Power Loss Minimization for Drag Reduction and Self-Propulsion using Surface Mass Transpiration
The remarkable efficacy with which normal surface mass transpiration (blowing and suction) alters a given base flow to achieve a desired predefined objective has motivated several investigations on drag reduction, self-propulsion and suppression of separation and wake unsteadiness in bluff body flows. However, the energetic efficiency, a critical parameter that determines the true efficacy and in particular practical feasibility of this control strategy, has received significantly less attention. In this work, we determine the optimal zero net mass transpiration blowing and suction profiles that minimize net power consumption while reducing drag or enabling self-propulsion in typical bluff body flows. We establish the influence of prescribed blowing and suction profiles on the hydrodynamic loads and net power consumption for a representative bluff body flow involving flow past a stationary two-dimensional circular cylinder. Using analysis based on Oseen’s equations, we find that all the symmetric modes, except the first one, lead to an increase in the net power consumption without affecting hydrodynamic drag. The optimal blowing and suction profile that yields minimum power consumption is such that the normal stress acting on the cylinder surface vanishes identically. Furthermore, we show that a self-propelling state corresponding to zero net drag force is attained when the first mode of blowing and suction profile is such that the flow field be-comes irrigational. Based on these findings we employ direct numerical simulation tools to decipher the Reynolds number dependence of the optimal profiles and the associated power consumption for both drag reduction and self-propulsion. For a typical Reynolds number, the time-averaged drag coefficient first decreases due to vortex shedding suppression, then increases and eventually decreases again after attaining a local maximum as the strength of the first mode is increased. The net power consumption continues to decrease with an increase in the strength of the first mode before reaching a minima after which it rises continuously. For a Reynolds number of 1000 over fifteen fold reduction in drag is achieved for an optimal blowing and suction profile with a maximum radial surface velocity that is nearly 1.97 times the free stream velocity. Next, to establish whether or not higher modes play a role in decreasing net power consumption at finite Reynolds number, we perform theoretical analysis of a configuration similar to the one described above for a spherical body. At zero Reynolds number, as a result of mode independence, we show that surface blow-ing and suction of any form that involves second or higher order axisymmetric or non-axisymmetric modes does not contribute to drag and only leads to an increase in total power consumption. However, at finite Reynolds number, using analysis based on Oseen’s equations, we find that the second and higher modes contribute substantially to the optimal profiles. Finally to understand the effects of a change in shape we consider generalization of the above analysis to axisymmetric prolate and oblate spheroidal bodies. We find that for a general axisymmetric body with non-constant curvature, the optimal drag reducing and self-propelling blowing and suction profiles for minimum power consumption contain second and higher-order modes along with the first mode even when the Reynolds number is zero. The net decrease in power consumption with the use of second and higher order modes exceeds 33% for a disk-like low aspect ratio self-propelling oblate spheroid. Moreover, we perform comparisons between blowing and suction and tangential surface velocity based boundary deformation propulsion mechanisms. Below an aspect ratio of 0.56 we find blowing and suction mechanism to be more efficient for self-propulsion of an oblate spheroid. In contrast, for a self-propelling pro-late spherical micro-swimmer, we show that the tangential surface tread milling consumes less power irrespective of the aspect ratio.