Research Projects

The linear stability of shear flow of two superposed fluid layers in a horizontal channel is studied. The lower fluid layer is populated with surfactants that appear either in the form of monomers or micelles and can also get adsorbed at the interface between the fluids. A mathematical model is formulated which combines the Navier-Stokes equations in each fluid layer, convection-diffusion equations for the concentration of monomers (at the interface and in the bulk fluid) and micelles (in the bulk), together with appropriate coupling conditions at the interface. The primary aim is to investigate when the system is unstable to arbitrary wavelength perturbations, and in particular, to determine the influence of surfactant solubility and/or sorption kinetics. A linear stability analysis is performed and the complex growth rates are obtained by solving an eigenvalue problem for Stokes flow, both numerically for disturbances of arbitrary wavelength and analytically using long-wave approximations.

A mathematical model describing the nonlinear evolution of water waves and their interaction with wave-energy devices and moving ships is presented (Kalogirou and Bokhove, 2016; Kalogirou et al., 2017). The model is derived variationally by directly extending Luke’s variational principle to include the ship dynamics, and this results in a coupled Hamiltonian system for the wave-ship dynamics. This nonlinear system consists of the classical potential flow water-wave equations, coupled to a set of equations describing the dynamics of the ship. The novelty of this work lies in the use of a Lagrange multiplier to impose a constraint on the water surface under the ship. We also present a model linearised around the rest steady state, which we consider a stepping stone towards solving for the nonlinear dynamics. The resulting linear variational system is discretised using (dis)continuous Galerkin finite element approximations, leading to a discrete algebraic variational principle.

Mathematical modelling of water waves in tanks with wave generators can be demonstrated by investigating variational methods asymptotically and numerically (Bokhove and Kalogirou, 2016; Gidel et al., 2017). A reduced potential flow water wave model was derived using variational techniques, based on the assumptions of waves with small amplitude and large wavelength. This model consists of a set of modified Benney-Luke equations describing the deviation from the still water surface and the bottom potential, and includes a time-dependent gravitational potential mimicking a removable sluice gate. The asymptotic model was solved numerically using a (dis)continuous Galerkin finite element method and a 2nd-order symplectic integrator for the time discretisation. The numerical results were compared to a soliton splash experiment in a long water channel with a contraction at its end, resulting after a sluice gate is removed at a finite time.

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 nonlinear stability of two-fluid Couette flows is studied using a novel evolution equation whose dynamics is validated by direct numerical simulation (Kalogirou et al., 2016). 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 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. Direct comparisons are also made with the available experimental results of Barthelet et al. when the thin layer occupies 1/5 of the channel height. Pointwise comparisons of the travelling wave shapes are carried out and once again the agreement is very good.

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 are two-dimensional and travel in the direction of the underlying shear (Kalogirou and Papageorgiou, 2016).

The Kuramoto-Sivashinsky (KS) equation 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). [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)]. 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 KS, 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.

An up-to-date list of publications can be found in the following link.