

the data necessary according to the geometry of the selected conduits.

The present study, gives to the engineer who is face to the installation or the use of the

conduits, the knowledge which will allow him in first part, to understand the studies

realized by the specialists to which he can be exposed, and in others parts, to use these

conduits correctly by knowing the phenomena and the constraints which made it possible to

realize them, as well as the practical methods of study of the conduits according to the

conditions of flow of the fluids transported.

Thus, our principal object is the study of the viscous, laminar and incompressible flow in

conduits of complex section, by the finite element method, by basing on the generalization

of the Poiseuille flow witch is theoretically study by Stocks for circular and plane sections,

in order to determine numerically the fields of the thermodynamic and dynamic parameters

through these sections.

Indeed, the complexity of the sections of the studied conduits induced that the analytical

solution (exact) does not exist, and obviously, our interest is directed towards the search for

the approximate numerical solutions.

The equations which control these flows in the general case, are complex and nolinear, but

with suitable assumptions for our study, these equations are reduced to the differential

equation of Poisson with second constant member, the solution of the equation represents

the distribution of the axial velocity through the section of control, and the boundary

conditions of the studied problem are of Dirichlet type, considering the presence of the

viscosity of the fluid and the forces of friction with walls.

The finite element method was applied successfully since it adapts for any section chosen,

while being based on an iterative representation, which uses the method of Khaletski for the

resolution of system.

The solution of the system gives us the distribution of velocity through the section chosen,

and once this speed is given, we can deduce from them the thermodynamic and dynamic

parameters such as the pressure, the speed max, the volume flow, the force friction to exert on

the wall, as well as the shear stress.

However, once this work is realized for a section quite selected such as the circular section, we can contribute with complex sections, and, since the analytical solution exists for some

geometry such as the rectangular and elliptic sections, a validation will make the object of a

comparison between these solutions and the results obtained by our developed dataprocessing

program, in order to pass to other more complex simulations.

These results allow, for example for engines constructor or another type of installation, and

in particular when they are interested in modifications of conduits, to have the results and

the data necessary according to the geometry of the selected conduits.

#### Mots-clés

conduits; finite elements; simulation; viscous and incompressible fluid.