Open-loop control of compressible afterbody flows using adjoint methods

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Abstract
The flow past an axisymmetric body is generically unstable to a steady and a time-periodic global instability, the latter being thought to lead the low-frequency unsteadiness of the wake even at larger Reynolds numbers. The present paper examines how the growth rate of the oscillatory unstable mode developing in the wake of bullet-shaped objects can be reduced by a steady forcing, whose effect is to modify the base flow. The use of the compressible Navier–Stokes equations allows to consider control through steady mass, momentum, and heat forcing applied in the bulk and at the wall. To do so, we extend to compressible flows and axisymmetric geometries the method first proposed by Hill (NASA Technical Report No. 103858, 1992) to analyze the control of the two-dimensional mode of the incompressible cylinder wake. This method aims at evaluating the sensitivity of one particular eigenvalue to forcing by resolution of adjoint equations. Considering control at the wall, it allows to compute directly the eigenvalue gradient with respect to the wall variables. We show that the oscillating mode can be stabilized by a steady blowing at the wall (the so-called base-bleed control). Expressing the gradient as a sum of production, streamwise advection, and cross-stream advection terms, we show that this stabilizing effect is due to cross-stream advection, in contradiction with the up to now accepted interpretation based on the local absolute and convective instability analysis of parallel profiles. The same technique allows to compute the gradient of the oscillatory eigenvalue to bulk mass, momentum, and heat sources. Momentum control can be achieved by placing a small ring in the lee of the afterbody. Similar to the two-dimensional case studied by Hill, the effect of such a ring is twofold, as it induces a steady drag force which modifies the base flow and a fluctuating drag force proportional to the perturbation momentum at the ring location. We show that the efficiency of the control can be improved by heating the ring, which then acts as an additional heat source.

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