Dynamics and heat transfer enhancement of MHD flows past a circular cylinder in a duct at high Hartmann number

A numerical study of magnetohydrodynamic flows and heat transfer past a circular cylinder in a duct under a strong magnetic field parallel to the cylinder axis is presented. In this configuration, the flow is quasi-two-dimensional and the modified Navier–Stokes equations are solved in a two–dimensional domain. The numerical simulations have been performed over a range of parameters including the Reynolds number 50 ≤ Re ≤ 3000, modified Hartmann number 50 ≤ Ha⋆ . 500, blockage ratio 0.1 ≤ β ≤ 0.5, offset ratio 0.25 ≤ γ ≤ 1, velocity amplitude 0 ≤ A ≤ 3 and forcing frequency 0 ≤ Ste ≤ 10. The primary aim of this study is to understand the fundamental mechanism that governs transition to unsteady flow in this system and exploit this for further improvements in heat transfer for MHD cooling duct flows. With this aim in mind, a spectral-element method is employed to compute the MHD flow and heat transfer past a confined circular cylinder in a rectangular duct. Meshes have been constructed to deal with the significant number of geometric flow parameters combinations. Thorough validation and grid resolution studies have been performed to ensure adequate domain sizes, and spatial and temporal resolutions to accurately resolve all flow and thermal features for the reported flow variable ranges. Studies show that the reported flow parameters are converged to better than 0.3% in terms of spatial and temporal accuracy and 1% with respect to the domain size. For the optimal growth studies, the dependence of energy growth on upstream domain length is also considered through the calculation of the energy growth over a fixed time span. This verifies that the effect of truncating the upstream length from 32d to 8d causes an error of less than 3% in the growth rate prediction. The critical Reynolds number Rec for the transition from steady to periodic flow is determined as a function of Ha⋆ and β, and this is found to increase with increasing Ha⋆ and β. In addition, the variation of the wake recirculation length in the steady flow regime is determined as a function of Reynolds number, Hartman number and blockage ratio, and a universal expression is proposed. The characteristics of heat transfer depend strongly on the proximity of the cylinder to the heated wall. For small blockage ratios, it increases significantly as the gap ratio decreases from 1 to 0.25. However, there is a substantial drop in heat transfer for high blockage ratio. Downstream cross-stream mixing induced by the cylinder wake is found to increase the heat transfer augmentation by more than a factor of two in some cases. The maximum gain in heat transfer generated by placing the cylinder in the channel near to the wall with that at the centerline was obtained for the lowest blockage ratio β = 0.1, as the cylinder is further approached the heated wall. For all β, a very significant transient energy growth is found in the subcritical regime below the onset of vortex shedding. This suggests a potential for the design of an actuation mechanism to invoke vortex shedding and then enhance heat transfer in these ducts. The energy amplification of the disturbances is found to decrease significantly with increasing Hartmann number and the peak growth shifts towards smaller times while it increases significantly with increasing blockage ratio. The structure of the optimal initial disturbance is found to be consistent across the all blockage ratios being tested. In line with similar problems, it convects along the separating region being amplified to the peak growth state downstream of the recirculation bubble. The maximum time for maximum energy growth τmax is found to increase significantly as recirculation length increases which demonstrates the amplifying nature of the separated shear layers in the wake. The critical Reynolds number for the onset of positive growth at different Hartmann numbers and blockage ratios is determined. It is found that it increases rapidly with increasing Hartmann number and blockage ratio. For all β, the peak energy amplification grows exponentially with Re from low Hartmann numbers. Direct numerical simulation studies in which the inflow is perturbed by random white noise confirms the predictions arising from the transient growth analysis: that is, the perturbation excites and feeds energy into the global mode. A considerable increase in heat transfer from the heated channel wall occurs from rotational oscillation of the cylinder, with a maximum enhancement of approximately 22% observed at higher amplitude over that for steady flow. The oscillation frequency range for effective enhancement is widened in both directions, while the frequency at which the peak of the Nusselt number occurs is shifted slightly to a lower frequency as A is increased. It is found that as the amplitude was reduced, the forcing frequency approaches the global frequency mode.