Hydrodynamic stability
Authors: Gepner Stan , Kumar Abhishek , Fabisiak Jakub 
Published: June 2023

Continuation methods and turning on bifurcations with Nektar++Continuation methods are often used when tracking interesting states of a dynamical system, such as unstable saddle points that are important for the organization of system’s dynamics. In turbulent flows and turbulent transition process itself such states play an organizing role and act by attracting the flow solution along their stable manifold only to eject it along respective unstable trajectories and resulting in complex evolution patterns. This fact alone makes such states an interesting research topic. With the use of continuation techniques one is able to determine conditions when such states are created as well as navigate respective branches of those solutions. One of the problems, though is the ability of standard tools to navigate and converge close to the bifurcation points, where systems often become singular. In this work we will present our approach and discuss the Nektar++ implementation of the methods used for the continuation and turning around bifurcations.Optimal inflow perturbation for the mixed baroclinic convectionIn this study, our objective is to mitigate the buoyant instability inherent in mixed baroclinic convection within a cavity by employing an advanced optimisation technique. We consider a nearly semi-cylindrical cavity featuring an upper free surface through which fluid can enter, and porous lower boundaries through which fluid can exit. The cavity consists of a semicircular lower boundary, two adiabatic sidewalls, and extends infinitely in the third direction. This specific configuration is pertinent to situations involving molten solid materials, such as in metallurgical casting processes.Our previous stability analysis of this problem revealed the presence of three-dimensional unstable modes [1]. Additionally, the receptivity analysis indicated that the through-flow at the centre of the inlet exhibits the highest receptivity. Given the inlet’s receptiveness, our focus is on identifying the optimal perturbation in the inflow profile capable of suppressing this instability. To accomplish this, we implemented an optimisation solver within the Nektar++ framework [2] to search for the optimal inflow perturbation with a prescribed time-dependence [3]. Preliminary results demonstrate that the optimised inflow profile effectively dampens the linear growth of the unstable mode for a specific duration.Quantification of mixing due to hydrodynamic instability invoked kinematics in corrugated channel flowsImproving mixing efficiency is a way to improve performance of numerous flow-based devices and a common approach is flow turbulisation, but however effective, this approach is not always desirable or realizable. Processing of highly viscous fluids, microfluidic applications or biological flows containing sheer-sensitive molecules calls for laminar mixing to be applied. Mixing in the laminar regime is particularly difficult since flows remain dominated by viscous effects and lacks strong advection and stirring. It is known that channel corrugation might result in the onset of hydrodynamic instabilities, leading to complex flow patterns and consequently improved mixing via principles of chaotic advection. In this work we perform Direct Numerical Simulation (DNS) of low Reynolds number, pressure driven flow in a doubly-periodic, corrugated channel (both symmetrically, and asymmetrically, with one wall corrugated and one flat) to analyse mixing inspired by hydrodynamic instabilities. Flows of varying Reynolds and Schmidt numbers are simulated to analyse mixing in different advective and diffusive conditions. We compare obtained asymmetric channel results with results in symmetry preserving corrugated channel using the rate of exponential decay of variance and negative index Sobolev mix-norm – norm similar to variance (and L2 norms in general), but overcoming its main flaw, which is performance in limit cases of low diffusivity. As the variance time derivative explicitly depends on diffusivity only, it can fail to properly measure mixing in cases where the advective stirring highly exceed the diffusion impact on mixing, while the mix-norm includes “smoothing” operator, overcoming this problem. Sobolev mix-norms, as well as the flow itself, were calculated with the Nektar++ tools. We also utilize and visualise the strange-eigenmodes – structures persistent in time in cases with laminar mixing induced by the chaotic advection and corresponding to eigen functions of the advection diffusion operator.