Introduction
In our previous work we stabilized the parabolic equilibrium profile in a 2D channel flow using actuators and sensors only at the wall. We demonstrated through numerical simulations that the proposed feedback law is effective for high values of the Reynolds number.
While the use of boundary control is practically more feasible than distributed control, it is even
more realistic to use control applied only on part of the boundary wall. In the present work we
compare several different configurations of the channel flow with boundary control restricted to
different parts of the boundary.
We consider the following cases:
- No control applied.
- Control applied over the full length of the walls.
- Subintervals (patches) of equal length distributed evenly on the two walls are controlled. (see Figure [fig.patch])
- The number of subintervals on each wall is one, two, four or eight.
- The total length of the patches is one fourth, one third, one half or one sixth of the total wall length.
- One control patch is applied on one wall only.
For these cases we present perturbation energy, vorticity and drag comparison, including the fully controlled case. Our general observations are the following.
- The stability properties of the flow can be improved even when the size of the controlled region is small relative to the uncontrolled region.
- The controlled and uncontrolled parts of the wall are competing against each other. The controlled part has to be large enough so that its effect has time to destroy the wall bounded turbulent structures, and the uncontrolled part has to be small enough so that instabilities does not grow over that region more than it decays over the controlled region.
- When the size of the controlled region is given, we achieve better performance with control distributed evenly along the wall than with control concentrated in one area.
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