The conservation principles are of utmost importance in fluid dynamics. They lead to sets of linear equations the solution of which yields the fields we are looking for (velocity, pressure, temperature...). The conservation of momentum can be written in the following component form (in spatial carthesian coordinates) [19]:
Since (definition of the total derivative):
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(586) |
and (conservation of mass) [19]
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(587) |
one can write
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(588) |
and consequently Equation (585) amounts to:
since the Cauchy stress
can be written as the viscous stress
minus the hydrostatic pressure
:
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(590) |
The viscous stress can be written as the sum of the laminar viscous
stress
and the turbulent viscous stress
satisfying [67]
( is the turbulent kinetic energy and
is the turbulent viscosity) leading to
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(593) |
Integrating Equation (589) over an element one obtains (using Gauss' theorem):
where is the volume of the element and
the external surface (which is
the sum of the area of all external faces of the element). The area of a face
is calculated by considering it as a 2-dimensional finite element and
calculating the Jacobian vector at the center (1-point integration). The
volume is obtained by replacing
by the coordinate
and
by
in
Equation (559):
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(595) |
Now, turning to Equation (594) each term is considered in detail for
element P and iteration of increment
.