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Abstract:
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In this scientific work, we use two effective methods : Lie groups theory and the finite
element method, to explain why the transition from laminar flow to turbulence flow
depends on the variation of the Reynolds number. We restrict ourselves to the case
of incompressible viscous Newtonian fluid flows. Their governing equations, i.e. the
continuity and Navier-Stokes equations are established and investigated. Their solutions
are expressed explicitly thanks to Lie's theory. The stability theory, which leads to an
eigenvalue problem is used together with the finite element method, showing a way to
compute the critical Reynolds number, for which the transition to turbulence occurs.
The stationary flow is also studied and a finite element method, the Newton method, is
used to prove the stability of its convergence, which is guaranteed for small variations of
the Reynolds number. |
