Exact solution of a stable inviscid vortical flow inside a two-dimensional rectangular chamber with inlet and outlet



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In this thesis, we analyze vortical circulation inside a rectangular enclosure where flow enters via an opening and exits to the ambient atmosphere through another slot situated at the opposite side of the inlet. The characteristic Reynolds number is considered to be high so that viscous effects are neglected. We present a semi-analytical procedure to determine the velocity field in such system for arbitrary chamber geometry specified by the aspect ratio of the enclosure as well as by the dimensions and positions of the inlet and outlet.

In this problem, one has to satisfy both inlet and exit conditions to find a unique flow solution. The inlet condition can easily be implemented from the velocity profile at the entry plane. The flow at the outlet is, however, difficult to account for. The most accurate exit condition is the continuity constraint between the velocities inside and outside the chamber. Such continuity is difficult to ensure because it requires additional computation of a semi-infinite flow-domain representing the ambient. Hence, generally, in numerical schemes either an arbitrary exit condition is assumed or the outside domain is truncated by a finite hypothetical boundary. In contrast, in the present study, the true exit-condition is enforced by expressing the fields as an expansion of proper basis solutions with unknown amplitudes which are evaluated from a system of linear equations consistent with the outlet geometry. As a result, we are able to accurately calculate velocity fields for different geometries and boundary conditions.

The stability of the obtained solution can be ensured by constraining the system parameters. Consequently, our results can be useful in analysis of steady cross-circulations in rectangular chambers where Reynold’s number is typically high.



Steady, Incompressible, Vorticity, Stability, Inviscid