Saturn’s moon Enceladus harbours a global subsurface ocean beneath its icy crust. Tidal dissipation within the moon’s core generates a substantial amount of heat which leads to ocean convection. Observations of the moon indicate ocean thickness variations of up to 20 km from equator to pole and heterogeneous heat generation within the core likely results in latitude-dependent temperature gradients. The effects of meridional thickness variations and heterogeneous temperature gradients on rotating thermal convection have not been simulated in previous studies. Here we simulate a non-uniform spherical shell employing a degree-2 zonal thickness profile. Using direct numerical simulations, we analysed various properties associated with heat transfer behaviour for flows in a uniform and non- uniform spherical shell domain driven by thermal convection with the Rayleigh number in the range 1.6×10⁵ ≤ Ra ≤ 5.0×10⁶ and constant Ekman number of Ek = 3×10^-4 and Prandtl number of Pr = 1. Our results demonstrate that different regimes of convection exist, which depend on the relative influence of rotation. With increasing thermal forcing, convection moves from being restricted to equatorial regions to filling the whole fluid domain. Global scaling behaviour for both domains was found to be consistent with literature, although weaker polar convection in non-uniform shells caused a decrease in heat transfer efficiency and thus a diminished heat transfer scaling behaviour. The diminishing transport of heat at the poles in the non-uniform shell deviates from the predicted heat flux profile at Enceladus, suggesting that stronger thermal heterogeneities are required to enhance polar heat transfer.

The whole construct of explicit algebraic Reynolds stress models, as they are known in RANS, is based on the weak equilibrium assumption. By employing this assumption, and in addition, incorporating models for terms in the Reynolds stress transport equations, the system of six partial differential equations can be simplified to an algebraic form. However, the resulting equations are non-linear in terms of the Reynolds stress anisotropy and therefore become implicit. The cause of the non-linearity is the presence of the ratio of kinetic energy production to dissipation (P/E) in the equations. In order to derive an explicit model for the Reynolds stress anisotropy, methods to determine P/E are required. For RANS, there are many variations of the final formulation based on the procedure by which this ratio is determined. Marstorp et al. (2009) provide an extension of this modelling framework used in RANS to LES in order to model subgrid-scale (SGS) stresses. In their formulation, the value of P/E, which in LES is the ratio of SGS kinetic energy production to dissipation, is specified as 1, thereby obviating any need for treating non-linearities in the equations.

In this thesis, the performance of the explicit algebraic subgrid-scale stress model (EASSM) developed by Marstorp et al. (2009) is tested in comparison with the dynamic Smagorinsky model (DSM) at an a priori level using DNS data for forced homogeneous turbulence with and without system rotation. Based on the results of the a priori analysis, a new model, termed as the non-equilibrium EASSM is introduced, which does not require the assumption that P/E = 1. The framework of EASSM demands the determination of the SGS kinetic energy and the time-scale in order to close the system of equation. While Marstorp et al. (2009) use algebraic expressions for determining these additional variables, for the non-equilibrium EASSM, avoiding the assumption that P/E = 1 enables the use of an evolution equation for the SGS kinetic energy. For the time-scale, an algebraic expression is derived as a function of SGS kinetic energy and dissipation. The performance of the non-equilibrium EASSM in comparison with the DSM and the EASSM of Marstorp et al. (2009) is evaluated by conducting LES of forced and decaying homogeneous turbulence, with and without rotation, using the finite volume code-INCA. For forced cases, the non-equilibrium model outperforms other models in terms of mean resolved and SGS kinetic energy predictions, and also gives a good match for the time averaged resolved spectrum with DNS. In the presence of strong rotation, however, all the models fail to capture the right decay rate as observed in DNS. On examining the DNS data for the two decaying cases, it appears that the isotropic scaling used for the modelling SGS kinetic energy dissipation in the non-equilibrium model deteriorates the model performance making it unable to provide the right level of inhibition in the decay of the resolved kinetic energy and dissipation.