My Research

The nonlinear stability of two-fluid Couette/Poiseuille flows is studied using a novel evolution equation whose dynamics is validated by direct numerical simulations (Kalogirou et al., 2016; Kalogirou et al., 2020). The evolution equation incorporates inertial effects at arbitrary Reynolds numbers through a non-local term arising from the coupling between the two fluid regions, and is valid when one of the layers is thin. The equation predicts asymmetric solutions and exhibits bistability, features that are essential observations in the Couette-flow experiments of Barthelet et al. (J. Fluid Mech., vol. 303, 1995, pp. 23–53). Comparisons between model solutions and direct numerical simulations show excellent agreement.

A related study on the Rayleigh–Taylor instability at the interface of two sheared fluid can be found in Kalogirou & Blyth, 2023. This work includes: mathematical modelling that reduces the dynamics of the full problem to a lubrication equation describing the evolution of the interface, numerical solutions thereof for adverse density stratifications, and asymptotic analysis that reveals the finer structure of interfacial waves. Comparisons between the lubrication model, direct numerical simulations, and the asymptotic solutions demonstrate an excellent agreement between the three in terms of interfacial structure, wave speed and film thicknesses.

Multilayer fluid flows with a surfactant-populated interface between the fluids has been studied using mathematical modelling and asymptotic techniques (Kalogirou, 2014). Following the flows into the nonlinear regime was of particular interest in order to understand nonlinear dynamics and underlying structures. The mathematical model derived was used to classify the dynamics and stability of the solutions, through analytical (asymptotic and linear stability analysis) and numerical computations. Extensive numerical experiments reveal that the dynamics are mostly organized into travelling or time-periodic travelling wave pulses, but spatiotemporal chaos is also supported when the length of the system is sufficiently large (Kalogirou et al., 2012). It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that travel in the direction of the underlying shear (Kalogirou and Papageorgiou, 2016). The interacting effect of surfactants, inertia, and gravity has also been investigated (Kalogirou, 2018).

The stability of a two-layer surfactant-laden Couette flow is considered, using linear stability analysis (Kalogirou & Blyth, 2019) and nonlinear asymptotic modelling (Kalogirou & Blyth, 2020). One of the fluids is contaminated with a soluble surfactant whose concentration may be above the critical micelle concentration, in which case micelles are formed in the bulk of the fluid. The primary aim of this study is to determine the influence of surfactant solubility on the linear stability of the flow and the nonlinear dynamics. The lubrication approximation is applied and a strongly nonlinear system of equations is derived for the evolution of the interface and surfactant concentration at the interface, as well as the vertically averaged monomer and micelle concentrations in the bulk (as a result of fast vertical diffusion). If the surfactant exists at large concentrations that exceed the critical micelle concentration, then long interfacial waves are stable at large times, unless density stratification effects overcome the stabilising influence of micelles.

A related study of the impact of surfactant solubility on the linear stability of a semi-infinite fluid undergoing a shearing motion can be found in Kalogirou & Blyth, 2021.

A novel wave-energy device is developed through an experimental setup comprising wave amplification in a contraction, a wave-activated buoy, and magnetically induced energy generation (Bokhove et al., 2019a). We develop a novel wave-to-wire mathematical model of the combined wave hydrodynamics, wave-activated buoy motion and electric power generation by magnetic induction, from first principles, satisfying one grand variational principle in its conservative limit. Wave and buoy dynamics are coupled via a Lagrange multiplier, which boundary value at the waterline is in a subtle way solved explicitly by imposing incompressibility in a weak sense. Dissipative features, such as electrical wire resistance and nonlinear LED loads, are added a posteriori. New is also the intricate and compatible finite-element space-time discretisation of the linearised dynamics, guaranteeing numerical stability and the correct energy transfer between the three subsystems. Preliminary simulations of our simplified and linearised wave-energy model are encouraging and involve a first study of the resonant behaviour and parameter dependence of the device (Bokhove et al., 2019a, 2019b, 2020; Bolton et al., 2021).

The damped motion of driven water waves in a vertical Hele-Shaw tank is investigated variationally and numerically (Kalogirou et al., 2016). The equations governing the hydrodynamics of the problem are derived from a variational principle for shallow water. The variational principle includes the effects of surface tension, linear momentum damping due to the proximity of the tank walls and incoming volume flux through one of the boundaries representing the generation of waves by a wave pump. The model equations are solved numerically using (dis)continuous Galerkin finite element methods as well as a standard finite volume solver and are compared to exact linear wave sloshing and driven wave sloshing results. Numerical solutions of the shallow-water-wave equations are also validated against laboratory experiments of artificially driven waves in the Hele-Shaw tank.

The Kuramoto-Sivashinsky equation (KSE) in one spatial dimension is one of the most well-known and well-studied partial differential equations; it exhibits spatio-temporal chaos that emerges through various bifurcations as the domain length increases. There have been several notable analytical studies aimed at understanding how this property extends to the case of two spatial dimensions. We performed an extensive numerical study of the Kuramoto-Sivashinsky equation to complement these analytical studies (Kalogirou et al., 2015; Tomlin et al., 2018). We explored in detail the statistics of chaotic solutions and classify the solutions that arise for domain sizes where the trivial solution is unstable and the long-time dynamics are completely two-dimensional. While we find that many of the features of the 1D KSE, including how the energy scales with system size, carry over to the 2D case, we also note several differences including the various paths to chaos that are not through period doubling.

A study on the numerical analysis of linearly-implicit schemes for multi-dimensional Kuramoto–Sivashinsky–type equations can be found in Akrivis et al. (2016).