Towards large eddy simulation of bubble dispersion in high Reynolds number wake flows

This study presents simulations of bubble dispersion in a free shear flow and cylinder wakes. The numerical method of a transient solver for two-dimensional incompressible flow with a subgrid-scale (SGS) turbulence model applied with the Lagrangian particle dynamics (LPD) method is proposed. The method is validated for the case of a free shear flow. The computed instantaneous bubble distribution and time averaged bubble concentrations agree well with both experimental and numerical data available in the open literature. The simulation of bubble dispersion over a cylinder is performed at Reynolds number of 9560. Bubble entrainment into the vortices in the cylinder wake is analysed. The results demonstrated the proposed method is able to reproduce the relevant physics of bubbly flows at high Reynolds numbers. Introduction Prediction of bubble distribution in ship wakes is an important naval research area. Bubbles trapped in the large vortical structures in the ship wake can form clusters that are able to persist for large distances, which increases the ship’s detectability, an important consideration in the defence environment. Complete analysis of the problem requires complex modelling of the physics for both liquid and gas phases. With recent developments in computational capability, understanding the behaviour of bubbly flows by numerical simulation is becoming more common. Two types of models are prevalent in the numerical simulation of dispersed bubbly flows, Eulerian-Lagrangian (particle tracking) and Eulerian-Eulerian (two-fluid) formulations. In the Eulerian-Lagrangian model, liquid is treated as a continuous phase that is solved using Navier-Stokes and continuity equations, and individual bubbles are tracked as dispersed phase. In the Eulerian-Eulerian model, each phase is treated as a continuous phase that is intermingled and interacting with the other phase based on the concept of volume or time averaging with different velocities and volume fractions. Both approaches have been used to model various flow applications, such as bubbly mixing layer [3], bubble columns ([8], [15]), and bubbly wake flows ([9], [12]). Considering relatively small amounts of bubbles (low void fractions) in the ship wakes, the method of Lagrangian particle tracking is most suitable for computing bubble distributions. It has advantages over the two-fluid method in the case of dilute suspensions or when large concentration variability of the discrete phase is present ([4], [5]). This situation is true for bubbly wake flows, such as those in ship wakes, where bubbles may experience preferential concentration and clustering effects in the near wake region but are rather dilute in the far wake [12]. Thus, the Eulerian-Lagrangian models are more suited for fundamental investigations of bubbly flow [10]. Simulations of bubbly turbulent flows require accurate representation of the fluctuating flow field that governs bubble dynamics. Direct numerical simulations (DNS) or large eddy simulations (LES) are often used to attain this accuracy [12]. DNS can provide detailed information of the flow field without the use of a turbulence model, but it is limited to low Reynolds number flows due to its high computational expenses. LES solves large scale turbulent eddy motions with a subgrid-scale (SGS) model that mimics the small scale motions. Thus, LES can handle high Reynolds number flows at a much reduced cost. A joint method of DNS combined with the Lagrangian particle dynamics (LPD) method was applied to simulate bubble dispersion over a cylinder and a hydrofoil at low Reynolds numbers in the previous work [16]. The simulations demonstrated that the interaction between the large scale eddies and the bubbles was successfully modelled and the bubble stream forms the characteristic “S” shape in the wake region that compares well with the corresponding experimental results [14]. However, no bubble entrainment into the vortex cores was observed in the numerical data as the vortices formed from the cylinder and the hydrofoil were not strong enough at low Reynolds numbers. In order to simulate bubbly flow at higher Reynolds numbers, a transient solver for two-dimensional incompressible flow with a SGS model is applied with the Lagrangian particle dynamics method. This is proposed as an intermediate step towards the LES of bubble dispersion in high Reynolds number wake flows. In this paper, the proposed method is validated for the case of bubble dispersion in a free shear flow, which has been previously studied by Rightley and Lasheras [11] and Smirnov et al. [12]. It is confirmed that the simulated distributions of bubbles agree well with the measurements. Bubble entrainment in the flow over a cylinder at a high Reynolds number is also investigated. Numerical Method for Incompressible Flow Incompressible Navier-Stokes Equations The analytical techniques applied in this numerical study are associated with time-integration of the incompressible NavierStokes and continuity equations ∂ui ∂t +u j ∂ui ∂x j = − 1 ρ ∂p ∂xi +ν ∂ui ∂x jx j (1)